1. 1 ECE 3336 Introduction to Circuits & Electronics Lecture Set #7 Inductors and Capacitors Fall 2012, TUE&TH 5:30-7:00pm Dr. Wanda Wosik Notes developed by Dr. Dave Shattuck

2. 2 Overview of this Part Inductors and Capacitors In this part, we will cover the following topics: Defining equations for inductors and capacitorsDefining equations for inductors and capacitors Power and energy storage in inductors and capacitorsPower and energy storage in inductors and capacitors Parallel and series combinations Basic Rules for inductors and capacitors

3. 3 Basic Elements, Review We are now going to pick up the remaining basic circuit elements that we will be covering in these modules.

4. 4 Circuit Elements In circuits, we think about basic circuit elements that are the basic “building blocks” of our circuits. This is similar to what we do in Chemistry with chemical elements like oxygen or nitrogen. A circuit element cannot be broken down or subdivided into other circuit elements. A circuit element can be defined in terms of the behavior of the voltage and current at its terminals.

5. 5 The 5 Basic Circuit Elements There are 5 basic circuit elements: 1.Voltage sources 2.Current sources 3.Resistors 4.Inductors 5.Capacitors We defined the first three elements in a previous module. We will now introduce inductors or capacitors.

6. 6 Inductors An inductor is a two terminal circuit element that has a voltage across its terminals which is proportional to the derivative of the current through its terminals. The coefficient of this proportionality (INDUCTANCE, L) is the defining characteristic of an inductor. An inductor is the device that we use to model the effect of magnetic fields on circuit variables. The energy stored In magnetic fields has effects on voltage and current. We use the inductor component to model these effects. In many cases a coil of wire can be modeled as an inductor.

7. A Current Can Generate a Magnetic Field Hans Christian Oersted (1777-1851)

8. 8 http://hyperphysics.phy- astr.gsu.edu/hbase/forces/funfor.html#c3http://hyperphysics.phy- astr.gsu.edu/hbase/forces/funfor.html#c3 http://hyperphysics.phy- astr.gsu.edu/hbase/magnetic/magfie.html#c1http://hyperphysics.phy- astr.gsu.edu/hbase/magnetic/magfie.html#c1 http://hyperphysics.phy- astr.gsu.edu/hbase/magnetic/amplaw.html#c1http://hyperphysics.phy- astr.gsu.edu/hbase/magnetic/amplaw.html#c1 http://hyperphysics.phy- astr.gsu.edu/hbase/electric/elefie.html#c1http://hyperphysics.phy- astr.gsu.edu/hbase/electric/elefie.html#c1

9. Faraday’s Law of Induction Faraday’s law: where Magnetic flux (unit: Weber) It has to be varying with time. The faster the change of flux is, the larger is the current. Electromagnetic induction gives us current in a conductor moving in the magnetic field Electromotive force A decade after Oersted’s discovery:

10. 10 Inductors There is an inductance whenever we have magnetic fields produced, and there are magnetic fields whenever current flows. ac current produces a voltage which counteracts the changes of this current Direction of B field and I current – right hand rule

11. 11 An inductor obeys the expression where v L is the voltage across the inductor, and i L is the current through the inductor, and L X is called the inductance. In addition, it works both ways. If something obeys this expression, we can think of it, and model it, as an inductor. The unit ([Henry] or [H]) is named for Joseph Henry, and is equal to a [Volt- second/Ampere]. [H]=[V][s]/[A] Inductors – Definition and Units There is an inductance whenever we have magnetic fields produced, and there are magnetic fields whenever current flows. However, this inductance is often negligible except in coils.

12. 12 Schematic Symbol for Inductors The schematic symbol that we use for inductors is shown here. The schematic symbol can be labeled either with a variable, like L X, or a value, with some number, and units. An example might be 390[mH].

13. 13 Inductor Polarities Previously, we have emphasized the important of reference polarities of current sources and voltages sources. There is no corresponding polarity to an inductor. As in Resistors. And as for resistor, direction matters with respect to the voltage and current in the passive sign convention.

14. The direction of currents induced by magnetic field Heinrich Friedrich Emil Lenz 1804 –1865 The induced current creates a magnetic field opposing the change of the inducing magnetic field The rule of resistance to change

15. 15

16. 16 Passive and Active Sign Convention for Inductors The sign of the equation that we use for inductors depends on whether we have used the passive sign convention or the active sign convention. Passive Sign ConventionActive Sign Convention

17. 17 Defining Equation, Integral Form, Derivation The defining equation for the inductor, can be rewritten in another way. If we want to express the current in terms of the voltage, we can integrate both sides. We get We pick t 0 and t for limits of the integral, where t is time, and t 0 is an arbitrary time value, often zero. The inductance, L X, is constant, and can be taken out of the integral. To avoid confusion, we introduce the dummy variable s in the integral. We get We finish the derivation in the next slide.

18. 18 We can take this equation and perform the integral on the right hand side. When we do this we get Thus, we can solve for i L (t), and we have two defining equations for the inductor, Defining Equations for Inductors and Remember that both of these are defined for the passive sign convention for i L and v L. If not, then we need a negative sign in these equations. Initial conditions

19. 19 Voltage v L ≠0 only if i L =f(t) The implications of these equations are significant. For example, if the current is not changing, then the voltage will be zero. This current could be a constant value, and large, and an inductor will have no voltage across it. This is counter-intuitive for many students. That is because they are thinking of actual coils, which have some finite resistance in their wires. For us, an ideal inductor has no resistance; it simply obeys the laws below. We might model a coil with both inductors and resistors, but for now, all we need to note is what happens with these ideal elements. and

20. 20 Current Change is Limited so Another implication is that we cannot change the current through an inductor instantaneously. If we were to make such a change, the derivative of current with respect to time would be infinity, and the voltage would have to be infinite. Since it is not possible to have an infinite voltage, it must be impossible to change the current through an inductor instantaneously. and But large voltages can be produced

21. 21 Dummy Variable s Replaces Variable t in the Integral Limits It is not really necessary to introduce a dummy variable. It really doesn ’ t matter what variable is integrated over, because when the limits are inserted, that variable goes away. and The independent variable t is in the limits of the integral. This is indicated by the i L (t) on the left-hand side of the equation. Remember, the integral here is not a function of s. It is a function of t. This is a constant.

22. Inductance for a Coil The role of core

23. 23 Energy in Inductors, Derivation We can find the energy stored in the magnetic field associated with the inductor. First, we note that the power is voltage times current, as it has always been. Now, we can multiply each side by dt, and integrate both sides to get We need limits. We know that when the current is zero, there is no magnetic field, and therefore there can be no energy in the magnetic field. That allowed us to use 0 for the lower limits. The upper limits came since we will have the energy stored, w L, for a given value of current, i L.

24. 24 Energy in Inductors, Formula We had the integral for the energy, Now, we perform the integration. Note that L X is a constant, independent of the current through the inductor, so we can take it out of the integral. We have We simplify this, and get the formula for energy stored in the inductor,

25. 25 Series Inductors Equivalent Circuits Two series inductors, L 1 and L 2, can be replaced with an equivalent circuit with a single inductor L EQ, as long as i L1 (t)=i L2 (t)=i LEQ (t) (∑L i is as for resistors) Obtained from KVL v LEQ (t)=V L1 (t)+V L2 (t)

26. 26 More than 2 Series Inductors This rule can be extended to more than two series inductors. In this case, for N series inductors, we have (∑L i reminds of ∑R i for resistors)

27. 27 Series Inductors Equivalent Circuits: A Reminder Two series inductors, L 1 and L 2, can be replaced with an equivalent circuit with a single inductor L EQ, as long as Remember that these two equivalent circuits are equivalent only with respect to the circuit connected to them. (In yellow here.)

28. 28 Series Inductors Equivalent Circuits: Initial Conditions Two series inductors, L 1 and L 2, can be replaced with an equivalent circuit with a single inductor L EQ, as long as To be equivalent with respect to the “rest of the circuit”, we must have any initial condition be the same as well. That is, i L1 (t 0 ) must equal i LEQ (t 0 ).

29. 29 Parallel Inductors Equivalent Circuits Two parallel inductors, L 1 and L 2, can be replaced with an equivalent circuit with a single inductor L EQ, as long as Obtained from KCL ∑(1/L i ) as for resistors ∑(1/R i ) i LEQ (t)=i L1 (t)+i L2 (t) v L1 (t)=v L2 (t)=v LEQ (t)

30. 30 More than 2 Parallel Inductors This rule can be extended to more than two parallel inductors. In this case, for N parallel inductors, we have The product over sum rule only works for two inductors. Analogy to RESISTORS

31. 31 Parallel Inductors Equivalent Circuits: A Reminder Remember that these two equivalent circuits are equivalent only with respect to the circuit connected to them. Two parallel inductors, L 1 and L 2, can be replaced with an equivalent circuit with a single inductor L EQ, as long as (In yellow here.)

32. 32 Parallel Inductors Equivalent Circuits: Initial Conditions To be equivalent with respect to the “rest of the circuit”, we must have any initial condition be the same as well. That is,

33. 33 Notes 1.We took some mathematical liberties in this derivation. For example, we do not really multiply both sides by dt, but the results that we obtain are correct here. 2.Note that the energy is a function of the current squared, which will be positive. We will assume that our inductance is also positive, and clearly ½ is positive. So, the energy stored in the magnetic field of an inductor will be positive. 3.These three equations are useful, and should be learned or written down. Go back to Overview slide. Overview