1 ECE 3144 Lecture 26 Dr. Rose Q. Hu Electrical and Computer Engineering Department Mississippi State University

2 Chapter 5: Capacitance and Inductance Two more passive circuit elements are introduced in this chapter: capacitors and inductors Unlike resistors, capacitors and inductors are able to absorb energy from the circuit, store it temporarily, and later return it. Unlike resistors, their terminal characteristics are described by linear differential equations. Capacitors are capable of storing energy when a voltage is present across the element. Inductors are capable of storing energy when a current is passing through them. In this chapter, we are particularly interested in the following objectives: –Finding the voltage-current relationship of ideal capacitors –Find the voltage-current relationship of ideal inductors –Calculating the energy stored in inductors and capacitors –Methods for reducing series/parallel combinations of inductors and capacitors –Predicting the behavior of op-amp circuits with capacitors

3 Ideal capacitors A capacitor is a circuit that consists of two conducting surface separated by a non- conducing, or dielectric, material. In the above figure, the positive charge is transferred to one plate and negative charges to the other. The charge on the capacitor is proportional to the voltage across it => q = Cv, where C is the proportionality factor known as the capacitance of the element. Capacitance unit –We now define the farad (F) as one coulomb per volt, and use this as our unit of capacitance. Due to the presence of dielectric, there is no conductance current flowing internally between the two plates of the capacitors. However, the equal currents are entering ands leaving the two terminals of the capacitor. The positive current entering one plate represents positive charge accumulated on the plate; the positive current leaving one plate represents negative charge accumulated on the plate. The current and the increasing charge on one plate are related as the familiar equation:

4 Ideal capacitors To complete KCL analysis in this case, we introduce one more concept: displacement current. –Via electromagnetic field theory, it can be shown that the conductance current is equal to the displacement current that flows internally between the two plates of the capacitors and is present any time that an electric field or voltage varies with time. KCL is therefore satisfied if we include both conductance and displacement currents. Like a resistor, a capacitor constructed of two parallel pales of area A, separated by a distance d, has a capacitance C=  A/d, where  is the permitivity, a constant of the insulating material between the plates.

5 Integral voltage-current relationship for a capacitor We know that q(t)=Cv(t) and i(t) = dq(t)/dt => (1) Equation (1) can be rewritten as (2) Integrate (2) between times t 0 and t Where v(t 0 ) is the voltage due to the charge that accumulates on the capacitor from time –  to t 0. Or we can express in another way

6 Energy Storage for a capacitor The power delivered to a capacitor is The energy stored in the electric field is therefore, v(-  ) = 0 The energy stored can also be written as by using q=Cv

7 Ideal inductors An inductor is a circuit element that consists of a conducing wire usually in the form of coil. We know that a current-carrying conductor produces a magnetic field, which is linearly related to the current that produces it. A changing magnetic field can induce a voltage in the neighboring circuit, which is proportional to the time rate change of the current producing the magnetic field. The above statements indicate: L, the constant of proportionality, is what we call the inductance. It is measured in the unit of henry. 1 henry (H) is equal to 1 volt-second per ampere. Like a capacitor, an inductor has an inductance of  N 2 A/s, where A is the cross-sectional area, s is the axial length of the helix, N is the number of complete turns of wire, and  is the a constant of the material inside the helix, call permeability. If the current is constant, then we have v=0. Thus we may view an inductor as a “short circuit to dc”. Another fact is that a sudden or discontinuous change in the current must be associated with an infinite voltage across the inductor.

8 Integral voltage-current relationships for an inductor We know that => Following development of the mathematical equations for the capacitor, we find Which lead to another equation

9 Energy storage for an inductor The power delivered to an inductor is given the current-voltage product as The energy w L accepted by the inductor is stored in the magnetic field around the coil, and expressed as The energy accepted by the inductor between t 0 and t is

10 Examples Provided during the class

11 Homework for lecture 26 Problems 5.4, 5.7, 5.17,5.18, 5.20, 5.23, 5.31 Due March 25