DC CIRCUITS: CHAPTER 4

Capacitors and Inductors Introduction Capacitors: terminal behavior in terms of current, voltage, power and energy Series and parallel capacitors Inductors: terminal behavior in terms of current, voltage, power and energy Series and parallel inductors

Introduction Two more linear, ideal basic passive circuit elements. Energy storage elements stored in both magnetic and electric fields. They found continual applications in more practical circuits such as filters, integrators, differentiators, circuit breakers and automobile ignition circuit. Circuit analysis techniques and theorems applied to purely resistive circuits are equally applicable to circuits with inductors and capacitors.

Capacitors Electrical component that consists of two conductors separated by an insulator or dielectric material. Its behavior based on phenomenon associated with electric fields, which the source is voltage. A time-varying electric fields produce a current flow in the space occupied by the fields. Capacitance is the circuit parameter which relates the displacement current to the voltage.

A capacitor with an applied voltage Plates – aluminum foil Dielectric – air/ceramic/paper/mica

Circuit symbols for capacitors (a) Fixed capacitor (b) Variable capacitor

Circuit parameters The amount of charge stored, q = CV. C is capacitance in Farad, ratio of the charge on one plate to the voltage difference between the plates. But it does not depend on q or V but capacitor’s physical dimensions i.e., (1)  = permeability of dielectric in Wb/Am A = surface area of plates in m2 d = distance btw the plates m

Current – voltage relationship of a capacitor To obtain the I-V characteristic of a capacitor, we differentiate both sides of eq.(1) . We obtain, Integrating both sides of eq.(2) we obtain, (2) (3)

Instantaneous power and energy for capacitors The instantaneous power delivered to a capacitor is, The energy stored in the capacitor, At V(-∞) = 0 (cap. uncharged at t = -∞, hence (4) (5) (6) or

Important properties of a capacitor A capacitor is an open circuit to dc. When the voltage across capacitor is not changing with time (constant), current thru it is zero. The voltage on a capacitor cannot change abruptly. - The voltage across capacitor must be continuous. Conversely, the current thru it can change instantaneously.

Practice problem 6.1 What is the voltage across a 3-F capacitor if the charge on one plate is 0.12 mC? How much energy is stored? (Ans: 40V, 2.4mJ)

Practice problem 6.2 If a 10-F capacitor is connected to a voltage source with v(t) = 50sin2000t V Calculate the current through it. (Ans: cos2000t A)

Practice problem 6.3 The current through a 100-F capacitor is i(t) = 50sin120t mA. Calculate the voltage across it at t = 1 ms and t = 5 ms Take v(0) = 0. (Ans: 93.137V, 1.736V)

Practice problem 6.4 An initially uncharged 1-mF capacitor has the current shown in Figure 6.11 across it. Calculate the voltage across it at t = 2 ms and t = 5 ms. (Ans: 100mV, 400mV)

Practice problem 6.5 Under dc conditions, find the energy stored in the capacitors in Fig. 6.13. (Ans: 405J, 90 J)

Series/parallel capacitances Series-parallel combination is powerful tool for circuit simplification. A group of capacitors can be combined to become a single equivalent capacitance using series-parallel rules.

Parallel capacitances The equivalent capacitance of N parallel-connected capacitors is the sum of the individual capacitances. (7) Parallel N-connected capacitors Equivalent circuit

Series capacitances The equivalent capacitance of series-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances. (8) Series N-connected capacitors Equivalent circuit

Practice problem 6.6 Find the equivalent capacitance seen at the terminals of the circuit in Fig. 6.17. (Ans: 40F)

Practice problem 6.7 Find the voltage across each of the capacitors in Fig. 6.20 (Ans: 30V, 30V, 10V, 20V)

Inductors Electrical component that opposes any change in electrical current. Composed of a coil or wire wound around a non-magnetic core/magnetic core. Its behavior based on phenomenon associated with magnetic fields, which the source is current. A time-varying magnetic fields induce voltage in any conductor linked by the fields. Inductance is the circuit parameter which relates the induced voltage to the current.

Typical form of an inductor

Circuit symbols for inductors Variable iron-core Air-core iron-core

Current – voltage relationship of an inductor The voltage across an inductor, L is the constant proportionality called inductance measured in Henry. To obtain current integrate eq. (7), (9) (10)

Instantaneous power and energy fir inductors The instantaneous power delivered to a capacitor is, The energy stored in the capacitor, At V(-∞) = 0 (ind. uncharged at t = -∞, hence (11) (12) (13)

Important properties of an inductor An inductor acts like a short circuit to dc. When the current thru inductor is not changing with time (constant), voltage across it is zero. The current thru an inductor cannot change instantaneously. - An important property is its opposition to the change in current flowing thru it. However the voltage across it can change abruptly.

Practice problem 6.8 If the current through a 1-mH inductor is i(t) = 20cos100t mA, find the terminal voltage and the energy stored. (Ans: -2sin100t mV, 0.2cos2100t J)

Practice problem 6.9 The terminal voltage of a 2-H inductor is V = 10(1 – t) V. find the current flowing thru it at t=4s and the energy stored in it within 0 < t < 4s. Assume i(0)=2A. (Ans: -18 A, 320 J)

Practice problem 6.10 Determine VC, iL and the energy stored in the capacitor and inductor in the circuit below under dc conditions. (Ans: 3V, 3A, 1.125J)

Series inductances The equivalent inductance of N series-connected inductors is the sum of the individual inductances. (14) Series N-connected inductors Equivalent circuit

Parallel inductances The equivalent inductance of series-connected inductors is the reciprocal of the sum of the reciprocals of the individual inductances. (15) Parallel N-connected inductors Equivalent circuit

Practice problem 6.11 Calculate the equivalent inductance for the inductive ladder network in Figure below. (Ans: 25 mH)

Practice problem 6.12 In the circuit of Figure below, given i1(t)=0.6e-2t. If i(0)=1.4A, find (a)i2(0); (b) i2(t) and i(t); (c) V1(t), V2(t) and V (t).