Inductor

Top of Form   Bottom of Form ONLINE ELECTRICAL ENGINEERING STUDY SITE HOME BASICS POWER SYSTEM MACHINES ELECTRONICS QUESTIONS MCQ VIDEOS COMMUNITY Top of Form Search for: Bottom of Form New Articles Hartley Oscillator Phototransistor Transistor Biasing What is an Oscillator? Inverting Amplifier Transistor Characteristics Closely Related Articles Ohms Law | Equation Formula and Limitation of Ohms Law Joules Law of Heating Faraday Law of Electromagnetic Induction Lenz Law of Electromagnetic Induction Faraday First and Second Laws of Electrolysis Coulombs Law | Explanation Statement Formulas Principle Limitation of Coulomb’s Law Biot Savart Law Gauss Theorem Fleming Left Hand rule and Fleming Right Hand rule Seebeck Effect and Seebeck Coefficient Faraday Law of Electromagnetic Induction « Previous Next » In 1831, Michael Faraday, an English physicist gave one of the most basic laws of electromagnetism called Faraday's law of electromagnetic induction. This law explains the working principle of most of theelectrical motors, generators, electrical transformers and inductors. This law shows the relationship between electric circuit and magnetic field. Faraday performs an experiment with a magnet and coil. During this experiment, he found how emf is induced in the coil when flux linked with it changes. He has also done experiments in electro-chemistry and electrolysis. Faraday's Experiment RELATIONSHIP BETWEEN INDUCED EMF AND FLUX In this experiment, Faraday takes a magnet and a coil and connects a galvanometer across the coil. At starting, the magnet is at rest, so there is no deflection in the galvanometer i.e needle of galvanometer is at the center or zero position. When the magnet is moved towards the coil, the needle of galvanometer deflects in one direction. When the magnet is held stationary at that position, the needle of galvanometer returns back to zero position. Now when the magnet is moved away from the coil, there is some deflection in the needle but in opposite direction and again when the magnet becomes stationary, at that point with respect to coil, the needle of the galvanometer returns back to the zero position. Similarly, if magnet is held stationary and the coil is moved away and towards the magnet, the galvanometer shows deflection in similar manner. It is also seen that, the faster the change in the magnetic field, the greater will be the induced emf or voltage in the coil. Position of magnet Deflection in galvanometer Magnet at rest No deflection in galvanometer Magnet moves towards the coil Deflection in galvanometer in one direction Magnet is held stationary at same position (near the coil) No deflection in galvanometer Magnet moves away from the coil Deflection in galvanometer but in opposite direction Magnet is held stationary at same position (away from the coil) No deflection in galvanometer Conclusion: From this experiment, Faraday concluded that whenever there is relative motion between conductor and a magnetic field, the flux linkage with a coil changes and this change in flux induces a voltage across a coil. Michael Faraday formulated two laws on the basis of above experiments. These laws are called Faraday's laws of electromagnetic induction. Faraday's Laws Faraday's First Law Any change in the magnetic field of a coil of wire will cause an emf to be induced in the coil. This emf induced is called induced emf and if the conductor circuit is closed, the current will also circulate through the circuit and this current is called induced current. Method to change magnetic field: By moving a magnet towards or away from the coil By moving the coil into or out of the magnetic field. By changing the area of a coil placed in the magnetic field By rotating the coil relative to the magnet. Faraday's Second Law It states that the magnitude of emf induced in the coil is equal to the rate of change of flux that linkages with the coil. The flux linkage of the coil is the product of number of turns in the coil and flux associated with the coil. Faraday Law Formula Consider a magnet approaching towards a coil. Here we consider two instants at time T1 and time T2. Flux linkage with the coil at time, T1 = NΦ1 Wb Flux linkage with the coil at time, T2 = NΦ2 wb Change in flux linkage = N(Φ2 - Φ1) Let this change in flux linkage be, Φ = Φ2 - Φ1 So, the Change in flux linkage = NΦ Now the rate of change of flux linkage = NΦ / t Take derivative on right hand side we will get The rate of change of flux linkage = NdΦ/dt But according to Faraday's law of electromagnetic induction, the rate of change of flux linkage is equal to induced emf. Considering Lenz's Law. Where, flux Φ in Wb = B.A B = magnetic field strength A = area of the coil HOW TO INCREASE EMF INDUCED IN A COIL By increasing the number of turns in the coil i.e N- From the formulae derived above it is easily seen that if number of turns of coil is increased, the induced emf also gets increased. By increasing magnetic field strength i.e B surrounding the coil- Mathematically if magnetic field increases, flux increases and if flux increases emf induced will also get increased. Theoretically, if the coil is passed through a stronger magnetic field, there will be more lines of force for coil to cut and hence there will be more emf induced. By increasing the speed of the relative motion between the coil and the magnet - If the relative speed between the coil and magnet is increased from its previous value, the coil will cut the lines of flux at a faster rate, so more induced emf would be produced. Applications of Faraday Law Faraday law is one of the most basic and important laws of electromagnetism . This law finds its application in most of the electrical machines, industries and medical field etc. Electrical Transformers It is a static ac device which is used to either step up or step down voltage or current. It is used in generating station, transmission and distribution system. The transformer works on Faraday's law. Electrical Generators The basic working principle of electrical generator is Faraday's law of mutual induction. Electric generator is used to convert mechanical energy into electrical energy. Induction Cookers The Induction cooker, is a most fastest way of cooking. It also works on principle of mutual induction. When current flows through the coil of copper wire placed below a cooking container, it produces a changing magnetic field. This alternating or changing magnetic field induces an emf and hence the current in the conductive container, and we know that flow of current always produces heat in it. Electromagnetic Flow Meters It is used to measure velocity of blood and certain fluids. When a magnetic field is applied to electrically insulated pipe in which conducting fluids are flowing, then according to Faraday's law, an electromotive force is induced in it. This induced emf is proportional to velocity of fluid flowing . Form the bases of Electromagnetic Theory Faraday's idea of lines of force is used in well known Maxwell's equations. According to Faraday's law, change in magnetic field gives rise to change in electric field and the converse of this is used in Maxwell's equations. Musical Instruments It is also used in musical instruments like electric guitar, electric violin etc. Faraday's Law-Video « Previous Next » Closely Related Articles Ohms Law | Equation Formula and Limitation of Ohms Law Joules Law of Heating Lenz Law of Electromagnetic Induction Faraday First and Second Laws of Electrolysis Coulombs Law | Explanation Statement Formulas Principle Limitation of Coulomb’s Law Biot Savart Law Gauss Theorem Fleming Left Hand rule and Fleming Right Hand rule Seebeck Effect and Seebeck Coefficient More Related Articles New Articles Hartley Oscillator Phototransistor Transistor Biasing What is an Oscillator? Inverting Amplifier Transistor Characteristics Please provide your valuable comments Top of Form Name : - Email : -  Location : - Your content : - Bottom of Form © 2011-2016 electrical4u.The content is copyrighted to electrical4uand may not be reproduced on other websites.  Connect with us     ARTICLES  FORUM  EDUCATION  MORE  IOT ARDUINO Top of Form Bottom of Form  Home   Textbook   Vol. II - Alternating Current (AC)   Transformers Mutual Inductance and Basic Operation Mutual Inductance and Basic Operation Chapter 9 - Transformers Suppose we were to wrap a coil of insulated wire around a loop of ferromagnetic material and energize this coil with an AC voltage source: (Figure below (a)) Insulated winding on ferromagnetic loop has inductive reactance, limiting AC current. As an inductor, we would expect this iron-core coil to oppose the applied voltage with its inductive reactance, limiting current through the coil as predicted by the equations XL = 2πfL and I=E/X (or I=E/Z). For the purposes of this example, though, we need to take a more detailed look at the interactions of voltage, current, and magnetic flux in the device. Kirchhoff’s voltage law describes how the algebraic sum of all voltages in a loop must equal zero. In this example, we could apply this fundamental law of electricity to describe the respective voltages of the source and of the inductor coil. Here, as in any one-source, one-load circuit, the voltage dropped across the load must equal the voltage supplied by the source, assuming zero voltage dropped along the resistance of any connecting wires. In other words, the load (inductor coil) must produce an opposing voltage equal in magnitude to the source, in order that it may balance against the source voltage and produce an algebraic loop voltage sum of zero. From where does this opposing voltage arise? If the load were a resistor (Figure above (b)), the voltage drop originates from electrical energy loss, the “friction” of electrons flowing through the resistance. With a perfect inductor (no resistance in the coil wire), the opposing voltage comes from another mechanism: the reaction to a changing magnetic flux in the iron core. When AC current changes, flux Φ changes. Changing flux induces a counter EMF. Michael Faraday discovered the mathematical relationship between magnetic flux (Φ) and induced voltage with this equation: The instantaneous voltage (voltage dropped at any instant in time) across a wire coil is equal to the number of turns of that coil around the core (N) multiplied by the instantaneous rate-of-change in magnetic flux (dΦ/dt) linking with the coil. Graphed, (Figure below) this shows itself as a set of sine waves (assuming a sinusoidal voltage source), the flux wave 90olagging behind the voltage wave: Magnetic flux lags applied voltage by 90o because flux is proportional to a rate of change, dΦ/dt. Magnetic flux through a ferromagnetic material is analogous to current through a conductor: it must be motivated by some force in order to occur. In electric circuits, this motivating force is voltage (a.k.a. electromotive force, or EMF). In magnetic “circuits,” this motivating force ismagnetomotive force, or mmf. Magnetomotive force (mmf) and magnetic flux (Φ) are related to each other by a property of magnetic materials known as reluctance (the latter quantity symbolized by a strange-looking letter “R”): In our example, the mmf required to produce this changing magnetic flux (Φ) must be supplied by a changing current through the coil. Magnetomotive force generated by an electromagnet coil is equal to the amount of current through that coil (in amps) multiplied by the number of turns of that coil around the core (the SI unit for mmf is the amp-turn). Because the mathematical relationship between magnetic flux and mmf is directly proportional, and because the mathematical relationship between mmf and current is also directly proportional (no rates-of-change present in either equation), the current through the coil will be in-phase with the flux wave as in (Figure below) Magnetic flux, like current, lags applied voltage by 90o. This is why alternating current through an inductor lags the applied voltage waveform by 90o: because that is what is required to produce a changing magnetic flux whose rate-of-change produces an opposing voltage in-phase with the applied voltage. Due to its function in providing magnetizing force (mmf) for the core, this current is sometimes referred to as themagnetizing current. It should be mentioned that the current through an iron-core inductor is not perfectly sinusoidal (sine-wave shaped), due to the nonlinear B/H magnetization curve of iron. In fact, if the inductor is cheaply built, using as little iron as possible, the magnetic flux density might reach high levels (approaching saturation), resulting in a magnetizing current waveform that looks something like Figure below As flux density approaches saturation, the magnetizing current waveform becomes distorted. When a ferromagnetic material approaches magnetic flux saturation, disproportionately greater levels of magnetic field force (mmf) are required to deliver equal increases in magnetic field flux (Φ). Because mmf is proportional to current through the magnetizing coil (mmf = NI, where “N” is the number of turns of wire in the coil and “I” is the current through it), the large increases of mmf required to supply the needed increases in flux results in large increases in coil current. Thus, coil current increases dramatically at the peaks in order to maintain a flux waveform that isn’t distorted, accounting for the bell-shaped half-cycles of the current waveform in the above plot. The situation is further complicated by energy losses within the iron core. The effects of hysteresis and eddy currents conspire to further distort and complicate the current waveform, making it even less sinusoidal and altering its phase to be lagging slightly less than 90o behind the applied voltage waveform. This coil current resulting from the sum total of all magnetic effects in the core (dΦ/dt magnetization plus hysteresis losses, eddy current losses, etc.) is called the exciting current. The distortion of an iron-core inductor’s exciting current may be minimized if it is designed for and operated at very low flux densities. Generally speaking, this requires a core with large cross-sectional area, which tends to make the inductor bulky and expensive. For the sake of simplicity, though, we’ll assume that our example core is far from saturation and free from all losses, resulting in a perfectly sinusoidal exciting current. As we’ve seen already in the inductors chapter, having a current waveform 90o out of phase with the voltage waveform creates a condition where power is alternately absorbed and returned to the circuit by the inductor. If the inductor is perfect (no wire resistance, no magnetic core losses, etc.), it will dissipate zero power. Let us now consider the same inductor device, except this time with a second coil (Figure below) wrapped around the same iron core. The first coil will be labeled the primary coil, while the second will be labeled thesecondary: Ferromagnetic core with primary coil (AC driven) and secondary coil. If this secondary coil experiences the same magnetic flux change as the primary (which it should, assuming perfect containment of the magnetic flux through the common core), and has the same number of turns around the core, a voltage of equal magnitude and phase to the applied voltage will be induced along its length. In the following graph, (Figure below) the induced voltage waveform is drawn slightly smaller than the source voltage waveform simply to distinguish one from the other: Open circuited secondary sees the same flux Φ as the primary. Therefore induced secondary voltage es is the same magnitude and phase as the primary voltage ep. This effect is called mutual inductance: the induction of a voltage in one coil in response to a change in current in the other coil. Like normal (self-) inductance, it is measured in the unit of Henrys, but unlike normal inductance it is symbolized by the capital letter “M” rather than the letter “L”: No current will exist in the secondary coil, since it is open-circuited. However, if we connect a load resistor to it, an alternating current will go through the coil, in-phase with the induced voltage (because the voltage across a resistor and the current through it are always in-phase with each other). (Figure below) Resistive load on secondary has voltage and current in-phase. At first, one might expect this secondary coil current to cause additional magnetic flux in the core. In fact, it does not. If more flux were induced in the core, it would cause more voltage to be induced voltage in the primary coil (remember that e = dΦ/dt). This cannot happen, because the primary coil’s induced voltage must remain at the same magnitude and phase in order to balance with the applied voltage, in accordance with Kirchhoff’s voltage law. Consequently, the magnetic flux in the core cannot be affected by secondary coil current. However, what does change is the amount of mmf in the magnetic circuit. Magnetomotive force is produced any time electrons move through a wire. Usually, this mmf is accompanied by magnetic flux, in accordance with the mmf=ΦR “magnetic Ohm’s Law” equation. In this case, though, additional flux is not permitted, so the only way the secondary coil’s mmf may exist is if a counteracting mmf is generated by the primary coil, of equal magnitude and opposite phase. Indeed, this is what happens, an alternating current forming in the primary coil—180o out of phase with the secondary coil’s current—to generate this counteracting mmf and prevent additional core flux. Polarity marks and current direction arrows have been added to the illustration to clarify phase relations: (Figure below) Flux remains constant with application of a load. However, a counteracting mmf is produced by the loaded secondary. If you find this process a bit confusing, do not worry. Transformer dynamics is a complex subject. What is important to understand is this: when an AC voltage is applied to the primary coil, it creates a magnetic flux in the core, which induces AC voltage in the secondary coil in-phase with the source voltage. Any current drawn through the secondary coil to power a load induces a corresponding current in the primary coil, drawing current from the source. Notice how the primary coil is behaving as a load with respect to the AC voltage source, and how the secondary coil is behaving as a source with respect to the resistor. Rather than energy merely being alternately absorbed and returned the primary coil circuit, energy is now being coupledto the secondary coil where it is delivered to a dissipative (energy-consuming) load. As far as the source “knows,” its directly powering the resistor. Of course, there is also an additional primary coil current lagging the applied voltage by 90o, just enough to magnetize the core to create the necessary voltage for balancing against the source (the exciting current). We call this type of device a transformer, because it transforms electrical energy into magnetic energy, then back into electrical energy again. Because its operation depends on electromagnetic induction between two stationary coils and a magnetic flux of changing magnitude and “polarity,” transformers are necessarily AC devices. Its schematic symbol looks like two inductors (coils) sharing the same magnetic core: (Figure below) Schematic symbol for transformer consists of two inductor symbols, separated by lines indicating a ferromagnetic core. The two inductor coils are easily distinguished in the above symbol. The pair of vertical lines represent an iron core common to both inductors. While many transformers have ferromagnetic core materials, there are some that do not, their constituent inductors being magnetically linked together through the air. The following photograph shows a power transformer of the type used in gas-discharge lighting. Here, the two inductor coils can be clearly seen, wound around an iron core. While most transformer designs enclose the coils and core in a metal frame for protection, this particular transformer is open for viewing and so serves its illustrative purpose well: (Figure below) Example of a gas-discharge lighting transformer. Both coils of wire can be seen here with copper-colored varnish insulation. The top coil is larger than the bottom coil, having a greater number of “turns” around the core. In transformers, the inductor coils are often referred to as windings, in reference to the manufacturing process where wire is wound around the core material. As modeled in our initial example, the powered inductor of a transformer is called the primary winding, while the unpowered coil is called the secondary winding. In the next photograph, Figure below, a transformer is shown cut in half, exposing the cross-section of the iron core as well as both windings. Like the transformer shown previously, this unit also utilizes primary and secondary windings of differing turn counts. The wire gauge can also be seen to differ between primary and secondary windings. The reason for this disparity in wire gauge will be made clear in the next section of this chapter. Additionally, the iron core can be seen in this photograph to be made of many thin sheets (laminations) rather than a solid piece. The reason for this will also be explained in a later section of this chapter. Transformer cross-section cut shows core and windings. It is easy to demonstrate simple transformer action using SPICE, setting up the primary and secondary windings of the simulated transformer as a pair of “mutual” inductors. (Figure below) The coefficient of magnetic field coupling is given at the end of the “k” line in the SPICE circuit description, this example being set very nearly at perfection (1.000). This coefficient describes how closely “linked” the two inductors are, magnetically. The better these two inductors are magnetically coupled, the more efficient the energy transfer between them should be. Spice circuit for coupled inductors. transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 l1 2 0 100 l2 3 5 100 ** This line tells SPICE that the two inductors ** l1 and l2 are magnetically “linked” together k l1 l2 0.999 vi1 3 4 ac 0 rload 4 5 1k .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end Note: the Rbogus resistors are required to satisfy certain quirks of SPICE. The first breaks the otherwise continuous loop between the voltage source and L1 which would not be permitted by SPICE. The second provides a path to ground (node 0) from the secondary circuit, necessary because SPICE cannot function with any ungrounded circuits. freq v(2) i(v1) 6.000E+01 1.000E+01 9.975E-03 Primary winding freq v(3,5) i(vi1) 6.000E+01 9.962E+00 9.962E-03 Secondary winding Note that with equal inductances for both windings (100 Henrys each), the AC voltages and currents are nearly equal for the two. The difference between primary and secondary currents is the magnetizing current spoken of earlier: the 90o lagging current necessary to magnetize the core. As is seen here, it is usually very small compared to primary current induced by the load, and so the primary and secondary currents are almost equal. What you are seeing here is quite typical of transformer efficiency. Anything less than 95% efficiency is considered poor for modern power transformer designs, and this transfer of power occurs with no moving parts or other components subject to wear. If we decrease the load resistance so as to draw more current with the same amount of voltage, we see that the current through the primary winding increases in response. Even though the AC power source is not directly connected to the load resistance (rather, it is electromagnetically “coupled”), the amount of current drawn from the source will be almost the same as the amount of current that would be drawn if the load were directly connected to the source. Take a close look at the next two SPICE simulations, showing what happens with different values of load resistors: transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 l1 2 0 100 l2 3 5 100 k l1 l2 0.999 vi1 3 4 ac 0 ** Note load resistance value of 200 ohms rload 4 5 200 .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end freq v(2) i(v1) 6.000E+01 1.000E+01 4.679E-02 freq v(3,5) i(vi1) 6.000E+01 9.348E+00 4.674E-02 Notice how the primary current closely follows the secondary current. In our first simulation, both currents were approximately 10 mA, but now they are both around 47 mA. In this second simulation, the two currents are closer to equality, because the magnetizing current remains the same as before while the load current has increased. Note also how the secondary voltage has decreased some with the heavier (greater current) load. Let’s try another simulation with an even lower value of load resistance (15 Ω): transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 l1 2 0 100 l2 3 5 100 k l1 l2 0.999 vi1 3 4 ac 0 rload 4 5 15 .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end freq v(2) i(v1) 6.000E+01 1.000E+01 1.301E-01 freq v(3,5) i(vi1) 6.000E+01 1.950E+00 1.300E-01 Our load current is now 0.13 amps, or 130 mA, which is substantially higher than the last time. The primary current is very close to being the same, but notice how the secondary voltage has fallen well below the primary voltage (1.95 volts versus 10 volts at the primary). The reason for this is an imperfection in our transformer design: because the primary and secondary inductances aren’t perfectly linked (a k factor of 0.999 instead of 1.000) there is “stray” or “leakage” inductance. In other words, some of the magnetic field isn’t linking with the secondary coil, and thus cannot couple energy to it: (Figure below) Leakage inductance is due to magnetic flux not cutting both windings. Consequently, this “leakage” flux merely stores and returns energy to the source circuit via self-inductance, effectively acting as a series impedance in both primary and secondary circuits. Voltage gets dropped across this series impedance, resulting in a reduced load voltage: voltage across the load “sags” as load current increases. (Figure below) Equivalent circuit models leakage inductance as series inductors independent of the “ideal transformer”. If we change the transformer design to have better magnetic coupling between the primary and secondary coils, the figures for voltage between primary and secondary windings will be much closer to equality again: transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 l1 2 0 100 l2 3 5 100 ** Coupling factor = 0.99999 instead of 0.999 k l1 l2 0.99999 vi1 3 4 ac 0 rload 4 5 15 .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end freq v(2) i(v1) 6.000E+01 1.000E+01 6.658E-01 freq v(3,5) i(vi1) 6.000E+01 9.987E+00 6.658E-01 Here we see that our secondary voltage is back to being equal with the primary, and the secondary current is equal to the primary current as well. Unfortunately, building a real transformer with coupling this complete is very difficult. A compromise solution is to design both primary and secondary coils with less inductance, the strategy being that less inductance overall leads to less “leakage” inductance to cause trouble, for any given degree of magnetic coupling inefficiency. This results in a load voltage that is closer to ideal with the same (high current heavy) load and the same coupling factor: transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 ** inductance = 1 henry instead of 100 henrys l1 2 0 1 l2 3 5 1 k l1 l2 0.999 vi1 3 4 ac 0 rload 4 5 15 .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end freq v(2) i(v1) 6.000E+01 1.000E+01 6.664E-01 freq v(3,5) i(vi1) 6.000E+01 9.977E+00 6.652E-01 Simply by using primary and secondary coils of less inductance, the load voltage for this heavy load (high current) has been brought back up to nearly ideal levels (9.977 volts). At this point, one might ask, “If less inductance is all that’s needed to achieve near-ideal performance under heavy load, then why worry about coupling efficiency at all? If its impossible to build a transformer with perfect coupling, but easy to design coils with low inductance, then why not just build all transformers with low-inductance coils and have excellent efficiency even with poor magnetic coupling?” The answer to this question is found in another simulation: the same low-inductance transformer, but this time with a lighter load (less current) of 1 kΩ instead of 15 Ω: transformer v1 1 0 ac 10 sin rbogus1 1 2 1e-12 rbogus2 5 0 9e12 l1 2 0 1 l2 3 5 1 k l1 l2 0.999 vi1 3 4 ac 0 rload 4 5 1k .ac lin 1 60 60 .print ac v(2,0) i(v1) .print ac v(3,5) i(vi1) .end freq v(2) i(v1) 6.000E+01 1.000E+01 2.835E-02 freq v(3,5) i(vi1) 6.000E+01 9.990E+00 9.990E-03 With lower winding inductances, the primary and secondary voltages are closer to being equal, but the primary and secondary currents are not. In this particular case, the primary current is 28.35 mA while the secondary current is only 9.990 mA: almost three times as much current in the primary as the secondary. Why is this? With less inductance in the primary winding, there is less inductive reactance, and consequently a much larger magnetizing current. A substantial amount of the current through the primary winding merely works to magnetize the core rather than transferuseful energy to the secondary winding and load. An ideal transformer with identical primary and secondary windings would manifest equal voltage and current in both sets of windings for any load condition. In a perfect world, transformers would transfer electrical power from primary to secondary as smoothly as though the load were directly connected to the primary power source, with no transformer there at all. However, you can see this ideal goal can only be met if there is perfectcoupling of magnetic flux between primary and secondary windings. Being that this is impossible to achieve, transformers must be designed to operate within certain expected ranges of voltages and loads in order to perform as close to ideal as possible. For now, the most important thing to keep in mind is a transformer’s basic operating principle: the transfer of power from the primary to the secondary circuit via electromagnetic coupling. REVIEW: Mutual inductance is where the magnetic flux of two or more inductors are “linked” so that voltage is induced in one coil proportional to the rate-of-change of current in another. A transformer is a device made of two or more inductors, one of which is powered by AC, inducing an AC voltage across the second inductor. If the second inductor is connected to a load, power will be electromagnetically coupled from the first inductor’s power source to that load. The powered inductor in a transformer is called the primary winding. The unpowered inductor in a transformer is called the secondary winding. Magnetic flux in the core (Φ) lags 90o behind the source voltage waveform. The current drawn by the primary coil from the source to produce this flux is called the magnetizing current, and it also lags the supply voltage by 90o. Total primary current in an unloaded transformer is called the exciting current, and is comprised of magnetizing current plus any additional current necessary to overcome core losses. It is never perfectly sinusoidal in a real transformer, but may be made more so if the transformer is designed and operated so that magnetic flux density is kept to a minimum. Core flux induces a voltage in any coil wrapped around the core. The induces voltage(s) are ideally in- phase with the primary winding source voltage and share the same waveshape. Any current drawn through the secondary winding by a load will be “reflected” to the primary winding and drawn from the voltage source, as if the source were directly powering a similar load. 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All rights reserved Privacy Policy · Terms of Service · User Agreement Connect with us     ARTICLES  FORUM  EDUCATION  MORE  IOT ARDUINO Top of Form Bottom of Form  Home   Textbook   Vol. I - Direct Current (DC)   Inductors Magnetic Fields and Inductance Magnetic Fields and Inductance Chapter 15 - Inductors Whenever electrons flow through a conductor, a magnetic field will develop around that conductor. This effect is called electromagnetism. Magnetic fields effect the alignment of electrons in an atom, and can cause physical force to develop between atoms across space just as with electric fields developing force between electrically charged particles. Like electric fields, magnetic fields can occupy completely empty space, and affect matter at a distance. Fields have two measures: a field force and a field flux. The field force is the amount of “push” that a field exerts over a certain distance. The fieldflux is the total quantity, or effect, of the field through space. Field force and flux are roughly analogous to voltage (“push”) and current (flow) through a conductor, respectively, although field flux can exist in totally empty space (without the motion of particles such as electrons) whereas current can only take place where there are free electrons to move. Field flux can be opposed in space, just as the flow of electrons can be opposed by resistance. The amount of field flux that will develop in space is proportional to the amount of field force applied, divided by the amount of opposition to flux. Just as the type of conducting material dictates that conductor’s specific resistance to electric current, the type of material occupying the space through which a magnetic field force is impressed dictates the specific opposition to magnetic field flux. Whereas an electric field flux between two conductors allows for an accumulation of free electron charge within those conductors, a magnetic field flux allows for a certain “inertia” to accumulate in the flow of electrons through the conductor producing the field. Inductors are components designed to take advantage of this phenomenon by shaping the length of conductive wire in the form of a coil. This shape creates a stronger magnetic field than what would be produced by a straight wire. Some inductors are formed with wire wound in a self-supporting coil. Others wrap the wire around a solid core material of some type. Sometimes the core of an inductor will be straight, and other times it will be joined in a loop (square, rectangular, or circular) to fully contain the magnetic flux. These design options all have an effect on the performance and characteristics of inductors. The schematic symbol for an inductor, like the capacitor, is quite simple, being little more than a coil symbol representing the coiled wire. Although a simple coil shape is the generic symbol for any inductor, inductors with cores are sometimes distinguished by the addition of parallel lines to the axis of the coil. A newer version of the inductor symbol dispenses with the coil shape in favor of several “humps” in a row: As the electric current produces a concentrated magnetic field around the coil, this field flux equates to a storage of energy representing the kinetic motion of the electrons through the coil. The more current in the coil, the stronger the magnetic field will be, and the more energy the inductor will store. Because inductors store the kinetic energy of moving electrons in the form of a magnetic field, they behave quite differently than resistors (which simply dissipate energy in the form of heat) in a circuit. Energy storage in an inductor is a function of the amount of current through it. An inductor’s ability to store energy as a function of current results in a tendency to try to maintain current at a constant level. In other words, inductors tend to resistchanges in current. When current through an inductor is increased or decreased, the inductor “resists” the change by producing a voltage between its leads in opposing polarity to the change. To store more energy in an inductor, the current through it must be increased. This means that its magnetic field must increase in strength, and that change in field strength produces the corresponding voltage according to the principle of electromagnetic self-induction. Conversely, to release energy from an inductor, the current through it must be decreased. This means that the inductor’s magnetic field must decrease in strength, and that change in field strength self-induces a voltage drop of just the opposite polarity. Just as Isaac Newton’s first Law of Motion (“an object in motion tends to stay in motion; an object at rest tends to stay at rest”) describes the tendency of a mass to oppose changes in velocity, we can state an inductor’s tendency to oppose changes in current as such: “Electrons moving through an inductor tend to stay in motion; electrons at rest in an inductor tend to stay at rest.” Hypothetically, an inductor left short-circuited will maintain a constant rate of current through it with no external assistance: Practically speaking, however, the ability for an inductor to self-sustain current is realized only with superconductive wire, as the wire resistance in any normal inductor is enough to cause current to decay very quickly with no external source of power. When the current through an inductor is increased, it drops a voltage opposing the direction of electron flow, acting as a power load. In this condition the inductor is said to be charging, because there is an increasing amount of energy being stored in its magnetic field. Note the polarity of the voltage with regard to the direction of current: Conversely, when the current through the inductor is decreased, it drops a voltage aiding the direction of electron flow, acting as a power source. In this condition the inductor is said to be discharging, because its store of energy is decreasing as it releases energy from its magnetic field to the rest of the circuit. Note the polarity of the voltage with regard to the direction of current. If a source of electric power is suddenly applied to an unmagnetized inductor, the inductor will initially resist the flow of electrons by dropping the full voltage of the source. As current begins to increase, a stronger and stronger magnetic field will be created, absorbing energy from the source. Eventually the current reaches a maximum level, and stops increasing. At this point, the inductor stops absorbing energy from the source, and is dropping minimum voltage across its leads, while the current remains at a maximum level. As an inductor stores more energy, its current level increases, while its voltage drop decreases. Note that this is precisely the opposite of capacitor behavior, where the storage of energy results in an increased voltage across the component! Whereas capacitors store their energy charge by maintaining a static voltage, inductors maintain their energy “charge” by maintaining a steady current through the coil. The type of material the wire is coiled around greatly impacts the strength of the magnetic field flux (and therefore the amount of stored energy) generated for any given amount of current through the coil. Coil cores made of ferromagnetic materials (such as soft iron) will encourage stronger field fluxes to develop with a given field force than nonmagnetic substances such as aluminum or air. The measure of an inductor’s ability to store energy for a given amount of current flow is called inductance. Not surprisingly, inductance is also a measure of the intensity of opposition to changes in current (exactly how much self-induced voltage will be produced for a given rate of change of current). Inductance is symbolically denoted with a capital “L,” and is measured in the unit of the Henry, abbreviated as “H.” An obsolete name for an inductor is choke, so called for its common usage to block (“choke”) high-frequency AC signals in radio circuits. Another name for an inductor, still used in modern times, is reactor, especially when used in large power applications. Both of these names will make more sense after you’ve studied alternating current (AC) circuit theory, and especially a principle known as inductive reactance. REVIEW: Inductors react against changes in current by dropping voltage in the polarity necessary to oppose the change. When an inductor is faced with an increasing current, it acts as a load: dropping voltage as it absorbs energy (negative on the current entry side and positive on the current exit side, like a resistor). When an inductor is faced with a decreasing current, it acts as a source: creating voltage as it releases stored energy (positive on the current entry side and negative on the current exit side, like a battery). The ability of an inductor to store energy in the form of a magnetic field (and consequently to oppose changes in current) is called inductance. It is measured in the unit of the Henry (H). Inductors used to be commonly known by another term: choke. In large power applications, they are sometimes referred to as reactors. Related Tools: STRIPLINE TRACE WIDTH CALCULATOR INVERTING OP-AMP RESISTOR CALCULATORBANDPASS FILTER CALCULATOR   ← Previous Page Textbook Index Next Page → You May Also Like: FIR Filter Design by Windowing: Concepts and the Rectangular Window ... Steve Arar Build an Arduino-Controlled AM/FM/SW Radio Combine the Si4844-A10 analog-tuned... Raymond Genovese Sinusoidal Steady-State Analysis Delve into phasor concepts and transforming a circuit with circuit analysis techniques to... Donald Krambeck PWM Digital-to-Analog Conversion with the SAM4S Xplained Pro In this article we’ll use the SAM4S Xplained... Robert Keim  Load More Share   Share   Share   Share   Share Published under the terms and conditions of the Design Science License   Pages in Chapter 15 Magnetic Fields and Inductance Inductors and Calculus Factors Affecting Inductance Series and Parallel Inductors Practical Considerations - Inductors  PDF Version ← Volume Index WHO WE ARE More about us  CONTENT News Projects Technical Articles Textbook Industry Articles Industry Webinars Code Library Video Lectures Worksheets Forum Tools Giveaways CATEGORIES Latest Analog Arduino Connectors Digital ICs Embedded IoT Power Sensors Test & Measurement Wearables Wireless RF CONNECT WITH US Facebook Twitter Contact Us YouTube LinkedIn Write For Us Advertise © EETech Media, LLC. All rights reserved Privacy Policy · Terms of Service · User Agreement Connect with us     ARTICLES  FORUM  EDUCATION  MORE  IOT ARDUINO Top of Form Bottom of Form  Home   Textbook   Vol. I - Direct Current (DC)   Inductors Magnetic Fields and Inductance Magnetic Fields and Inductance Chapter 15 - Inductors Whenever electrons flow through a conductor, a magnetic field will develop around that conductor. This effect is called electromagnetism. Magnetic fields effect the alignment of electrons in an atom, and can cause physical force to develop between atoms across space just as with electric fields developing force between electrically charged particles. Like electric fields, magnetic fields can occupy completely empty space, and affect matter at a distance. Fields have two measures: a field force and a field flux. The field force is the amount of “push” that a field exerts over a certain distance. The fieldflux is the total quantity, or effect, of the field through space. Field force and flux are roughly analogous to voltage (“push”) and current (flow) through a conductor, respectively, although field flux can exist in totally empty space (without the motion of particles such as electrons) whereas current can only take place where there are free electrons to move. Field flux can be opposed in space, just as the flow of electrons can be opposed by resistance. The amount of field flux that will develop in space is proportional to the amount of field force applied, divided by the amount of opposition to flux. Just as the type of conducting material dictates that conductor’s specific resistance to electric current, the type of material occupying the space through which a magnetic field force is impressed dictates the specific opposition to magnetic field flux. Whereas an electric field flux between two conductors allows for an accumulation of free electron charge within those conductors, a magnetic field flux allows for a certain “inertia” to accumulate in the flow of electrons through the conductor producing the field. Inductors are components designed to take advantage of this phenomenon by shaping the length of conductive wire in the form of a coil. This shape creates a stronger magnetic field than what would be produced by a straight wire. Some inductors are formed with wire wound in a self-supporting coil. Others wrap the wire around a solid core material of some type. Sometimes the core of an inductor will be straight, and other times it will be joined in a loop (square, rectangular, or circular) to fully contain the magnetic flux. These design options all have an effect on the performance and characteristics of inductors. The schematic symbol for an inductor, like the capacitor, is quite simple, being little more than a coil symbol representing the coiled wire. Although a simple coil shape is the generic symbol for any inductor, inductors with cores are sometimes distinguished by the addition of parallel lines to the axis of the coil. A newer version of the inductor symbol dispenses with the coil shape in favor of several “humps” in a row: As the electric current produces a concentrated magnetic field around the coil, this field flux equates to a storage of energy representing the kinetic motion of the electrons through the coil. The more current in the coil, the stronger the magnetic field will be, and the more energy the inductor will store. Because inductors store the kinetic energy of moving electrons in the form of a magnetic field, they behave quite differently than resistors (which simply dissipate energy in the form of heat) in a circuit. Energy storage in an inductor is a function of the amount of current through it. An inductor’s ability to store energy as a function of current results in a tendency to try to maintain current at a constant level. In other words, inductors tend to resistchanges in current. When current through an inductor is increased or decreased, the inductor “resists” the change by producing a voltage between its leads in opposing polarity to the change. To store more energy in an inductor, the current through it must be increased. This means that its magnetic field must increase in strength, and that change in field strength produces the corresponding voltage according to the principle of electromagnetic self-induction. Conversely, to release energy from an inductor, the current through it must be decreased. This means that the inductor’s magnetic field must decrease in strength, and that change in field strength self-induces a voltage drop of just the opposite polarity. Just as Isaac Newton’s first Law of Motion (“an object in motion tends to stay in motion; an object at rest tends to stay at rest”) describes the tendency of a mass to oppose changes in velocity, we can state an inductor’s tendency to oppose changes in current as such: “Electrons moving through an inductor tend to stay in motion; electrons at rest in an inductor tend to stay at rest.” Hypothetically, an inductor left short-circuited will maintain a constant rate of current through it with no external assistance: Practically speaking, however, the ability for an inductor to self-sustain current is realized only with superconductive wire, as the wire resistance in any normal inductor is enough to cause current to decay very quickly with no external source of power. When the current through an inductor is increased, it drops a voltage opposing the direction of electron flow, acting as a power load. In this condition the inductor is said to be charging, because there is an increasing amount of energy being stored in its magnetic field. Note the polarity of the voltage with regard to the direction of current: Conversely, when the current through the inductor is decreased, it drops a voltage aiding the direction of electron flow, acting as a power source. In this condition the inductor is said to be discharging, because its store of energy is decreasing as it releases energy from its magnetic field to the rest of the circuit. Note the polarity of the voltage with regard to the direction of current. If a source of electric power is suddenly applied to an unmagnetized inductor, the inductor will initially resist the flow of electrons by dropping the full voltage of the source. As current begins to increase, a stronger and stronger magnetic field will be created, absorbing energy from the source. Eventually the current reaches a maximum level, and stops increasing. At this point, the inductor stops absorbing energy from the source, and is dropping minimum voltage across its leads, while the current remains at a maximum level. As an inductor stores more energy, its current level increases, while its voltage drop decreases. Note that this is precisely the opposite of capacitor behavior, where the storage of energy results in an increased voltage across the component! Whereas capacitors store their energy charge by maintaining a static voltage, inductors maintain their energy “charge” by maintaining a steady current through the coil. The type of material the wire is coiled around greatly impacts the strength of the magnetic field flux (and therefore the amount of stored energy) generated for any given amount of current through the coil. Coil cores made of ferromagnetic materials (such as soft iron) will encourage stronger field fluxes to develop with a given field force than nonmagnetic substances such as aluminum or air. The measure of an inductor’s ability to store energy for a given amount of current flow is called inductance. Not surprisingly, inductance is also a measure of the intensity of opposition to changes in current (exactly how much self-induced voltage will be produced for a given rate of change of current). Inductance is symbolically denoted with a capital “L,” and is measured in the unit of the Henry, abbreviated as “H.” An obsolete name for an inductor is choke, so called for its common usage to block (“choke”) high-frequency AC signals in radio circuits. Another name for an inductor, still used in modern times, is reactor, especially when used in large power applications. Both of these names will make more sense after you’ve studied alternating current (AC) circuit theory, and especially a principle known as inductive reactance. REVIEW: Inductors react against changes in current by dropping voltage in the polarity necessary to oppose the change. When an inductor is faced with an increasing current, it acts as a load: dropping voltage as it absorbs energy (negative on the current entry side and positive on the current exit side, like a resistor). When an inductor is faced with a decreasing current, it acts as a source: creating voltage as it releases stored energy (positive on the current entry side and negative on the current exit side, like a battery). The ability of an inductor to store energy in the form of a magnetic field (and consequently to oppose changes in current) is called inductance. It is measured in the unit of the Henry (H). Inductors used to be commonly known by another term: choke. In large power applications, they are sometimes referred to as reactors. Related Tools: STRIPLINE TRACE WIDTH CALCULATOR INVERTING OP-AMP RESISTOR CALCULATORBANDPASS FILTER CALCULATOR   ← Previous Page Textbook Index Next Page → You May Also Like: FIR Filter Design by Windowing: Concepts and the Rectangular Window ... Steve Arar Build an Arduino-Controlled AM/FM/SW Radio Combine the Si4844-A10 analog-tuned... Raymond Genovese Sinusoidal Steady-State Analysis Delve into phasor concepts and transforming a circuit with circuit analysis techniques to... Donald Krambeck PWM Digital-to-Analog Conversion with the SAM4S Xplained Pro In this article we’ll use the SAM4S Xplained... Robert Keim  Load More Share   Share   Share   Share   Share Published under the terms and conditions of the Design Science License   Pages in Chapter 15 Magnetic Fields and Inductance Inductors and Calculus Factors Affecting Inductance Series and Parallel Inductors Practical Considerations - Inductors  PDF Version ← Volume Index WHO WE ARE More about us  CONTENT News Projects Technical Articles Textbook Industry Articles Industry Webinars Code Library Video Lectures Worksheets Forum Tools Giveaways CATEGORIES Latest Analog Arduino Connectors Digital ICs Embedded IoT Power Sensors Test & Measurement Wearables Wireless RF CONNECT WITH US Facebook Twitter Contact Us YouTube LinkedIn Write For Us Advertise © EETech Media, LLC. All rights reserved Privacy Policy · Terms of Service · User Agreement Connect with us     ARTICLES  FORUM  EDUCATION  MORE  IOT ARDUINO Top of Form Bottom of Form  Home   Textbook   Vol. I - Direct Current (DC)   Inductors Magnetic Fields and Inductance Magnetic Fields and Inductance Chapter 15 - Inductors Whenever electrons flow through a conductor, a magnetic field will develop around that conductor. This effect is called electromagnetism. Magnetic fields effect the alignment of electrons in an atom, and can cause physical force to develop between atoms across space just as with electric fields developing force between electrically charged particles. Like electric fields, magnetic fields can occupy completely empty space, and affect matter at a distance. Fields have two measures: a field force and a field flux. The field force is the amount of “push” that a field exerts over a certain distance. The fieldflux is the total quantity, or effect, of the field through space. Field force and flux are roughly analogous to voltage (“push”) and current (flow) through a conductor, respectively, although field flux can exist in totally empty space (without the motion of particles such as electrons) whereas current can only take place where there are free electrons to move. Field flux can be opposed in space, just as the flow of electrons can be opposed by resistance. The amount of field flux that will develop in space is proportional to the amount of field force applied, divided by the amount of opposition to flux. Just as the type of conducting material dictates that conductor’s specific resistance to electric current, the type of material occupying the space through which a magnetic field force is impressed dictates the specific opposition to magnetic field flux. Whereas an electric field flux between two conductors allows for an accumulation of free electron charge within those conductors, a magnetic field flux allows for a certain “inertia” to accumulate in the flow of electrons through the conductor producing the field. Inductors are components designed to take advantage of this phenomenon by shaping the length of conductive wire in the form of a coil. This shape creates a stronger magnetic field than what would be produced by a straight wire. Some inductors are formed with wire wound in a self-supporting coil. Others wrap the wire around a solid core material of some type. Sometimes the core of an inductor will be straight, and other times it will be joined in a loop (square, rectangular, or circular) to fully contain the magnetic flux. These design options all have an effect on the performance and characteristics of inductors. The schematic symbol for an inductor, like the capacitor, is quite simple, being little more than a coil symbol representing the coiled wire. Although a simple coil shape is the generic symbol for any inductor, inductors with cores are sometimes distinguished by the addition of parallel lines to the axis of the coil. A newer version of the inductor symbol dispenses with the coil shape in favor of several “humps” in a row: As the electric current produces a concentrated magnetic field around the coil, this field flux equates to a storage of energy representing the kinetic motion of the electrons through the coil. The more current in the coil, the stronger the magnetic field will be, and the more energy the inductor will store. Because inductors store the kinetic energy of moving electrons in the form of a magnetic field, they behave quite differently than resistors (which simply dissipate energy in the form of heat) in a circuit. Energy storage in an inductor is a function of the amount of current through it. An inductor’s ability to store energy as a function of current results in a tendency to try to maintain current at a constant level. In other words, inductors tend to resistchanges in current. When current through an inductor is increased or decreased, the inductor “resists” the change by producing a voltage between its leads in opposing polarity to the change. To store more energy in an inductor, the current through it must be increased. This means that its magnetic field must increase in strength, and that change in field strength produces the corresponding voltage according to the principle of electromagnetic self-induction. Conversely, to release energy from an inductor, the current through it must be decreased. This means that the inductor’s magnetic field must decrease in strength, and that change in field strength self-induces a voltage drop of just the opposite polarity. Just as Isaac Newton’s first Law of Motion (“an object in motion tends to stay in motion; an object at rest tends to stay at rest”) describes the tendency of a mass to oppose changes in velocity, we can state an inductor’s tendency to oppose changes in current as such: “Electrons moving through an inductor tend to stay in motion; electrons at rest in an inductor tend to stay at rest.” Hypothetically, an inductor left short-circuited will maintain a constant rate of current through it with no external assistance: Practically speaking, however, the ability for an inductor to self-sustain current is realized only with superconductive wire, as the wire resistance in any normal inductor is enough to cause current to decay very quickly with no external source of power. When the current through an inductor is increased, it drops a voltage opposing the direction of electron flow, acting as a power load. In this condition the inductor is said to be charging, because there is an increasing amount of energy being stored in its magnetic field. Note the polarity of the voltage with regard to the direction of current: Conversely, when the current through the inductor is decreased, it drops a voltage aiding the direction of electron flow, acting as a power source. In this condition the inductor is said to be discharging, because its store of energy is decreasing as it releases energy from its magnetic field to the rest of the circuit. Note the polarity of the voltage with regard to the direction of current. If a source of electric power is suddenly applied to an unmagnetized inductor, the inductor will initially resist the flow of electrons by dropping the full voltage of the source. As current begins to increase, a stronger and stronger magnetic field will be created, absorbing energy from the source. Eventually the current reaches a maximum level, and stops increasing. At this point, the inductor stops absorbing energy from the source, and is dropping minimum voltage across its leads, while the current remains at a maximum level. As an inductor stores more energy, its current level increases, while its voltage drop decreases. Note that this is precisely the opposite of capacitor behavior, where the storage of energy results in an increased voltage across the component! Whereas capacitors store their energy charge by maintaining a static voltage, inductors maintain their energy “charge” by maintaining a steady current through the coil. The type of material the wire is coiled around greatly impacts the strength of the magnetic field flux (and therefore the amount of stored energy) generated for any given amount of current through the coil. Coil cores made of ferromagnetic materials (such as soft iron) will encourage stronger field fluxes to develop with a given field force than nonmagnetic substances such as aluminum or air. The measure of an inductor’s ability to store energy for a given amount of current flow is called inductance. Not surprisingly, inductance is also a measure of the intensity of opposition to changes in current (exactly how much self-induced voltage will be produced for a given rate of change of current). Inductance is symbolically denoted with a capital “L,” and is measured in the unit of the Henry, abbreviated as “H.” An obsolete name for an inductor is choke, so called for its common usage to block (“choke”) high-frequency AC signals in radio circuits. Another name for an inductor, still used in modern times, is reactor, especially when used in large power applications. Both of these names will make more sense after you’ve studied alternating current (AC) circuit theory, and especially a principle known as inductive reactance. REVIEW: Inductors react against changes in current by dropping voltage in the polarity necessary to oppose the change. When an inductor is faced with an increasing current, it acts as a load: dropping voltage as it absorbs energy (negative on the current entry side and positive on the current exit side, like a resistor). When an inductor is faced with a decreasing current, it acts as a source: creating voltage as it releases stored energy (positive on the current entry side and negative on the current exit side, like a battery). The ability of an inductor to store energy in the form of a magnetic field (and consequently to oppose changes in current) is called inductance. It is measured in the unit of the Henry (H). Inductors used to be commonly known by another term: choke. In large power applications, they are sometimes referred to as reactors. Related Tools: STRIPLINE TRACE WIDTH CALCULATOR INVERTING OP-AMP RESISTOR CALCULATORBANDPASS FILTER CALCULATOR   ← Previous Page Textbook Index Next Page → You May Also Like: FIR Filter Design by Windowing: Concepts and the Rectangular Window ... Steve Arar Build an Arduino-Controlled AM/FM/SW Radio Combine the Si4844-A10 analog-tuned... Raymond Genovese Sinusoidal Steady-State Analysis Delve into phasor concepts and transforming a circuit with circuit analysis techniques to... Donald Krambeck PWM Digital-to-Analog Conversion with the SAM4S Xplained Pro In this article we’ll use the SAM4S Xplained... Robert Keim  Load More Share   Share   Share   Share   Share Published under the terms and conditions of the Design Science License   Pages in Chapter 15 Magnetic Fields and Inductance Inductors and Calculus Factors Affecting Inductance Series and Parallel Inductors Practical Considerations - Inductors  PDF Version ← Volume Index WHO WE ARE More about us  CONTENT News Projects Technical Articles Textbook Industry Articles Industry Webinars Code Library Video Lectures Worksheets Forum Tools Giveaways CATEGORIES Latest Analog Arduino Connectors Digital ICs Embedded IoT Power Sensors Test & Measurement Wearables Wireless RF CONNECT WITH US Facebook Twitter Contact Us YouTube LinkedIn Write For Us Advertise © EETech Media, LLC. All rights reserved Privacy Policy · Terms of Service · User Agreement Connect with us     ARTICLES  FORUM  EDUCATION  MORE  IOT ARDUINO Top of Form Bottom of Form  Home   Textbook   Vol. I - Direct Current (DC)   Inductors Inductors and Calculus Inductors and Calculus Chapter 15 - Inductors Inductors do not have a stable “resistance” as conductors do. However, there is a definite mathematical relationship between voltage and current for an inductor, as follows: You should recognize the form of this equation from the capacitor chapter. It relates one variable (in this case, inductor voltage drop) to a rate of change of another variable (in this case, inductor current). Both voltage (v) and rate of current change (di/dt) are instantaneous: that is, in relation to a specific point in time, thus the lower-case letters “v” and “i”. As with the capacitor formula, it is convention to express instantaneous voltage as vrather than e, but using the latter designation would not be wrong. Current rate-of-change (di/dt) is expressed in units of amps per second, a positive number representing an increase and a negative number representing a decrease. Like a capacitor, an inductor’s behavior is rooted in the variable of time. Aside from any resistance intrinsic to an inductor’s wire coil (which we will assume is zero for the sake of this section), the voltage dropped across the terminals of an inductor is purely related to how quickly its current changes over time. Suppose we were to connect a perfect inductor (one having zero ohms of wire resistance) to a circuit where we could vary the amount of current through it with a potentiometer connected as a variable resistor: If the potentiometer mechanism remains in a single position (wiper is stationary), the series-connected ammeter will register a constant (unchanging) current, and the voltmeter connected across the inductor will register 0 volts. In this scenario, the instantaneous rate of current change (di/dt) is equal to zero, because the current is stable. The equation tells us that with 0 amps per second change for a di/dt, there must be zero instantaneous voltage (v) across the inductor. From a physical perspective, with no current change, there will be a steady magnetic field generated by the inductor. With no change in magnetic flux (dΦ/dt = 0 Webers per second), there will be no voltage dropped across the length of the coil due to induction. If we move the potentiometer wiper slowly in the “up” direction, its resistance from end to end will slowly decrease. This has the effect of increasing current in the circuit, so the ammeter indication should be increasing at a slow rate: Assuming that the potentiometer wiper is being moved such that the rate of current increase through the inductor is steady, the di/dt term of the formula will be a fixed value. This fixed value, multiplied by the inductor’s inductance in Henrys (also fixed), results in a fixed voltage of some magnitude. From a physical perspective, the gradual increase in current results in a magnetic field that is likewise increasing. This gradual increase in magnetic flux causes a voltage to be induced in the coil as expressed by Michael Faraday’s induction equation e = N(dΦ/dt). This self-induced voltage across the coil, as a result of a gradual change in current magnitude through the coil, happens to be of a polarity that attempts to oppose the change in current. In other words, the induced voltage polarity resulting from an increase in current will be oriented in such a way as to push against the direction of current, to try to keep the current at its former magnitude. This phenomenon exhibits a more general principle of physics known as Lenz’s Law, which states that an induced effect will always be opposed to the cause producing it. In this scenario, the inductor will be acting as a load, with the negative side of the induced voltage on the end where electrons are entering, and the positive side of the induced voltage on the end where electrons are exiting. Changing the rate of current increase through the inductor by moving the potentiometer wiper “up” at different speeds results in different amounts of voltage being dropped across the inductor, all with the same polarity (opposing the increase in current): Here again we see the derivative function of calculus exhibited in the behavior of an inductor. In calculus terms, we would say that the induced voltage across the inductor is the derivative of the current through the inductor: that is, proportional to the current’s rate-of-change with respect to time. Reversing the direction of wiper motion on the potentiometer (going “down” rather than “up”) will result in its end-to-end resistance increasing. This will result in circuit current decreasing (a negative figure for di/dt). The inductor, always opposing any change in current, will produce a voltage drop opposed to the direction of change: How much voltage the inductor will produce depends, of course, on how rapidly the current through it is decreased. As described by Lenz’s Law, the induced voltage will be opposed to the change in current. With adecreasing current, the voltage polarity will be oriented so as to try to keep the current at its former magnitude. In this scenario, the inductor will be acting as a source, with the negative side of the induced voltage on the end where electrons are exiting, and the positive side of the induced voltage on the end where electrons are entering. The more rapidly current is decreased, the more voltage will be produced by the inductor, in its release of stored energy to try to keep current constant. Again, the amount of voltage across a perfect inductor is directly proportional to the rate of current change through it. The only difference between the effects of a decreasing current and an increasing current is thepolarity of the induced voltage. For the same rate of current change over time, either increasing or decreasing, the voltage magnitude (volts) will be the same. For example, a di/dt of -2 amps per second will produce the same amount of induced voltage drop across an inductor as a di/dt of +2 amps per second, just in the opposite polarity. If current through an inductor is forced to change very rapidly, very high voltages will be produced. Consider the following circuit: In this circuit, a lamp is connected across the terminals of an inductor. A switch is used to control current in the circuit, and power is supplied by a 6 volt battery. When the switch is closed, the inductor will briefly oppose the change in current from zero to some magnitude, but will drop only a small amount of voltage. It takes about 70 volts to ionize the neon gas inside a neon bulb like this, so the bulb cannot be lit on the 6 volts produced by the battery, or the low voltage momentarily dropped by the inductor when the switch is closed: When the switch is opened, however, it suddenly introduces an extremely high resistance into the circuit (the resistance of the air gap between the contacts). This sudden introduction of high resistance into the circuit causes the circuit current to decrease almost instantly. Mathematically, the di/dt term will be a very large negative number. Such a rapid change of current (from some magnitude to zero in very little time) will induce a very high voltage across the inductor, oriented with negative on the left and positive on the right, in an effort to oppose this decrease in current. The voltage produced is usually more than enough to light the neon lamp, if only for a brief moment until the current decays to zero: For maximum effect, the inductor should be sized as large as possible (at least 1 Henry of inductance). Related Tools: POWER ADDED EFFICIENCY CALCULATOR TWISTED PAIR IMPEDANCE CALCULATORFRIIS TRANSMISSION CALCULATOR   ← Previous Page Textbook Index Next Page → You May Also Like: Using Scilab GUI for RGB and Lux Measurements Gather data via USB from a BH1745NUC... Robert Keim How to Display an Image on an LCD using an EFM8 Microcontroller Use the EFM8’s USB functionality to... Robert Keim Electronics Design on Linux Interested in using Linux for your everyday tasks, but worried about compatibility? This... Trevor Gamblin Design Your Own Controller for a Solder Reflow Oven In the last installment, we built the... Patrick Lloyd  Load More Share   Share   Share   Share   Share Published under the terms and conditions of the Design Science License   Pages in Chapter 15 Magnetic Fields and Inductance Inductors and Calculus Factors Affecting Inductance Series and Parallel Inductors Practical Considerations - Inductors  PDF Version ← Volume Index WHO WE ARE More about us  CONTENT News Projects Technical Articles Textbook Industry Articles Industry Webinars Code Library Video Lectures Worksheets Forum Tools Giveaways CATEGORIES Latest Analog Arduino Connectors Digital ICs Embedded IoT Power Sensors Test & Measurement Wearables Wireless RF CONNECT WITH US Facebook Twitter Contact Us YouTube LinkedIn Write For Us Advertise © EETech Media, LLC. All rights reserved Privacy Policy · Terms of Service · User Agreement Connect with us     ARTICLES  FORUM  EDUCATION  MORE  IOT ARDUINO Top of Form Bottom of Form  Home   Textbook   Vol. II - Alternating Current (AC) Reactance and Impedance -- Inductive   AC Inductor Circuits AC Inductor Circuits Chapter 3 - Reactance and Impedance -- Inductive Inductors do not behave the same as resistors. Whereas resistors simply oppose the flow of electrons through them (by dropping a voltage directly proportional to the current), inductors oppose changes in current through them, by dropping a voltage directly proportional to the rate of change of current. In accordance with Lenz’s Law, this induced voltage is always of such a polarity as to try to maintain current at its present value. That is, if current is increasing in magnitude, the induced voltage will “push against” the electron flow; if current is decreasing, the polarity will reverse and “push with” the electron flow to oppose the decrease. This opposition to current change is called reactance, rather than resistance. Expressed mathematically, the relationship between the voltage dropped across the inductor and rate of current change through the inductor is as such: The expression di/dt is one from calculus, meaning the rate of change of instantaneous current (i) over time, in amps per second. The inductance (L) is in Henrys, and the instantaneous voltage (e), of course, is in volts. Sometimes you will find the rate of instantaneous voltage expressed as “v” instead of “e” (v = L di/dt), but it means the exact same thing. To show what happens with alternating current, let’s analyze a simple inductor circuit: (Figure below) Pure inductive circuit: Inductor current lags inductor voltage by 90o. If we were to plot the current and voltage for this very simple circuit, it would look something like this: (Figure below) Pure inductive circuit, waveforms. Remember, the voltage dropped across an inductor is a reaction against the change in current through it. Therefore, the instantaneous voltage is zero whenever the instantaneous current is at a peak (zero change, or level slope, on the current sine wave), and the instantaneous voltage is at a peak wherever the instantaneous current is at maximum change (the points of steepest slope on the current wave, where it crosses the zero line). This results in a voltage wave that is 90o out of phase with the current wave. Looking at the graph, the voltage wave seems to have a “head start” on the current wave; the voltage “leads” the current, and the current “lags” behind the voltage. (Figure below) Current lags voltage by 90o in a pure inductive circuit. Things get even more interesting when we plot the power for this circuit: (Figure below) In a pure inductive circuit, instantaneous power may be positive or negative Because instantaneous power is the product of the instantaneous voltage and the instantaneous current (p=ie), the power equals zero whenever the instantaneous current or voltage is zero. Whenever the instantaneous current and voltage are both positive (above the line), the power is positive. As with the resistor example, the power is also positive when the instantaneous current and voltage are both negative (below the line). However, because the current and voltage waves are 90o out of phase, there are times when one is positive while the other is negative, resulting in equally frequent occurrences of negative instantaneous power. But what does negative power mean? It means that the inductor is releasing power back to the circuit, while a positive power means that it is absorbing power from the circuit. Since the positive and negative power cycles are equal in magnitude and duration over time, the inductor releases just as much power back to the circuit as it absorbs over the span of a complete cycle. What this means in a practical sense is that the reactance of an inductor dissipates a net energy of zero, quite unlike the resistance of a resistor, which dissipates energy in the form of heat. Mind you, this is for perfect inductors only, which have no wire resistance. An inductor’s opposition to change in current translates to an opposition to alternating current in general, which is by definition always changing in instantaneous magnitude and direction. This opposition to alternating current is similar to resistance, but different in that it always results in a phase shift between current and voltage, and it dissipates zero power. Because of the differences, it has a different name: reactance. Reactance to AC is expressed in ohms, just like resistance is, except that its mathematical symbol is X instead of R. To be specific, reactance associated with an inductor is usually symbolized by the capital letter X with a letter L as a subscript, like this: XL. Since inductors drop voltage in proportion to the rate of current change, they will drop more voltage for faster-changing currents, and less voltage for slower-changing currents. What this means is that reactance in ohms for any inductor is directly proportional to the frequency of the alternating current. The exact formula for determining reactance is as follows: If we expose a 10 mH inductor to frequencies of 60, 120, and 2500 Hz, it will manifest the reactances in Table Figure below. Reactance of a 10 mH inductor: Frequency (Hertz) Reactance (Ohms) 60 3.7699 120 7.5398 2500 157.0796 In the reactance equation, the term “2πf” (everything on the right-hand side except the L) has a special meaning unto itself. It is the number of radians per second that the alternating current is “rotating” at, if you imagine one cycle of AC to represent a full circle’s rotation. A radian is a unit of angular measurement: there are 2π radians in one full circle, just as there are 360oin a full circle. If the alternator producing the AC is a double-pole unit, it will produce one cycle for every full turn of shaft rotation, which is every 2π radians, or 360o. If this constant of 2π is multiplied by frequency in Hertz (cycles per second), the result will be a figure in radians per second, known as the angular velocity of the AC system. Angular velocity may be represented by the expression 2πf, or it may be represented by its own symbol, the lower-case Greek letter Omega, which appears similar to our Roman lower-case “w”: ω. Thus, the reactance formula XL = 2πfL could also be written as XL = ωL. It must be understood that this “angular velocity” is an expression of how rapidly the AC waveforms are cycling, a full cycle being equal to 2π radians. It is not necessarily representative of the actual shaft speed of the alternator producing the AC. If the alternator has more than two poles, the angular velocity will be a multiple of the shaft speed. For this reason, ω is sometimes expressed in units of electrical radians per second rather than (plain) radians per second, so as to distinguish it from mechanical motion. Any way we express the angular velocity of the system, it is apparent that it is directly proportional to reactance in an inductor. As the frequency (or alternator shaft speed) is increased in an AC system, an inductor will offer greater opposition to the passage of current, and vice versa. Alternating current in a simple inductive circuit is equal to the voltage (in volts) divided by the inductive reactance (in ohms), just as either alternating or direct current in a simple resistive circuit is equal to the voltage (in volts) divided by the resistance (in ohms). An example circuit is shown here: (Figurebelow) Inductive reactance However, we need to keep in mind that voltage and current are not in phase here. As was shown earlier, the voltage has a phase shift of +90owith respect to the current. (Figure below) If we represent these phase angles of voltage and current mathematically in the form of complex numbers, we find that an inductor’s opposition to current has a phase angle, too: Current lags voltage by 90o in an inductor. Mathematically, we say that the phase angle of an inductor’s opposition to current is 90o, meaning that an inductor’s opposition to current is a positive imaginary quantity. This phase angle of reactive opposition to current becomes critically important in circuit analysis, especially for complex AC circuits where reactance and resistance interact. It will prove beneficial to represent any component’s opposition to current in terms of complex numbers rather than scalar quantities of resistance and reactance. REVIEW: Inductive reactance is the opposition that an inductor offers to alternating current due to its phase-shifted storage and release of energy in its magnetic field. Reactance is symbolized by the capital letter “X” and is measured in ohms just like resistance (R). Inductive reactance can be calculated using this formula: XL = 2πfL The angular velocity of an AC circuit is another way of expressing its frequency, in units of electrical radians per second instead of cycles per second. It is symbolized by the lower-case Greek letter “omega,” or ω. Inductive reactance increases with increasing frequency. In other words, the higher the frequency, the more it opposes the AC flow of electrons. Related Tools: 555 TIMER MONOSTABLE CIRCUIT TRACE RESISTANCE CALCULATOREMBEDDED MICROSTRIP IMPEDANCE CALCULATOR   ← Previous Page Textbook Index Next Page → You May Also Like: How to Build a Robot - Design and Schematic Part one of a series of articles on building a... Travis Fagerness Everything You Need to Know About Direct Digital Synthesis Direct Digital Synthesis is used to generate... Marie Christiano What are Integrated Development Environments? Integrated Development Environments... Marie Christiano Beginner’s Guide to the Arduino This project will help you get started with the Arduino, including a description of the... Editorial Team  Load More Share   Share   Share   Share   Share Published under the terms and conditions of the Design Science License   Pages in Chapter 3 AC Resistor Circuits (Inductive) AC Inductor Circuits Series Resistor-Inductor Circuits Parallel Resistor-Inductor Circuits Inductor Quirks More on the “Skin Effect’’  PDF Version ← Volume Index WHO WE ARE More about us  CONTENT News Projects Technical Articles Textbook Industry Articles Industry Webinars Code Library Video Lectures Worksheets Forum Tools Giveaways CATEGORIES Latest Analog Arduino Connectors Digital ICs Embedded IoT Power Sensors Test & Measurement Wearables Wireless RF CONNECT WITH US Facebook Twitter Contact Us YouTube LinkedIn Write For Us Advertise © EETech Media, LLC. All rights reserved Privacy Policy · Terms of Service · User Agreement Capacitors and batteries 7-9-99 Capacitors: devices for storing charge A capacitor is a device for storing charge. It is usually made up of two plates separated by a thin insulating material known as the dielectric. One plate of the capacitor is positively charged, while the other has negative charge. The charge stored in a capacitor is proportional to the potential difference between the two plates. For a capacitor with charge Q on the positive plate and -Q on the negative plate, the charge is proportional to the potential: If C is the capacitance, Q = CV The capacitance is a measure of the amount of charge a capacitor can store; this is determined by the capacitor geometry and by the kind of dielectric between the plates. For a parallel plate capacitor made up of two plates of area A and separated by a distance d, with no dielectric material, the capacitance is given by : Note that capacitance has units of farads (F). A 1 F capacitor is exceptionally large; typical capacitors have capacitances in the pF - microfarad range. Dielectrics, insulating materials placed between the plates of a capacitor, cause the electric field inside the capacitor to be reduced for the same amount of charge on the plates. This is because the molecules of the dielectric material get polarized in the field, and they align themselves in a way that sets up another field inside the dielectric opposite to the field from the capacitor plates. The dielectric constant is the ratio of the electric field without the dielectric to the field with the dielectric: Note that for a set of parallel plates, the electric field between the plates is related to the potential difference by the equation: for a parallel-plate capacitor: E = V / d For a given potential difference (i.e., for a given voltage), the higher the dielectric constant, the more charge can be stored in the capacitor. For a parallel-plate capacitor with a dielectric between the plates, the capacitance is: Energy stored in a capacitor The energy stored in a capacitor is the same as the work needed to build up the charge on the plates. As the charge increases, the harder it is to add more. Potential energy is the charge multiplied by the potential, and as the charge builds up the potential does too. If the potential difference between the two plates is V at the end of the process, and 0 at the start, the average potential is V / 2. Multiplying this average potential by the charge gives the potential energy : PE = 1/2 Q V. Substituting in for Q, Q = CV, gives: The energy stored in a capacitor is: U = 1/2 C V2 Capacitors have a variety of uses because there are many applications that involve storing charge. A good example is computer memory, but capacitors are found in all sorts of electrical circuits, and are often used to minimize voltage fluctuations. Another application is a flash bulb for a camera, which requires a lot of charge to be transferred in a short time. Batteries are good at providing a small amount of charge for a long time, so charge is transferred slowly from a battery to a capacitor. The capacitor is discharged quickly through a flash bulb, lighting the bulb brightly for a short time. If the distance between the plates of a capacitor is changed, the capacitance is changed. For a charged capacitor, a change in capacitance correspond to a change in voltage, which is easily measured. This is exploited in applications ranging from certain microphones to the the keys in some computer keyboards. Playing with a capacitor To help understand how a capacitor works, we can experiment using a power supply, a capacitor, and a piece of dielectric material. The power supply provides the voltage, or potential difference, that causes charge to build up on the capacitor plates. With the power supply connected to the capacitor, a constant difference in potential is maintained between the two plates. This results in a certain amount of charge moving on to the plates from the power supply, and there is a particular electric field between the plates. When some dielectric material is inserted between the plates, the field can not change because the potential difference is constant, and E = V / d. To ensure that the field does not change, charge flows from the power supply to the plates of the capacitor. Removing the dielectric causes the charge to flow back to the power supply, keeping the field constant. To summarize, when the voltage is fixed but the capacitance changes, the amount of charge on the plates changes. On the other hand, if the power supply is connected to the capacitor briefly and then removed, it will be the charge that stays constant. If a dielectric material is inserted between the plates in this case, the field between the plates will be reduced, as will the potential difference. Removing the dielectric increases the field, and therefore increases the voltage. Electric fields and potentials in the human body The body is full of electrical impulses, and we can measure these signals using electrodes placed on the skin. The rhythmic contractions of the heart, for example, are caused by carefully timed electrical impulses. These can be measured with an electrocardiogram (ECG or EKG). If the heart is malfunctioning, this will usually produce a change in the electrical activity of the heart, with particular changes corresponding to particular problems. Similar analysis can be done on the brain using an electroencephalogram (EEG). Batteries and EMF Capacitors are very good at storing charge for short time periods, and they can be charged and recharged very quickly. There are many applications, however, where it's more convenient to have a slow-but-steady flow of charge; for these applications batteries are used. A battery is another device for storing charge (or, put another way, for storing electrical energy). A battery consists of two electrodes, the anode (negative) and cathode (positive. Usually these are two dissimilar metals such as copper and zinc. These are immersed in a solution (sometimes an acid solution). A chemical reaction results in a potential difference between the two terminals. When the battery is connected to a circuit, electrons produced by the chemical reaction at the anode flow through the circuit to the cathode. At the cathode, the electrons are consumed in another chemical reaction. The circuit is completed by positive ions (H+, in many cases) flowing through the solution in the battery from the anode to the cathode. The voltage of a battery is also known as the emf, the electromotive force. This emf can be thought of as the pressure that causes charges to flow through a circuit the battery is part of. This flow of charge is very similar to the flow of other things, such as heat or water. A flow of charge is known as a current. Batteries put out direct current, as opposed to alternating current, which is what comes out of a wall socket. With direct current, the charge flows only in one direction. With alternating current, the charges slosh back and forth, continually reversing direction. Current and Drift velocity An electric current, which is a flow of charge, occurs when there is a potential difference. For a current to flow also requires a complete circuit, which means the flowing charge has to be able to get back to where it starts. Current (I) is measured in amperes (A), and is the amount of charge flowing per second. current : I = q / t, with units of A = C / s When current flows through wires in a circuit, the moving charges are electrons. For historical reasons, however, when analyzing circuits the direction of the current is taken to be the direction of the flow of positive charge, opposite to the direction the electrons go. We can blame Benjamin Franklin for this. It amounts to the same thing, because the flow of positive charge in one direction is equivalent to the flow of negative charge in the opposite direction. When a battery or power supply sets up a difference in potential between two parts of a wire, an electric field is created and the electrons respond to that field. In a current-carrying conductor, however, the electrons do not all flow in the same direction. In fact, even when there is no potential difference (and therefore no field), the electrons are moving around randomly. This random motion continues when there is a field, but the field superimposes onto this random motion a small net velocity, the drift velocity. Because electrons are negative charges, the direction of the drift velocity is opposite to the electric field. In a typical case, the drift velocity of electrons is about 1 mm / s. The electric field,on the other hand, propagates much faster than this, more like 108 m / s. Storing Energy in a Capacitor The energy stored on a capacitor can be expressed in terms of the work done by the battery. Voltagerepresents energy per unit charge, so the work to move a charge element dq from the negative plate to the positive plate is equal to V dq, where V is the voltage on the capacitor. The voltage V is proportional to the amount of charge which is already on the capacitor. Element of energy stored: If Q is the amount of charge stored when the whole battery voltage appears across the capacitor, then the stored energy is obtained from the integral: More detail Calculation This energy expression can be put in three equivalent forms by just permutations based on the definition of capacitance C=Q/V. But the battery energy output is QV! Where did half of the energy go? Index Capacitor Concepts   HyperPhysics***** Electricity and Magnetism R Nave Go Back Storing Energy in a Capacitor When the switch is closed to connect the battery to thecapacitor, there is zero voltage across the capacitor since it has no charge buildup. The voltage on the capacitor is proportional to the charge Storing energy on the capacitor involves doing work to transport charge from one plate of the capacitor to the other against the electrical forces. As the charge builds up in the charging process, each successive element of charge dq requires more work to force it onto the positive plate. Summing these continuously changing quantities requires an integral. Calculation More detail on integral Note that the total energy stored QV/2 is exactly half of the energy QV which is supplied by the battery, independent of R! Index Capacitor Concepts   HyperPhysics***** Electricity and Magnetism R Nave Go Back Capacitor Energy Integral Transporting differentialcharge dq to the plate of thecapacitor requires work But as the voltage rises toward the battery voltage in the process of storing energy, each successive dq requires more work. Summing all these amounts of work until the total charge is reached is an infinite sum, the type of task anintegral is essential for. The form of the integral shown above is a polynomial integral and is a good example of the power of integration. Charging a capacitor Calculation Index Capacitor Concepts   HyperPhysics***** Electricity and Magnetism R Nave Go Back Where did half of the capacitor charging energy go? The problem of the "energy stored on a capacitor" is a classic one because it has some counterintuitive elements. To be sure, the battery puts out energy QVb in the process of charging the capacitor to equilibrium at battery voltage Vb. But half of that energy is dissipated in heat in the resistance of the charging pathway, and only QVb/2 is finally stored on the capacitor at equilibrium. The counter-intuitive part starts when you say "That's too much loss to tolerate. I'm just going to lower the resistance of the charging pathway so I will get more energy on the capacitor." This doesn't work, because the energy loss rate in the resistance I2R increases dramatically, even though you do charge the capacitor more rapidly. It's not at all intuitive in this exponential charging process that you will still lose half the energy into heat, so this classic problem becomes an excellent example of the value of calculus and the integral as an engineering tool. Part of the intuitive part that goes into setting up the integral is that getting the first element of charge dq onto the capacitor plates takes much less work because most of the battery voltage is dropping across the resistance R and only a tiny energy dU = dqV is stored on the capacitor. Proceeding with the integral, which takes a quadratic form in q, gives a summed energy on the capacitor Q2/2C = CVb2/2 = QVb/2 where the Vb here is the battery voltage. So the bottom line is that you have to put out 2 joules from the battery to put 1 joule on the capacitor, the other joule having been irretrievably lost to heat - the 2nd Law of Thermodynamics bites you again, regardless of your charging rate. The non-intuitive nature of this problem is the reason that the integral approach is valuable. Transporting differentialcharge dq to the plate of thecapacitor requires work Though it will not be shown here, if you proceed further with this problem by making the charging resistance so small that the initial charging current is extremely high, a sizable fraction of the charging energy is actually radiated away as electromagnetic energy. This crosses the threshold into antenna theory because not all the loss in charging was thermodynamic - but still the loss in the process was half the energy supplied by the battery in charging the capacitor. So the energy supplied by the battery is E = CVb2, but only half that is on the capacitor - the other half has been lost to heat, or in the extremely low charging resistance case, to heat and electromagnetic energy. Charging a capacitor Calculation Index Capacitor Concepts   HyperPhysics***** Electricity and Magnetism R Nave Go Back