Sub: Circuits and Networks (2130901 ) Topic: Time Domain response of second order linear circuit Branch: Electrical Engineering

1. Active Learning Assignment
Sub: Circuits and Networks (2130901 )
Topic: Time Domain response of second order linear circuit
Guided By: Prof. Sweta Khakhakhar
Branch: Electrical Engineering
Batch : B1
Prepared By: (1) Abhishek Choksi 140120109005
(2) Himal Desai 140120109008
(3) Harsh Dedakiya 140120109012

2. Contents:
• Discharging of a capacitor through an inductor
• Source free series RLC circuit
• Source free parallel RLC circuit
• Types of Response
• Second order linear network with constant input

3. Discharging of a capacitor through an inductor
• Consider LC network as shown
in fig.
• Initially the switch is at point A.
So the voltage source Vs is
connected across capacitor. Due
to this a capacitor is fully
charged to Vs volts.
• At time t = 0 the switch is moved
to position ‘B’. An equivalent
network is shown in fig.

4. Equation in Term of Capacitor Voltage:
• The capacitor current can be expressed in term of voltage
across capacitor as,
𝑖 𝑐 = 𝐶
𝑑𝑉𝑐
𝑑𝑡
So,
𝑖 𝑐
𝐶
=
𝑑𝑉𝑐
𝑑𝑡
---------- (1)
• Now as shown in fig. 𝑖 𝑐 = −𝑖 𝐿
−𝑖 𝐿
𝐶
=
𝑑𝑉𝑐
𝑑𝑡
---------- (2)
• Differentiating both side with respect to ‘t’ we get,
𝑑2
𝑑𝑡2 𝑉𝑐 = −
1
𝐶
𝑑𝑖 𝐿
𝑑𝑡
---------- (3)

5. • The voltage across inductor is given by,
𝑉𝐿 = 𝐿
𝑑𝑖 𝐿
𝑑𝑡
So,
𝑉 𝐿
𝐿
=
𝑑𝑖 𝐿
𝑑𝑡
---------- (4)
• But from fig 𝑉𝐿 = 𝑉𝑐
𝑉𝑐
𝐿
=
𝑑𝑖 𝐿
𝑑𝑡
---------- (5)
• Putting Equation (5) in equation (3) we get,
𝑑2
𝑑𝑡2 𝑉𝑐 = −
1
𝐶
𝑉𝑐
𝐿
𝑑2
𝑑𝑡2 𝑉𝑐 = −
𝑉𝑐
𝐶𝐿
---------- (6)
• This is second order differential equation in term of
capacitor voltage 𝑉𝑐

6. Equation in Term of Inductor Current
• Consider Equation (5)
Vc
L
=
diL
dt
• Differentiating both the side with respect to time we get,
d2iL
dt
=
1
L
dVc
dt
---------- (7)
• Putting Equation (2) in Equation (7) we get,
d2iL
dt
= −
iL
CL
---------- (8)
• This is the Second order differential equation in term of
inductor current.

7. Source Free Series RLC Circuit
• Consider a source free series RLC circuit as shown in
fig.

8. Equation in Term of
• Applying KVL to the given network,
VR + VL + VC = 0 ---------- (1)
Now,
VR = Voltage across resistor = IR R ---------- (2)
VL = Voltage across inductor = 𝐿
𝑑𝑖 𝐿
𝑑𝑡
---------- (3)
VC = Voltage across capacitor =
1
𝐶 −∞
𝜏
𝑖 𝐶 𝜏 𝑑𝜏 ---------- (4)
• Putting these value in Equation (1) we get,
IR R + 𝐿
𝑑𝑖 𝐿
𝑑𝑡
+
1
𝐶 −∞
𝜏
𝑖 𝐶 𝜏 𝑑𝜏 = 0 ---------- (5)
• Now referring fig (1) IR = IL = IC . We want equation in term
of IL . So replacing IR and IC by IL in Equation (5) we get,

9. IL R + 𝐿
𝑑𝑖 𝐿
𝑑𝑡
+
1
𝐶 −∞
𝜏
𝑖 𝐶 𝜏 𝑑𝜏 = 0 ---------- (6)
• To eliminate integration sign, differentiate with respect to
time ’t’
R
𝑑𝑖 𝐿
𝑑𝑡
+ 𝐿
𝑑2 𝑖 𝐿
𝑑𝑡2 +
𝑖 𝐿
𝐶
= 0
• Rearranging the equation,
𝐿
𝑑2 𝑖 𝐿
𝑑𝑡2
+ R
𝑑𝑖 𝐿
𝑑𝑡
+
𝑖 𝐿
𝐶
= 0
• To make coefficient of
𝑑2
𝑑𝑡2 equal to 1, divide both the side
by 1,
𝑑2 𝑖 𝐿
𝑑𝑡2 +
𝑅
𝐿
𝑑𝑖 𝐿
𝑑𝑡
+
𝑖 𝐿
𝐿𝐶
= 0 ---------- (7)
• This is a second order differential equation of source free
RLC circuit in term of 𝑖 𝐿.

10. Equation in Term of Vc
• Consider Equation 3, we have to express every current in
term of 𝑖 𝐶. So put IR = IC , IL = IC . The last term in the
equation 5 indicates, voltage across capacitor, Vc.
𝑅 𝑖 𝑐 + 𝐿
𝑑𝑖 𝑐
𝑑𝑡
+ 𝑉𝑐 = 0 ---------- (8)
• Now we want equation in term of Vc. The capacitor current
IC can be expressed in term of capacitor voltage Vc as,
𝑖 𝑐 = 𝐶
𝑑𝑉𝑐
𝑑𝑡
• Putting this value in the equation 8 we get,
RC
𝑑𝑉 𝐶
𝑑𝑡
+ 𝐿
𝑑
𝑑𝑡
𝐶
𝑑𝑉𝑐
𝑑𝑡
+ 𝑉𝑐 = 0

11. RC
𝑑𝑉 𝐶
𝑑𝑡
+ 𝐿𝐶
𝑑2 𝑉𝑐
𝑑𝑡2 + 𝑉𝐶 = 0
• Rearranging the equation,
𝐿𝐶
𝑑2 𝑉𝑐
𝑑𝑡2 + RC
𝑑𝑉 𝐶
𝑑𝑡
+ 𝑉𝐶 = 0
• To obtain the coefficient of
𝑑2 𝑉𝑐
𝑑𝑡2 , equal to 1, divide both
the side by LC,
𝑑2 𝑉𝑐
𝑑𝑡2 +
𝑅
𝐿
𝑑𝑉 𝐶
𝑑𝑡
+
1
𝐿𝐶
𝑉𝐶 = 0 ----------(9)
• This is the second order differential equation, for source
free RLC circuit in term of capacitor voltage 𝑉𝑐.

12. Source free Parallel RLC Circuit
• Consider source free parallel RLC circuit as shown in
fig

13. Equation in Term of 𝑉𝐶
• As shown in fig. R, L and C are connected in parallel, so
voltage across each element is same.
VR = VL = VC ---------- (1)
• Now applying KCL we get,
iR + iL + iC = 0
• Expressing every current in term of corresponding voltage
we get,
𝑉 𝑅
𝑅
+
1
𝐿 −∞
𝑡
𝑉𝐿 𝜏 𝑑𝜏 + 𝐶
𝑑𝑉 𝐶
𝑑𝑡
= 0 ---------- (2)
• But we want every voltage in term of 𝑉𝐶. Now from the
equation (1) VR = VC and VL = VC
• Putting these values in equation (2) we get,

14. 𝑉 𝐶
𝑅
+
1
𝐿 −∞
𝑡
𝑉𝐶 𝜏 𝑑𝜏 + 𝐶
𝑑𝑉 𝐶
𝑑𝑡
= 0
• To eliminate integration sign, take derivatives with respect
to time,
1
𝑅
𝑑
𝑑𝑡
𝑉𝐶 +
1
𝐿
𝑉𝐶 + 𝐶
𝑑2
𝑑𝑡2 𝑉𝐶
2
= 0
• Rearranging the terms,
𝐶
𝑑2
𝑑𝑡2 𝑉𝐶
2
+
1
𝑅
𝑑
𝑑𝑡
𝑉𝐶 +
1
𝐿
𝑉𝐶 = 0
• To make coefficient of equal to 1, divide both side by C
𝑑2
𝑑𝑡2 𝑉𝐶
2
+
1
𝑅𝐶
𝑑
𝑑𝑡
𝑉𝐶 +
1
𝐿𝐶
𝑉𝐶 = 0 ---------- (3)
• This is the second order differential equation, for source free
parallel RLC circuit in term of capacitor voltage 𝑉𝐶.

15. Equation in Term of iL
• Consider Equation (2) it is,
𝑉 𝑅
𝑅
+
1
𝐿 −∞
𝑡
𝑉𝐿 𝜏 𝑑𝜏 + 𝐶
𝑑𝑉 𝐶
𝑑𝑡
= 0
• Now the second term indicate the equation of that means,
iL =
1
𝐿 −∞
𝑡
𝑉𝐿 𝜏 𝑑𝜏
• Thus we can write,
𝑉 𝑅
𝑅
+ iL + 𝐶
𝑑𝑉 𝐶
𝑑𝑡
= 0
• From the equation (1), ) VR = VL and VL = VC
𝑉 𝐿
𝑅
+ iL + 𝐶
𝑑𝑉 𝐿
𝑑𝑡
= 0 ---------- (4)

16. • But the voltage across inductor is,
VL = L
𝑑𝑖 𝐿
𝑑𝑡
---------- (5)
• Putting Equation (5) in Equation (4) we get,
1
𝑅
𝐿
𝑑𝑖 𝐿
𝑑𝑡
+ 𝑖 𝐿 + 𝐶𝐿
𝑑2
𝑑𝑡2 𝑖 𝐿 = 0
• Rearranging the terms,
𝐶𝐿
𝑑2
𝑑𝑡2 𝑖 𝐿 +
1
𝑅
𝐿
𝑑𝑖 𝐿
𝑑𝑡
+ 𝑖 𝐿 = 0
• To make the coefficient of first term equal to 1, divide both
side by CL,
𝑑2
𝑑𝑡2 𝑖 𝐿 +
1
𝑅𝐶
𝐿
𝑑𝑖 𝐿
𝑑𝑡
+
1
𝐿𝐶
𝑖 𝐿 = 0 ---------- (6)
• This is the second order differential equation, for parallel
RLC circuit in term of inductance current

17. Type of Response
• The output of source free second order linear system is an
oscillatory waveform.
• The work ‘damping’ indicates, decrease in the peak
amplitude of oscillation. It is due to the effect of energy that
is absorbed by elements of system.
• Depending on the damping, the response of the system is
classified as follows:
1. Critical damping response
2. Under damping response
3. Over damping response
4. Undamped response

18. Critical Damping Response
• If the damping factor is
sufficient to prevent
oscillations then a system
response is called as
critical response.
• Consider that, the current
is exponentially increasing
as shown in the fig. Here
the response is not
oscillating. So it is
critically damped response.

19. Under Damping Response
• If the amount of damping is
less than the response is
called as under damped
response.
• The oscillation are present
but eventually decays out as
shown in fig.

20. Over Damped Response
• In this case, amount of
damping is large. Any small
change in the circuit
parameter, will prevent the
damping.

21. Undamped Response
• If there is no damping then
it is called as undamped
response. Here the
oscillation are called as
sustained oscillations. It is
shown in fig.

22. Second Order Linear Circuit with Constant
Input
• When some independent sources are present in RLC circuit
then its differential equation is,
•
𝑑2 𝑥
𝑑𝑡2 + 𝑏
𝑑
𝑑𝑡
𝑥 + 𝑐𝑥 = 𝑓(𝑡)
• Compared to the equation of the source free circuits, this
equation are extra term f(t).
• Here the term f(t) depend on the source input and the
derivative of input.
• If f(t) is not constant then the solution becomes
complicated, so such differential equations are solved using
laplace transform.
.

23. • If f(t) is having some constant value that means, f(t) = F
and the solution can be easily obtained
• Consider the general solution of differential equation,
•
𝑑2 𝑥
𝑑𝑡2 + 𝑏
𝑑
𝑑𝑡
𝑥 + 𝑐𝑥 = 𝐹
• Here the roots are,
• 𝑠1, 𝑠2 =
−𝑏± 𝑏2−4𝑎𝑐
2
• And the general solution is given by,
• In equation 3 is the general solution when f(t) = 0.
• The value of X(F) is independent of the root and it is
given by,
• 𝑋 𝐹 =
𝐹
𝐶