COME 212

Network Analysis

Dr. Hiba S. Abdallah

Assistant Professor of Communications and
Electronics

Electrical and Computer Engineering

Department
Beirut Arab University
 First Order And Second Order
Response Of RL And RC Circuit
 First-Order and Second-Order
Response of RL and RC Circuit

• Natural response of RL and RC Circuit

• Step Response of RL and RC Circuit
• General solutions for natural and step response
• Introduction to the natural and step response of
RLC circuit
• Natural response of series and parallel RLC circuit
• Step response of series and parallel RLC circuit
 Natural response of RL and RC
Circuit
• RL- resistor-inductor
• RC-resistor-capacitor
• First-order circuit: RL or RC circuit
because their voltages and currents
are described by first-order
differential equation.
• Natural response: refers to the
behavior (in terms of voltages and
currents) of the circuit, with no
external sources of excitation.
Natural response of RC circuit
Consider the conditions below:
1. At t < 0, switch is in a closed position
for along time.
2. At t=0, the instant when the switch is
opened
3. At t > 0, switch is not close for along
time
• For t ≤ 0, v(t) = V0.
For t ≥ 0: du 1
 dv
u RC
ic  iR  0
v (t ) 1 1 t

C
dv(t ) v(t )
 0 V0 u du   RC  0
dv
dt R
1
dv(t ) v(t )
 0 ln v(t )  ln V0   (t  0)
dt RC RC
dv(t )

v(t )  v(t )  t voltan
dt RC ln    
dv(t ) 1  V0  RC
 dt  t RC
v(t ) RC v(t )  V0 e
• Thus for t > 0,

 t RC
v(t )  V0 e
v(t ) V0 t
ic (t )    e
RC

R R
W (t )  C v(t )   C V0 e
1 2 1 2
 2t
RC

2 2
The graph of the natural response
of RC circuit

v(t )  V0 t 0
 t RC
 V0 e t 0
• The time constant, τ = RC and thus,


v(t )  V0 e
t

• The time constant, τ determine how
fast the voltage reach the steady state:
Natural response of RL circuit
Consider the conditions below:
1. At t < 0, switch is in a closed position
for along time.
2. At t=0, the instant when the switch is
opened
3. At t > 0, switch is not close for along
time
• For t ≤ 0, i(t) = I0
For t > 0,
v(t )  R i (t )  0 i (t ) 1 R t
L
di (t )
 R i (t )  0
i (0) u
du    dv
L 0

dt R
di (t ) ln i (t )  ln i (0)   (t  0)
L   R i (t ) L
dt
di (t ) R  i (t )  R Current
  dt ln    t
i (t ) L  i (0)  L
du R t R L
  dv i (t )  i (0) e
u L
 • Thus for t > 0,

t R L
i(t )  I 0 e w(t )  Li (t ) 
1 2

2
v(t )  i (t ) R 1 2  2t R L
t R L  LI 0 e
  RI 0 e 2
 Ex.
The switch in the circuit has been closed
for along time before is opened at t=0.
Find
a) IL (t) for t ≥ 0
b) I0 (t) for t ≥ 0+
c) V0 (t) for t ≥ 0+
d) The percentage of the total energy
stored in the 2H inductor that is
dissipated in the 10Ω resistor.
 Ans.
a) The switch has been closed for along
time prior to t=0, so voltage across the
inductor must be zero at t = 0-.
Therefore the initial current in the
inductor is 20A at t = 0-. Hence iL (0+)
also is 20A, because an instantaneous
change in the current cannot occur in
an inductor.
• The equivalent resistance and time
constant:

Req  2  40 10  10

L 2
   0.2 saat
Req 10
• The expression of inductor current, iL(t)
as,

  t
i L (t )  i(0 ) e
5t
 20 e A t0
b) The current in the 40Ω resistor
can be determine using current
division,

 10 
i0  i L  
 10  40 
• Note that this expression is valid for
t ≥ 0+ because i0 = 0 at t = 0-.
• The inductor behaves as a short
circuit prior to the switch being
opened, producing an instantaneous
change in the current i0. Then,

5t 
i0 (t )  4e A t0
c) The voltage V0 directly obtain
using Ohm’s law

V0 (t )  40i0
5t 
 160e V t0
d) The power dissipated in the 10Ω
resistor is
2
V0
p10 (t ) 
10
10t 
 2560 e W t0
• Thetotal energy dissipated in the
10Ω resistor is

W10 (t )   2560e 10t
dt
0

 256 J
• Theinitial energy stored in the 2H
inductor is

1 2
W ( 0)  L i ( 0)
2
 2 400   400 J
1
2
• Therefore the percentage of
energy dissipated in the 10Ω
resistor is,

256
100  64%
400
 First-Order and Second-Order
Response of RL and RC Circuit

• Natural response of RL and RC Circuit

• Step Response of RL and RC Circuit
• General solutions for natural and step response
• Introduction to the natural and step response of
RLC circuit
• Natural response of series and parallel RLC circuit
• Step response of series and parallel RLC circuit
 Step response of RC circuit
• The step response of a circuit is its behavior
when the excitation is the step function, which
maybe a voltage or a current source.
Consider the conditions below:
1. At t < 0, switch is in a closed and
opened position for along time.
2. At t=0, the instant when the switch is
opened and closed
3. At t > 0, switch is not close and
opened for along time
 • For t ≤ 0, v(t)=V0
For t > 0,
1 du
 dv 
Vs  v(t )  Ri (t ) RC u  Vs
dv(t )  t  ln v(t )  V   ln V  V 
Vs  v(t )  RC RC
s 0 s
dt
t  v(t )  Vs  volt
1
dt 
dv(t )   ln  
RC Vs  v(t ) RC  V 0  V s 

1 dv(t ) v(t )  Vs  V0  Vs e  t RC

 dt 
RC v(t )  Vs  Vs  V0  Vs e  t
• Thus for t >0

V  Vs  V0  Vs e  t

 V f  Vn
Where V f V s
Vn  V0  Vs e  t
• Vf = force voltage or also known
as steady state response
• Vn = transient voltage is the
circuit’s temporary response that
will die out with time
Step Response: RC circuit
force

total

Natural
• The current for step response of RC circuit
dv
i (t )  C
dt
 1  t 
 C   (V0  Vs )e 
  
  V0  Vs e
1  t

R
 Vs V0   t   t
   e 
i (t )  i (0 )e
R R
Step response of RL circuit
Consider the conditions below:
1. At t < 0, switch is in a opened
position for along time.
2. At t=0, the instant when the switch is
closed
3. At t > 0, switch is not open for along
time
 •i(t)=I0 for t ≤ 0.
 For t > 0,
R du
Vs  Ri (t )  v(t )  dv 
L u  Vs R
di (t )
Vs  Ri (t )  L R t i (t ) du
dt   dv  
L 0 I 0 u  Vs
R
Vs L di (t )
 i (t )   t  ln i (t )  Vs R   ln I 0  Vs R 
R
R R dt L
R di (t )  i (t )  Vs R 
dt  V R
 t  ln   Current
L R  i (t )  I0  R 
s Vs
L

 I 0  e
R di t R L
 dt  i(t )  Vs Vs
L i  Vs R R R
• Thus,

i(t )  I 0 t 0
 Vs
R  I 0  Vs
R e t R L
t 0
di (t )
v(t )  L t0
dt
 Vs  R I 0 e t R L
t0
 Ex.
The switch in the circuit has been
open for along time. The initial charge
on the capacitor is zero. At t = 0, the
switch is closed. Find the expression for
a) i(t) for t ≥ 0
b) v(t) when t ≥ 0+
 Answer (a)
• Initial voltage on the capacitor is
zero. The current in the 30kΩ
resistor is

 (7.5)( 20)
i (0 )   3mA
50
• The final value of the capacitor
current will be zero because the
capacitor eventually will appear as an
open circuit in terms of dc current.
Thus if = 0.
• The time constant, τ is

6
  (20  30)10 (0.1) 10
3

 5ms
• Thus, the expression of the current
i(t) for t ≥ 0 is
  t
i ( t )  i (0 )e
t
 3e 5103

 200t 
 3e mA t0
 Answer (b)
• Theinitial value of voltage is zero
and the final value is

Vf  (7.5)(20)  150V
• The capacitor vC(t) is
v C ( t )  Vf  V0  Vf e  t

 200t
 150  (0  150)e
 200t
 150  150e V t0

• Thus, the expression of v(t) is

200t 200t
v( t )  150  150e  (30)(3)e
 200t 
 (150  60e )V t  0
 First-Order and Second-Order
Response of RL and RC Circuit

• Natural response of RL and RC Circuit

• Step Response of RL and RC Circuit
• General solutions for natural and step response
• Introduction to the natural and step response of
RLC circuit
• Natural response of series and parallel RLC circuit
• Step response of series and parallel RLC circuit
When computing the step and natural responses of
circuits, it may help to follow these steps:
1. Identify the variable of interest for the circuit. For
RC circuits, it is most convenient to choose the
capacitive voltage, for RL circuits, it is best to
choose the inductive current.
2. Determine the initial value of the variable, which is
its value at t0.
3. Calculate the final value of the variable, which is its
value as t→∞.
4. Calculate the time constant of the circuit, τ.
 First-Order and Second-Order
Response of RL and RC Circuit
• Natural response of RL and RC Circuit
• Step Response of RL and RC Circuit
• General solutions for natural and step response
• Introduction to the natural and step response of
RLC circuit
• Natural response of series and parallel RLC circuit
• Step response of series and parallel RLC circuit
 First-Order and Second-Order
Response of RL and RC Circuit

• Natural response of RL and RC Circuit

• Step Response of RL and RC Circuit
• General solutions for natural and step response
• Sequential switching
• Introduction to the natural and step response of
RLC circuit
• Natural response of series and parallel RLC circuit
• Step response of series and parallel RLC circuit
 Second order response for RLC
• RLC circuit: consists of resistor,
inductor and capacitor
• Second order response : response
from RLC circuit
• Type of RLC circuit:
1. Series RLC
2. Parallel RLC
Natural response of parallel RLC
• Summing all the currents away
from node,
V 1 t dv
  vd  I 0  C 0
R L 0 dt
• Differentiating once with respect
to t,
2
1 dv v d v
 C 2  0
R dt L dt
2
d v 1 dv v
2
   0
dt RC dt LC
• Assume that v  Ae st

As st A st
As e 
2 st
e  e 0
RC LC
 2 s 1 
Ae  s 
st
 0
 RC
LC 
characteristic equation
• Characteristic equation is zero:

 2 s 1 
s   0
 RC LC 
• The two roots:

2
1  1  1
s1      
2 RC  2 RC  LC
2
1  1  1
s2      
2 RC  2 RC  LC
• The
natural response of series
RLC:

v  A1 e  A2 e
s1t s2t
• The two roots:

s1      0
2 2

s2      0
2 2

• where:
1 1
 0 
2 RC LC
• Summary
Parameter Terminology Value in natural
response
Characteristic s1     2  0
s1, s2 equation
2

s2     2  0
2

α Neper frequency

1
2 RC
Resonant radian
0 frequency 1
0 
LC
• The two roots, s1 and s2 depend on
the value of α and ωo.
• 3 possible conditions are:
1. If ωo < α2 , the voltage response is
overdamped
2. If ωo > α2 , the voltage response is
underdamped
3. If ωo = α2 , the voltage response is
critically damped
Overdamped voltage response
• Overdamped voltage expression:

v  A1 e  A2 e
s1t s2t
• The constants of A1 and A2 can be
obtained from these two equations:


v(0 )  A1  A2  (1)

dv(0 )
 s1 A1  s2 A2  (2)
dt
• Whereby, vo+ is obtained from the circuit at
the instant when the switching occurred.
• The value of v(0+) = V0 and the initial
value of dv/dt is

 
dv(0 ) iC (0 )

dt C
The process for finding the overdamped
response, v(t) :
1. Find the roots of the characteristic
equation, s1 dan s2, using the value of
R, L and C.
2. Find v(0+) and dv(0+)/dt using circuit
analysis.
3. The values of A1 and A2 is obtained by
solving the equation below:

v(0 )  A1  A2
 
dv(0 ) iC (0 )
  s1 A1  s2 A2
dt C
4. Substitute the value for s1, s2, A1 dan A2
to determine the expression for v(t) for t
≥ 0.
• Theresponse of overdamped
voltage for v(0) = 1V and i(0) = 0
 Underdamped voltage response

• When ωo2 > α2, the roots of the

characteristic equation are
complex and the response is
underdamped.
• The roots s1 and s2 as,

s1     (0   )
2 2

   j 0  
2 2

   jd
s2    jd
• ωd : damped radian frequency
• Theunderdamped voltage
response of a parallel RLC circuit
is
 t
v( t )  B1 e cos d t
 t
 B2 e sin d t
 • The
constants B1 dan B2 are real
number.
The two simultaneous equation that
determine B1 and B2 are:

v(0 )  V0  B1
 
dv(0 ) iC (0 )
  1B1  d B2
dt C
Response of underdamped voltage
for v(0) = 1V and i(0) = 0
 Critically Damped voltage
response
• A circuitis critically damped when
ωo2 = α2 ( ωo = α). The two roots of
the characteristic equation are real
and equal:
1
s1  s2    
2 RC
• The solution for the voltage is
t t
v(t )  D1t e  D2 e
•The two simultaneous equation that
determine D1 and D2 are,

v(0 )  V0  D2
 
dv(0 ) iC (0 )
  D1  D2
dt C
Response of the critically damped
voltage for v(0) = 1V and i(0) = 0
The step response of a
parallel RLC circuit
• From the KCL,

iL  iR  iC  I
v dv
iL   C I
R dt
 di
• Because
vL
dt
2
dv d iL
L 2
• We get

dt dt
• Thus,

2
L diL d iL
iL   LC 2  I
R dt dt
2
d iL 1 diL iL I
2
  
dt RC dt LC LC
• Thereare two approaches to solve
the equation:
– directapproach
– indirect approach.
 Indirect Approach

• From the KCL:

1 t v dv
L  0
vd 
R
 C
dt
 I
• Differentiate once with respect to
t:
2
v 1 dv d v
 C 2  0
L R dt dt
2
d v 1 dv v
2
   0
dt RC dt LC
• The solution for v depends on the
roots of the characteristic equation:

v  A1 e  A2 e
s1t s2t

 t
v  B1 e cos d t
t
 B2 e sin d t
t t
v  D1t e  D2 e
• Substitute into KCL equation :

 
iL  I  A1 e  A2 es1t s2t

 t
iL  I  B1 e cos d t
 t
 B2 e sin d t
  t   t
iL  I  D1 t e  D2 e
 Direct Approach

• Itis much easier to find the primed

constants directly in terms of the
initial values of the response
function.

     
A1 , A2 , B1 , B2 , D1 , D2
• Theprimed constants could be
determined from:

diL (0)
iL (0) and
dt
• The solution for a second-order
differential equation consists of
the forced response and the
natural response.
 • If If and Vf is the final value of the
response function, the solution for the
step function can be written as:

Function of the same form 

i  If   
as the natural response 
function of the same form 
v  Vf   
as the natural response 
 Natural response of a series RLC

• The procedures for finding the

natural response of a series RLC
circuit is similar as the natural
response of a parallel RLC circuit as
both circuits are described by the
similar form of differential equations.
Series RLC circuit
• Summing the voltage around the
loop,

di 1 t
Ri  L   i d  V0  0
dt C 0
• Differentiate once with respect to
t,
2
di d i i
R L 2  0
dt dt C
2
d i R di i
2
   0
dt L dt LC
• The characteristic equation for the
series RLC circuit is,

R 1
s  s
2
0
L LC
• Theroots of the characteristic
equation are,
2
R  R  1
s1, 2     
2L  2 L  LC
@

s1, 2      0
2 2
• Neper frequency (α) for series
RLC,
R
 rad / s
2L
And the resonant radian frequency,

1
0  rad / s
LC
The current response will be
overdamped, underdamped or
critically damped according to,

0  
2 2
0  
2 2

0   2 2
• Thus the three possible solutions for
the currents are,

i(t )  A1 e  A2 e
s1t s2t

t
i(t )  B1e cos d t
t
 B2e sin d t
 t  t
i(t )  D1t e  D2 e
 Step response of series RLC

• Theprocedures are the same as

the parallel circuit.
Series RLC circuit
• Use KVL,

di
v  Ri  L  vC
dt
• Thecurrent, i is related to the
capacitor voltage (vC ) by
expression,

dv C
iC
dt
• Differentiate once i with respect
to t
2
di d vC
C 2
dt dt
• Substitute into KVL equation,

2
d vC R dvC vC V
2
  
dt L dt LC LC
• Three possible solutions for vC are,
 
vC  V f  A1 e  A2 e s1t s2t

 t
vC  V f  B1 e cos d t
 t
 B2 e sin d t
  t   t
vC  V f  D1 t e  D2 e
 Example 1
(Step response of parallel RLC)
The initial energy stored in the circuit is zero.
At t = 0, a DC current source of 24mA is
applied to the circuit. The value of the resistor
is 400Ω.
1. What is the initial value of iL?
2. What is the initial value of diL/dt?
3. What is the roots of the characteristic
equation?
4. What is the numerical expression for iL(t)
when t ≥ 0?
 Solution
1. No energy is stored in the circuit
prior to the application of the DC
source, so the initial current in the
inductor is zero. The inductor
prohibits an instantaneous change in
inductor current, therefore iL(0)=0
immediately after the switch has
been opened.
2. The initial voltage on the capacitor is
zero before the switch has been
opened, therefore it will be zero
immediately after. Because


di L thus di L (0 )
vL 0
dt dt
 3. From the circuit elements,
12
1 10
0 2
   16  10 8

LC (25)(25)
9
1 10
 
2 RC (2)( 400)( 25)
 5 10 rad / s
4   25  10
2 8
• Thusthe roots of the
characteristic equation are real,

s1  5  10  3  10
4 4

 20 000 rad / s

s 2  5  10  3  10
4 4

 80 000 rad / s

 4. The inductor current response
will be overdamped.

 
i L  I f  A1 e  A2 es1t s2t
• Two simultaneous equation:
 
i L (0)  I f  A1  A2  0
di L (0)  
 s1 A1  s 2 A2  0
dt
 
A1  32mA A2  8mA
• Numerical solution:

 24  32e 20000t

iL (t )   80000t
mA

  8e 
untuk t0
 Example 2
(step response of series
RLC)

• No energy is stored in the

100mH inductor or 0.4µF
capacitor when switch in the
circuit is closed. Find vC(t) for
t ≥ 0.
 Solution
• The roots of the characteristic equation:

2
 280 
6
280 10
s1      
0. 2  0.2  0.10.4
  1400  j 4800 rad / s
s 2   1400  j 4800 rad / s
• Theroots are complex, so the
voltage response is
underdamped. Thus:

 1400t
vC  48  B1 e cos 4800t
 1400t
 B2 e sin 4800t t0
 • No energy is stored in the circuit
initially, so both vC(0) and
dvC(0+)/dt are zero. Then:


vC (0)  0  48  B1

dvC (0 )  
 0  4800 B2  1400 B1
dt
• Solving for B1’and B2’yields,


B1  48V

B2  14V
• Thus, the solution for vC(t),

 48  48 e 1400t cos 4800 t 

vC (t )   V
  14 e 1400t sin 4800 t 
 
for t0