Source Free RC Circuit a electrical engineering topics.

1. Welcome to our presentation
Team
1

2.  Md. Masud Rana
ID: 221902112
Afroza Akter Naznin
ID: 221902132
 Maria Akter Rimi
ID:221902098
Mohibul Haque
ID: 221902199
Miss. Sumaiya Shoshi
Lecterer
BSc in Computer science and Engineering(CSE)
Green University of Bangladesh
Presented by Presented to
Topic: Source free RC circuit
2

3. Source free RC circuit Index
 Source free RC circuit
 Derivation
 Voltage response
 Time constant
 Practice problem
 Source free RC circuit 7.1
3

4. Source free RC circuit
● A source-free RC circuit occurs when its dc source is suddenly
disconnected. The energy already stored in the capacitor is released
to the Resistors.
● RC source-free circuit is analyzed from its initial voltage v(0) = 𝑣° and
time constant τ
● Circuits based on resistor-capacitor combinations are more common than
their resistor-inductor analogs. The principal reasons for this are the smaller losses
present in a physical capacitor, lower cost, better agreement between the simple
mathematical model and the actual device behavior, and also smaller size and lighter weight,
both of which are particularly important for integrated-circuit applications.
4

5. Source free RC circuit
● The energy already stored in the capacitor(s) is
released to the resistor(s) & dissipated.
●The Key to Working with a Source-Free RC Circuit
is Finding:
1.The initial voltage across the capacitor.
2.The time constant t.
5

6. 6
DERIVATION :
V(t) across the capacitor. Since the capacitor is initially
charged, we
can assume that at time t=0, the initial voltage
v(0) =V0
with the corresponding value of the energy stored as
𝑤(0) =
1
2
𝑐𝑣0
2
Applying KCL at the top node of the circuit in Fig. 7.1 yields
𝑖𝑐 + 𝑖𝑅 = 0
By definition,
𝑖𝑐 = 𝑐
𝑑𝑣
𝑑𝑡
𝑎𝑛𝑑 𝑖𝑅 =
𝑌
𝑅
, Thus
=>
𝑐𝑑𝑣
𝑑𝑡
+
𝑉
𝑅
= 0
=>
𝑑𝑣
𝑑𝑡
+
𝑉
𝑅𝑐
= 0
This is a first-order differential equation, since only the first
derivative of v is involved. To solve it, we rearrange the
terms as
Source free RC circuit

7. 7
𝑑𝑣
𝑣
= −
1
𝑅𝐶
𝑑𝑡
.
𝑑𝑣
𝑣
= −
1
𝑅𝐶
𝑑𝑡
ln 𝑣 = −
𝑡
𝑅𝐶
+ ln 𝐴
Integrating both sides, we get
ln.
𝑉
𝐴
= −
𝑡
𝑅𝐶
=> 𝑉 𝑡 = 𝐴 𝑒−𝑡/𝑅𝐶
=>𝑉(𝑡) = 𝑒−𝑡/𝑅𝐶
This shows that the voltage response of the RC circuit is an exponential
decay of the initial voltage. Since the response is due to the initial
energy stored and the physical characteristics of the circuit and not due
to some external voltage or current source, it is called the natural
response of the circuit.
The natural response is illustrated graphically in Fig. 7.1. Note that at t= 0, we
have the correct initial condition as in Eq. (7.1). As t increases, the voltage
decreases toward zero. The rapidity with which the voltage decreases is
expressed in terms of the time constant, denoted by 𝑉 𝑡 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑒−𝑡/𝑅𝐶 the
lowercase.
V0
𝑣°
𝑣°

8. Source free RC circuit
VOLTAGE RESPONSE:
● As t increases, the voltage decreases exponentially towards zero.The
rapidity with which the voltage
decreases is expressed in terms of the time constant, denoted by τ
8

9. Source free RC circuit
TIME CONSTANT:
● Graphical determination of the time constant τ from the
response curve.
9

10. 10

11. 𝑣𝑐 0 = 60𝑣
∴ 12116 =
12 × 6
12 + 6
= 4𝛺
Re-draw the circuit.
Again, 4+4= 12𝛺
Re-draw the circuit.
→
4𝛺
1
8
8𝛺
+
V
-
7.1
12𝛺
1
3
F
A
𝑖°
𝑖°
→
+
𝑣𝑐
-
.
Source free RC circuit:
11

12. Using KCL at node A
𝑖𝑖𝑛= 𝑖𝑜𝑢𝑡
0 − 𝑣𝑎
12
=
1
3
𝑑
𝑑𝑡
(𝑣𝑎 − 0)
=>
𝑣𝑐
12
=
1
3
𝑑𝑣𝑐
𝑑𝑡
=>
𝑑𝑣𝑐
𝑑𝑡
= −3 ×
𝑣𝑐
12
→
8𝛺
𝑖°
4𝛺
𝑣𝑥
𝑖°
→
𝑣𝑐
=>To find 𝑣𝑥 , voltage divider: 12

13. 𝑣𝑥 =
4
4 + 8
𝑣𝑐
𝑣𝑥(𝑡) =
1
3
× 𝑣𝑐 (𝑡)
𝑣𝑥(𝑡) =
1
3
× 60𝑒−0.25𝑡
𝑣𝑥(𝑡) =
1
3
× 60𝑒−0.25𝑡
𝑣𝑥(𝑡) = 20𝑒−0.25𝑡
Again, To find 𝑖°
𝑖° =
0 − 𝑣𝑥
4
=
−𝑣𝑥
4
=>𝑖° (𝑡) =
20𝑒−0.25𝑡
4
=>𝑖° (𝑡) = 5𝑒−0.25𝑡𝐴 Ans.
Ans.
13

14. ANY QUESTION ?
16
14

15. THANK YOU
15