Module2-Transient Response of first
order, second order circuits and
Resonance
Module 2
Time response in inductance (L) and capacitance (C),
steady state response of circuits with RLC components.
Response (forced & natural) of first order circuits (RL &
RC): series, parallel, source free, complex circuits with
more than one resistance, power sources and switches.
Response of second order circuit (RLC): series, parallel
and complex circuits. Series and parallel resonance
condition.
Transient & Steady State Response
Transient Response
1. Switch Operation
Electrical circuits are connected to supply by closing
the switch and disconnected from the supply by
opening the switch. This switching operation will
change the current and voltage in the device.
2. Energy Storage Elements
An inductive device will not allow sudden change in
current.
A capacitive device will not allow sudden change in
voltage
Transient & Steady State Response
1V
I
I
1Ω
1Ω
2V
I = 1A
I = 2A
Voltage source change from 1V to 2V immediately
Does the current change immediately too?
Transient & Steady State Response
Voltage
2V
1V
time
Current
2A
1A
time
Transient & Steady State Response
1V
I
I
1Ω
1Ω
I = 1A
L
2V
I = 2A
Voltage source change from 1V to 2V immediately
Does the current change immediately too?
L
Voltage
2V
1V
time
Current
2A
1A
time
Transient + Steady State Response
R,L,C Characteristics
Element
V/I Relation
DC Steady-State
Resistor
vR (t ) = R iR (t )
V=IR
Capacitor
d vC (t )
iC (t ) = C
dt
I = 0; open
Inductor
d iL (t )
vL (t ) = L
dt
V = 0; short
First Order Circuits
◼
◼
Any circuit with a single energy storage
element, an arbitrary number of sources, and
an arbitrary number of resistors is a circuit of
order 1
Any voltage or current in such a circuit is the
solution to a 1st order differential equation
First-Order Circuits
◼
◼
◼
A circuit that contains only sources, resistors
and an inductor is called an RL circuit.
A circuit that contains only sources, resistors
and a capacitor is called an RC circuit.
RL and RC circuits are called first-order
circuits because their voltages and currents
are described by first-order differential
equations.
Second Order Circuits
◼
◼
Any circuit with a single capacitor, a single
inductor, an arbitrary number of sources, and
an arbitrary number of resistors is a circuit of
order 2
Any voltage or current in such a circuit is the
solution to a 2nd order differential equation
Transient Response
• RL Circuit
First-order differential equation
• RC Circuit
• RLC Circuit
Second-order differential equation
Forced & Source Free Response
Source Free Response of an RL circuit
Consider the following circuit, for which the switch
is closed for t<0, and then opened at t = 0
t=0
Is
Ro
i
L
+
R
V
–
The dc voltage V, has been supplying the RL
circuit with constant current for a long time
18
Source Free Response of an RL circuit
Consider the series connection of a
resistor and an inductor, as shown
in Fig.
Our goal is to determine the current
through the inductor.
idea - inductor current cannot
change instantaneously.
At we assume that the inductor has
an initial current or I0 = i (t)
Source Free Response of an RL circuit
Source Free Response of an RL circuit
Source Free Response of an RL circuit
Time constant, τ determines the rate at which the current or
voltage approaches zero.
Time constant for RL circuit is
in seconds.
Source Free Response of an RL circuit
Current through the Inductor is:
Voltage across the resistor R is:
v(t ) = i(t ) R = I 0 Re − ( R / L )t
Voltage across the Inductor is:
Power dissipated in the resistor is:
p = vRi(t ) = I Re
2
0
−2 ( R / L ) t
Energy absorbed by the resistor or stored by the inductor
1 2
wL (t ) =  pdt =  Lidi = Li (t )
2
Time constant is
in seconds
The 12-V battery in Fig. is disconnected
at t = 0. Find the inductor current and
voltage v for all times.
The 12-V battery in Fig. is disconnected
at t = 0. Find the inductor current and
voltage v for all times.