# Response of First-Order Circuits - UMass D

Response of First-Order Circuits RL Circuits RC Circuits ECE 201 Circuit Theory I 1 The Natural Response of a Circuit The currents and voltages that arise when energy stored in an inductor or capacitor is suddenly released into a resistive circuit. These signals are determined by the circuit itself, not by external sources! ECE 201 Circuit Theory I 2

Step Response The sudden application of a DC voltage or current source is referred to as a step. The step response consists of the voltages and currents that arise when energy is being absorbed by an inductor or capacitor. ECE 201 Circuit Theory I 3 Circuits for Natural Response Energy is stored in an inductor (a) as an initial current. Energy is stored in a capacitor (b) as an initial voltage. ECE 201 Circuit Theory I

4 General Configurations for RL If the independent sources are equal to zero, the circuits simplify to ECE 201 Circuit Theory I 5 Natural Response of an RL Circuit Consider the circuit shown. Assume that the switch has been closed for a long time, and is opened at t=0. ECE 201 Circuit Theory I

6 What does for a long time Mean? All of the currents and voltages have reached a constant (dc) value. What is the voltage across the inductor just before the switch is opened? ECE 201 Circuit Theory I 7 Just before t = 0 The voltage across the inductor is equal to zero. There is no current in either resistor. The current in the inductor is equal to IS.

ECE 201 Circuit Theory I 8 Just after t = 0 The current source and its parallel resistor R0 are disconnected from the rest of the circuit, and the inductor begins to release energy. ECE 201 Circuit Theory I 9 The expression for the current di L Ri 0 dt

ECE 201 Circuit Theory I 10 di L Ri 0 dt A first-order ordinary differential equation with constant coefficients. How do we solve it? di R dt idt dt L ECE 201 Circuit Theory I 11

di R dt idt dt L di R dt i L dx R dy x L i (t) R

ln (t t ) i (t ) L i (t) R ln t i (0) L i (t ) t i ( t0 ) t0 0

0 i (t) i (0)e ( R )t L ECE 201 Circuit Theory I 12 The current in an inductor cannot change instantaneously Let the time just before switching be called t(0-).

The time just after switching will be called t(0+). For the inductor, i (0 ) i (0) I ECE 201 Circuit Theory I 0 13 The Complete Solution R t L i (t) I e ,t 0 0

ECE 201 Circuit Theory I 14 The voltage drop across the resistor v iR R t L v I Re ,t 0 . 0 v(0 ) 0 v(0) I R 0 ECE 201 Circuit Theory I

15 The Power Dissipated in the Resistor v p vi i R R 2 2 R 2 t L p I Re 2

0 ,t 0 ECE 201 Circuit Theory I 16 The Energy Delivered to the Resistor t t 0 0 R

2 x L w pdx I Re dx 2 0 1 w I R(1 e R 2 L 1 t , w LI 2 R

2 t L 2 0 ),t 0. 2 0 ECE 201 Circuit Theory I 17

Time Constant The rate at which the current or voltage approaches zero. L R ECE 201 Circuit Theory I 18 Rewriting in terms of Time Constant i (t) I e t

0 v(t) I Re t 0 t 2 p I Re 2

0 1 w LI (1 e ) 2 2 t 2 0 ECE 201 Circuit Theory I 19 Table 7.1 page 217 of the text

ECE 201 Circuit Theory I 20 Graphical Interpretation of Time Constant Determine the time constant from the plot of the circuits natural response. i (t) I e t 0 di 1

I e dt di I (0) dt I Straight Line Approximation i (t) I t ECE 201 Circuit Theory I t 0

0 0 0 21 Graphical Interpretation Tangent at t = 0 intersects the time axis at the time constant ECE 201 Circuit Theory I 22 Procedure to Determine the Natural Response of an RL Circuit Find the initial current through the

inductor. Find the time constant,, of the circuit (L/R). Generate i(t) from I0 and using i (t) I e t 0 ECE 201 Circuit Theory I 23

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