Electromagnetism Lecture#12 Part 2 MUHAMMAD MATEEN YAQOOB THE UNIVERSITY OF LAHORE SARGODHA CAMPUS

THE SOURCE-FREE RC CIRCUIT MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation. The voltage response of the RC circuit. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

The time constant of a circuit is the time required for the response to decay by a factor of 1/e or 36.8 percent of its initial value. The voltage response of the RC circuit. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

capacitor is fully discharged (or charged) after five time constants To find τ from the response curve, draw the tangent to the curve. The tangent intercepts with the time axis at t = τ. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

Plot of v/V 0 = e −t/τ for various values of the time constant MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

The Key to Working with a Source - free RC Circuit is Findin g : 1. The initial voltage v(0) = V 0 across the capacitor. 2. The time constant τ. v C (t) = v(t) = v(0)e −t/τ other variables Capacitor current i C Resistor voltage v R Resistor current i R can be determined. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

THE SOURCE FREE RL CIRCUIT MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

The smaller the time constant τ of a circuit, the faster the rate of decay of the response. The larger the time constant, the slower the rate of decay of the response. At any rate, the response decays to less than 1 percent of its initial value (i.e., reaches steady state) after 5τ. The current response of the RL circuit MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

The Key to Working with a Source - free RL Circuit is to Find : 1. The initial current i(0) = I 0 through the inductor. 2. The time constant τ of the circuit. i L (t) =i(t) = i(0)e −t/τ. other variables Inductor voltage v L Resistor voltage v R Resistor current i R can be obtained. MATEEN YAQOOB DEPARTMENT OF COMPUTER SCIENCE

SECOND-ORDER CIRCUITS A second-order circuit is characterized by a second-order differential equation. It consists of resistors and the equivalent of two energy storage elements.

Circuit Excitation Two ways of excitation 1. By initial conditions of the storage elements (These source free circuits will give natural responses as expected) 2.By step inputs: Circuits are excited by independentsources. These circuits will give both the natural response and the forced response

Finding Initial and Final Values We begin by learning how to obtain the initial conditions for the circuit variables and their derivatives, as this is crucial to analyze second order circuits. Perhaps the major problem students face in handling second-order circuits is finding the initial and final conditions on circuit variables. Students are usually comfortable getting the initial and final values of v and i but often have difficulty finding the initial values of their derivatives: dv/dt and di/dt.

There are two key points to keep in mind in determining the initial conditions. First—as always in circuit analysis—we must carefully handle the polarity of voltage v(t) across the capacitor and the direction of the current i(t) through the inductor. Keep in mind that v and i are defined strictly according to the passive sign convention. One should carefully observe how these are defined and apply them accordingly.

Second, keep in mind that the capacitor voltage is always continuous so that v(0 + ) = v(0 − ) (a) and the inductor current is always continuous so that i(0 + ) = i(0 − ) (b) where t = 0 − denotes the time just before a switching event and t = 0 + is the time just after the switching event, assuming that the switching event takes place at t = 0. Therefore, in finding initial conditions, we first focus on those variables that cannot change abruptly, capacitor voltage and inductor current, by applying Eq. (a & b).