ECE 201 Circuit Theory I 1

Response of First-Order Circuits

RL Circuits

RC Circuits

ECE 201 Circuit Theory I 2

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 3

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 4

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 5

General Configurations for RL

• If the independent sources are equal to zero, the circuits simplify to

ECE 201 Circuit Theory I 6

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 7

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 8

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 9

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 10

The expression for the current

0di

L Ridt

ECE 201 Circuit Theory I 11

0di

L Ridt

A first-order ordinary differential equation with constant coefficients.

How do we solve it?

di Rdt idt

dt L

ECE 201 Circuit Theory I 12

0 0

( )

( )

0

0

( )

( )ln ( )

( )

( )ln

(0)

( ) (0)

i t t

i t t

Rt

L

di Rdt idt

dt Ldi R

dti Ldx R

dyx Li t R

t ti t L

i t Rt

i L

i t i e

ECE 201 Circuit Theory I 13

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,

0(0 ) (0 )i i I

ECE 201 Circuit Theory I 14

The Complete Solution

0( ) , 0

Rt

Li t I e t

ECE 201 Circuit Theory I 15

The voltage drop across the resistor

0

0

, 0 .

(0 ) 0

(0 )

Rt

L

v iR

v I Re t

v

v I R

ECE 201 Circuit Theory I 16

The Power Dissipated in the Resistor

2

2

22

0, 0

Rt

L

vp vi i R

R

p I Re t

ECE 201 Circuit Theory I 17

The Energy Delivered to the Resistor

22

00 0

22

0

2

0

1(1 ), 0.

2

1,

2

Rt t xL

Rt

L

w pdx I Re dx

w I R e tRL

t w LI

ECE 201 Circuit Theory I 18

Time Constant

• The rate at which the current or voltage approaches zero.

LR

ECE 201 Circuit Theory I 19

Rewriting in terms of Time Constant

0

0

22

0

22

0

( )

( )

1(1 )

2

t

t

t

t

i t I e

v t I Re

p I Re

w LI e

ECE 201 Circuit Theory I 20

Table 7.1 page 233 of the text

ECE 201 Circuit Theory I 21

Graphical Interpretation of Time Constant

• Determine the time constant from the plot of the circuit’s natural response.

0

0

0

0

0

( )

1

(0 )

( )

t

t

i t I e

diI e

dtdi Idt

Ii t I t

Straight Line Approximation

ECE 201 Circuit Theory I 22

Graphical Interpretation

Tangent at t = 0 intersects the time axis at the time constant

ECE 201 Circuit Theory I 23

Procedure to Determine the Natural Response of an RL Circuit

• Find the initial current through the inductor.

• Find the time constant,τ, of the circuit (L/R).

• Generate i(t) from I0 and τ using

0( )

t

i t I e