BASICS OF ELECTRICAL CIRCUITS
EHB 211 E
RL & RC Circuits
Asst. Prof. Onur Ferhanoğlu
1
Natural Response of an RL Circuit
• t< 0, switch was ON for a long time
• All currents & voltages have reached a
constant value
• Inductor acts as a short circuit before the
switch is opened
After switch is turned OFF
KVL
(1st order differential equation)
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
2
Natural Response of an RL Circuit
Voltage across resistor
Asst. Prof. Onur Ferhanoğlu
In an inductor:
• Change of current is not instantaneous
i(0-) = i(0+)
• Change of voltage can be instantaneous
v(0-) ≠ v(0+)
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
3
Natural Response of an RL Circuit
POWER:
Asst. Prof. Onur Ferhanoğlu
ENERGY (delivered to resistor):
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
4
Natural Response of an RL Circuit – time constant
Rate at which current
& voltage changes:
After 5τ, current & voltages reach their final values
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
5
Natural Response of an RL Circuit – time constant
Rate at which current
& voltage changes:
After 5τ, current & voltages reach their final values
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
6
Exercise
The switch in the below circuit has been closed for a long time before it was opened
at t = 0. Find
a) iL(t)
b) io(t)
c) vo(t)
a) iL(0-) = iL(0+) = 20A (inductor is short at t = 0-)
τ = L/Req = 0.2
b)
Asst. Prof. Onur Ferhanoğlu
c)
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
7
Exercise 2
For the below circuit find:
a) i1,, i2 and i3 for t>0 (attention to their initial values)
b) Initial stored energy in the parallel inductors
c) How much energy is stored in the inductors for t->∞
a) τ = 4/8 = 0.5
v = 8i
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
8
Exercise 2 – continued…
For the below circuit find:
a) i1,, i2 and i3 for t>0 (attention to their initial values)
b) Initial stored energy in the parallel inductors
c) How much energy is stored in the inductors for t->∞
b) 0.5L1i2 + 0.5L2i2
c)i1 -> 1.6 A , i2 -> -1.6 A
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
9
Natural response of an RC circuit
KCL
Similar to RL circuit..
In an capacitor:
• Change of voltage is not instantaneous
v(0-) = v(0+)
• Change of current can be instantaneous
i(0-) ≠ i(0+)
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
10
Natural response of an RC circuit
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
11
Exercise 3
For the circuit find:
a) vc(t), for t> 0
b) vo(t), for t> 0
c) io(t), for t> 0
a) t<0 -> capacitor will charge to 100V (V0 = 100V)
b)
c)
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
12
Exercise 4
For the circuit find:
a) v1(t), v2(t), v(t), i(t), for t> 0
b) Calculate the initial energy stored in the capacitors
c) Determine the energy stored in the capacitors for t -> ∞
a)
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
13
Exercise 4
For the circuit find:
a) v1(t), v2(t), v(t), i(t), for t> 0
b) Calculate the initial energy stored in the capacitors
c) Determine the energy stored in the capacitors for t -> ∞
b)
c)
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
14
The Step response of RL circuits
Step response: response of the circuit when a source
is suddenly applied
KVL
if I0 = 0
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
15
The Step response of RL circuits
if I0 = 0
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
16
Exercise 5
For the circuit, find
a) i(t)
b) v(0+)
c) When does v = 24V
d) Plot i(t) and v(t) vs. t
a) At position a, inductor behaves as a short circuit
I0 = -8 A (all the source current passes through the inductor)
τ = 200/2 = 100ms = 0.1 s
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
17
Exercise 5 continued..
For the circuit, find
a) i(t)
b) v(0+)
c) When does v = 24V
d) Plot i(t) and v(t) vs. t
b)
i=
Alternatively:
v = (24 – (-8.2))e-10t
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
18
Exercise 5 continued..
For the circuit, find
a) i(t)
b) v(0+)
c) When does v = 24V
d) Plot i(t) and v(t) vs. t
c)
Asst. Prof. Onur Ferhanoğlu
d)
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
19
The Step response of RC circuits
KCL
Recall from RL circuits
therefore
i = C.dv/dt
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
20
Exercise 6
For the circuit, find
a) vo(t), for t > 0
b) io(t), for t > 0
a) Initial value of vo is 40.(60/80) = 30V (voltage divider)
Transform the circuit so that we can take advantage of deduced equations (previous page)
NORTON EQUIVALENT for t>0
Inorton = -60 / 40 = -1.5 mA
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
21
Exercise 6 continued..
For the circuit, find
a) vo(t), for t > 0
b) io(t), for t > 0
a)
b)
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
22
General solution for step and natural responses
4 possible 1st order circuits
-> For all 4 circuits, the differential equation is of the following
form:
Because the sources are DC (constant), final value of x (current
or voltage) will be constant, dx/dt -> 0
where xf is the final value of the variable (current & voltage)
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
23
Exercise 7
For the circuit, find
a) Initial value of vC
b) Final value of vC
c) Time constant when switch is at b
d) vC(t) when t>0
e) i(t) when t>0
a) Voltage divider:
b) Long after switching from a to b -> capacitor will act as an open circuit
vC(∞) = 90V
c)
d)
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
24
Exercise 7 continued…
For the circuit, find
a) Initial value of vC
b) Final value of vC
c) Time constant when switch is at b
d) vC(t) when t>0
e) i(t) when t>0
e) Find initial and final values of i(t)
i(∞) = 0 (open circuit)
i(0+) = (initial value) = current in 400kΩ resistor:
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
25
Exercise 8
For the circuit, the initial charge on
the capacitor is 0. Find:
a) i(t) for t > 0
b) v(t) for t > 0
a)
i(∞) = 0 (open circuit)
b) First find vC(t)
vC(0) = 0 (given in the question)
vC(∞) = 7.5mA * 20k = 150V
(0 volts across 30k resistor)
v(t) = vC(t) + v30kΩ
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
26
Exercise 9
For the circuit, find:
a) v(t) for t > 0
b) i(t) for t > 0
a) i(0) = 20V/4Ω = 5A
v(0) = 20V – 5A*1 = 15V
v(∞) = 0V (inductor: short circuit)
b) i(∞) = 20V / 1 Ω = 20A
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
27
Sequential switching
Switching occurs more than once
In the circuit, at t=0 the switch is moved from
a to b, and at t=15ms, from b to c:
(Initial voltage on capacitor = 0)
Switch ->b
v(0) = 0
v(∞) = 400 V, if the switch remained at b
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
Switch -> c
v(15ms) = 310.75V
v(∞) = 0 (capacitor: open circuit)
28
Sequential switching
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
29
Integrating amplifier
If initial voltage is 0
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
30
Integrating amplifier + switching
The initial voltage on the capacitor is 5V.
If switch stays in terminal a for 9ms,
when does saturation take place
At terminal a
At terminal b
vo(9ms) = -5 + 9 = 4V
Asst. Prof. Onur Ferhanoğlu
RL & RC Circuits/ BASICS OF ELECTRICAL CIRCUITS
31