![Page 1: First Order RC and RL Transient Circuitseee.guc.edu.eg/Courses/Electronics/ELCT301 Electric Circuits I/Lectu… · •For the RL circuit in the previous example, it was determined](https://reader030.vdocuments.us/reader030/viewer/2022040605/5eacdc346dd39f73476c1939/html5/thumbnails/1.jpg)
First Order RC and RL Transient
Circuits
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Objectives
• To introduce the transients’ phenomena.
• To analyze step and natural responses of first order RC circuits.
• To analyze step and natural responses of first order RL circuits.
![Page 3: First Order RC and RL Transient Circuitseee.guc.edu.eg/Courses/Electronics/ELCT301 Electric Circuits I/Lectu… · •For the RL circuit in the previous example, it was determined](https://reader030.vdocuments.us/reader030/viewer/2022040605/5eacdc346dd39f73476c1939/html5/thumbnails/3.jpg)
Transients
• In circuits with inductors and capacitors voltage and current cannot
change instantaneously.
• The application or removal of sources or circuit elements creates a
transient behavior.
• Transient is the process of going from one steady state to another
steady state following a sudden change in the circuit configuration.
• Sudden changes are mainly due to switching process or faults.
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FIRST ORDER CIRCUITS
• Circuits that contain a single energy storing elements.
• Either a capacitor or an inductor.
SECOND ORDER CIRCUITS
• Circuits with two independent energy storing elements in any
combination
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Transient Analysis
• The circuit is modeled in time domain using differential equations.
• The order of the differential equation equals the number of
independent energy storing elements in the circuits.
• Currents and Voltages of circuits with just one C or One L can be
obtained using first order differential equations.
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First Order ODF
00)y(t YBAy
dt
dy
The first order ordinary differential equation in the form
Has a solution
A
1T ,Y
where,
)eY - ( Y(t)
F
T
t-
0
A
B
YyFF
Transient partSteady state
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Initial Conditions
Initial conditions are the values of the capacitors voltage or the inductor
current at starting instant of the transient period.
• t = 0- is the instant just before switching.
• t = 0+ is the instant just after switching.
• In Capacitors VC(0-) = VC(0+). Where VC is the capacitor voltage.
• In inductors iL(0-) = iL(0+). Where iL is the Inductor current.
![Page 8: First Order RC and RL Transient Circuitseee.guc.edu.eg/Courses/Electronics/ELCT301 Electric Circuits I/Lectu… · •For the RL circuit in the previous example, it was determined](https://reader030.vdocuments.us/reader030/viewer/2022040605/5eacdc346dd39f73476c1939/html5/thumbnails/8.jpg)
First Order RC circuit
0cTHRTH
VRiV
R
C
ci
dt
dvCi
VTH
RTH
C
+
vc_
THC
C
THvv
dt
dvCR
A. Step Response
The response of the circuit to sudden application of an energy supply.
t=0
TH
TH
C
TH
C vCR
vCRdt
dv 11
For t = 0+ to inf.
Since,
Then,
![Page 9: First Order RC and RL Transient Circuitseee.guc.edu.eg/Courses/Electronics/ELCT301 Electric Circuits I/Lectu… · •For the RL circuit in the previous example, it was determined](https://reader030.vdocuments.us/reader030/viewer/2022040605/5eacdc346dd39f73476c1939/html5/thumbnails/9.jpg)
TH
TH
C
TH
C vCR
vCRdt
dv 11
o The following is the first order differential equation describing the
Change in the capacitor voltage during the transient period.
o The solution of this equation requires the initial conditions of the
capacitor voltage VC(0). Note that VC(0+) = VC(0-) = VCo
)eV -(V V(t) -
C0CC
t
FFCV
o The general solution of this equation is given by:
Where,
- VCF is the final value of VC
- is the circuit time constant
First Order RC circuit
![Page 10: First Order RC and RL Transient Circuitseee.guc.edu.eg/Courses/Electronics/ELCT301 Electric Circuits I/Lectu… · •For the RL circuit in the previous example, it was determined](https://reader030.vdocuments.us/reader030/viewer/2022040605/5eacdc346dd39f73476c1939/html5/thumbnails/10.jpg)
Final Value of VC
• The final value of VC occurs when the capacitor is fully charge i.e.
the rate of change of Vc = 0
• Using the differential equation,
• Then, VCF = VTH
• VCF is usually refers to as VC(∞)
• VCF can be obtained by replacing the capacitor with an open circuit
since iCF = 0
TH
TH
CF
TH
vCR
vCR
110
VTH
RTH
+
vcF_
t = ∞
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Time constant
The time constant is a measure of how fast is the charging process of
the Capacitor.
Mathematically it’s the time required to reach 63% of the final value.
CRTH
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Example
6kv
i0tFor CO
kkkRTH 36||6
CFTHVVv 6
+_ 0t
k6
k6
k6Fm100
V12
)(tiO
0t(t),iFind O
Hence, if the capacitor voltage is known the problem is solved
+
- 0t
k6
k6
k6
k6V12
)(tiO
a b
TH
v
3.0101003000 6 CRTH
VVVV CCC 246)0()0( 0
mA e 0.6667-1(t)i
)e6 -2( 6(t)
3.0-
0
3.0-
C
t
t
V
4V+_
4V+_
1) Initial Conditions
2) Time Constant
3) Final Value
4) Diff. equation
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0),( FIND tti
mAk
Vi 2
12
24
][32)2)(2(36)0( VkmAVvc
)0()0( cC vv
capacitor across voltageInitial :1 STEP
USE CIRCUIT IN STEADY STATE PRIOR TO THE SWITCHING
)( Determine :2 STEP C V
+
VCF
_
VV
mAi
CF 275.4236
5.48
036)(
constant Time :3 STEP
CRTH :circuit Capacitive
kkkRTH 5.16||2 FC m100
sF 15.0)10100)(105.1( 63
e 6
5
6
27
6
)()(
7)e2 -23( 27(t)
15.0-
15.0-
C
t
C
t
tVti
V
equation aldifferenti The :4 STEP
Example
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B. Natural Response of RC circuits
• The response of the circuit due to the energy stored in the capacitor is known as
natural response (no sources in the circuit).
• For t >0 KVL equation around the circuit yields,
t=0
0
0
CC
C
vdt
dvRC
viR
e
)e0 -(V 0(t)
-
-
0CC
R
V
dt
dVC(t)i
V
t
CoCC
t
iC
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Natural Response
Capacitor Voltage
CR :constant Time
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In the circuit shown in the figure , the switch opens at t = 0.
Find the numerical expression for i(t).
Example
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Solution
• Before t = 0, the circuit has reached steady state so that the capacitor
acts like an open circuit. The circuit is equivalent to that shown in Fig.
(a) after transforming the voltage source.
Initial Conditions
![Page 18: First Order RC and RL Transient Circuitseee.guc.edu.eg/Courses/Electronics/ELCT301 Electric Circuits I/Lectu… · •For the RL circuit in the previous example, it was determined](https://reader030.vdocuments.us/reader030/viewer/2022040605/5eacdc346dd39f73476c1939/html5/thumbnails/18.jpg)
Time Constant
![Page 19: First Order RC and RL Transient Circuitseee.guc.edu.eg/Courses/Electronics/ELCT301 Electric Circuits I/Lectu… · •For the RL circuit in the previous example, it was determined](https://reader030.vdocuments.us/reader030/viewer/2022040605/5eacdc346dd39f73476c1939/html5/thumbnails/19.jpg)
Final Value = 0
KCL yields
Solution
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First Order R-L circuits
0)0()0( LL ii
The differential equation is
Solving the equation yields
At t = 0 the switch
closes,
The initial conditions
tLRsL
tLRSS
tLRLFLLF
eVdt
tdiLtv
eR
V
R
Vti
IIIti
)/(
)/(
)/(
)()(
)0()(
))0(()(
The final conditions
ILF = Vs/R
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R-L Step Response
First Order R-L circuits step response
EQ
L =
R
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Equivalent Resistance seen by an Inductor
• For the RL circuit in the previous example,
it was determined that = L/R.
• As with the RC circuit, the value of R should actually be the equivalent
(or Thevenin) resistance seen by the inductor.
• In general, a first-order RL circuit has the following time constant:
seen from the terminals of the inductor for t > 0EQwith independent sources killed
R = REQ
L =
R
LCircuit
Circuit
t > 0
independent
sources killed
REQ
![Page 23: First Order RC and RL Transient Circuitseee.guc.edu.eg/Courses/Electronics/ELCT301 Electric Circuits I/Lectu… · •For the RL circuit in the previous example, it was determined](https://reader030.vdocuments.us/reader030/viewer/2022040605/5eacdc346dd39f73476c1939/html5/thumbnails/23.jpg)
Natural Response of R-L Circuit
• When the switch is closed (ON) – the inductor will store the
energy. As a result, the inductor is said to be charged.
• When the switch is opened (OFF), this will result in the
instantaneous change in the circuit. The inductor will supply
the energy stored to the resistor.
![Page 24: First Order RC and RL Transient Circuitseee.guc.edu.eg/Courses/Electronics/ELCT301 Electric Circuits I/Lectu… · •For the RL circuit in the previous example, it was determined](https://reader030.vdocuments.us/reader030/viewer/2022040605/5eacdc346dd39f73476c1939/html5/thumbnails/24.jpg)
Using the Kirchhoff’s voltage law (KVL), to find the differential
equation of the loop:
By using differential and integral equations technique:
0 iRdt
diL
/)0()( teIti
The circuit when the switch is opened
where = L/R is a time constant.
Natural Response of R-L Circuit
/)0()( tL eRItv
![Page 25: First Order RC and RL Transient Circuitseee.guc.edu.eg/Courses/Electronics/ELCT301 Electric Circuits I/Lectu… · •For the RL circuit in the previous example, it was determined](https://reader030.vdocuments.us/reader030/viewer/2022040605/5eacdc346dd39f73476c1939/html5/thumbnails/25.jpg)
The current generated against time shows the inductor loses its energy
exponentially
/)0()( teIti
Natural Response of R-L Circuit
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Example
• Assuming that i(0) = 10 A, calculate i(t) and ix (t) in the circuit.
The equivalent resistance is the same as
the Thevenin resistance at the inductor
terminals.
Because of the dependent source, we
insert a voltage source with vo = 1 V
![Page 27: First Order RC and RL Transient Circuitseee.guc.edu.eg/Courses/Electronics/ELCT301 Electric Circuits I/Lectu… · •For the RL circuit in the previous example, it was determined](https://reader030.vdocuments.us/reader030/viewer/2022040605/5eacdc346dd39f73476c1939/html5/thumbnails/27.jpg)
• Since there is no energy source in the circuit, the final value of the inductor current is
zero i.e. ILF = 0
• The time constant = L/R = 1.5 sec.
5.1/
/
10)(
)0()(
t
t
eti
eIti
The natural response equation yields
5.1/
/
3
10)(
)0()1
(5.0)(
tL
tL
etv
eIdt
diLtv
The inductor voltage
Ix (t)= VL(t)/2
Solution
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28
The switch in the circuit shown below has been closed for a long time.
It opens at t = 0. Find i(t) for t > 0.
teti 102)(
Example
Answer:
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29
For the circuit, find i(t) for t > 0.
Answer: i(t) = 2e–2t A