23
1 st Order Circuits
1st Order Circuits
Objective of the Lecture Explain the transient response of a RC circuit
As the capacitor stores energy when there is:
a transition in a unit step function source, u(t-to)
or a voltage or current source is switched into the circuit.
Explain the transient response of a RL circuit
As the inductor stores energy when there is:
a transition in a unit step function source, u(t-to)
or a voltage or current source is switched into the circuit.
Also known as a forced response to an independent source
RC Circuit
IC = 0A when t < to
VC = 0V when t < to
Because I1 = 0A (replace it with an open circuit).
RC Circuit Find the final condition
of the voltage across the
capacitor.
Replace C with an open
circuit and determine the
voltage across the terminal.
IC = 0A when t ~ ∞ s VC = VR = I1R when t ~ ∞ s
RC CircuitIn the time between to and t = ∞ s, the capacitor stores energy and currents flow through R and C.
RCeRItV
Idt
dVC
R
V
III
R
VI
dt
dVCI
VV
tt
C
CC
CR
RR
CC
RC
1)(
0
0
0
1
1
1
RL Circuit
RL Circuit (con’t) Initial condition is not important as the magnitude of
the voltage source in the circuit is equal to 0V when t ≤ to.
Since the voltage source has only been turned on at t = to, the circuit at t ≤ to is as shown below.
As the inductor has not stored any energy because no power source has been connected to the circuit as of yet, all voltages and currents are equal to zero.
RL Circuit So, the final condition of the inductor current needs to
be calculated after the voltage source has switched on.
Replace L with a short circuit and calculate IL(∞).
Final Condition
R
VI
II
VV
R
RL
L
1
)(
0)(
RL Circuit
/)(1
1
1
1)(
0
0
ott
L
LL
RL
eR
VtI
L
VI
L
R
dt
dI
VRIdt
dI
R
L
dt
dILV
RVII
VVV
LL
RRL
RL
/
01
Electronic Response Typically, we say that the currents and voltages in a
circuit have reached steady-state once 5 have passed after a change has been made to the value of a current or voltage source in the circuit.
In a circuit with a forced response, percentage-wise how close is the value of the voltage across a capacitor in an RC circuit to its final value at 5?
Complete Response Is equal to the natural response of the circuit plus the
forced response
Use superposition to determine the final equations for voltage across components and the currents flowing through them.
Example #1 Suppose there were two unit step function sources in
the circuit.
Example #1 (con’t) The solution for Vc would be the result of
superposition where:
I2 = 0A, I1 is left on
The solution is a forced response since I1 turns on at t = t1
I1 = 0A, I2 is left on
The solution is a natural response since I2 turns off at t = t2
Example #1 (con’t)
1
)t-(t-
1
1
when e-1 )(
when 0)(
1
ttRItV
ttVtV
RCC
C
Example #1 (con’t)
2
)t-(t-
2
22
when e)(
when )(
2
ttRItV
ttRItV
RCC
C
Example #1 (con’t) If t1 < t2
2
)t-(t-
2
)t-(t-
1
212
)t-(t-
1
12
when e e-1 )(
when e1)(
hen w 0)(
21
1
ttRIRItV
tttRIRItV
ttRIVtV
RCRCC
RCC
C
General Equations
RL
eIIL
tV
eIIItI
RC
eVVC
tI
eVVVtV
t
LLL
t
LLLL
t
CCC
t
CCCC
/
)0()()(
)()0()()(
)0()()(
)()0()()(
/
/
/
/
When a voltage or current
source changes its magnitude
at t= 0s in a simple RC or
RL circuit.
Equations for a simple RC circuit
Equations for a simple RL circuit
Needed to Complete HW # 21
How to renew your MatLAB licensehttp://swat.eng.vt.edu/Matlabtutorial.html
If you would like to contact them directly, SWAT is located at:
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Introductory Tutorials MathWorks (www.mathworks.com) has
On-line tutorials including A Very Elementary MATLAB Tutorial
http://www.mathworks.com/academia/student_center/tutorials/intropage.html
Videos (look at the ones below Getting Started)http://www.mathworks.com/products/matlab/demos.html
Worked examples (further down the demos page)http://www.mathworks.com/products/matlab/demos.html
Textbook has a MatLAB tutorial in Appendix E.
Summary The final condition for:
the capacitor voltage (Vo) is determined by replacing the capacitor with an open circuit and then calculating the voltage across the terminals.
The inductor current (Io) is determined by replacing the inductor with a short circuit and then calculating the current flowing through the short.
The time constant for: an RC circuit is RC and an RL circuit is L/R
The general equations for the forced response of: the voltage across a capacitor is
the current through an inductor is
o
tt
oL
o
tt
oC
tteItI
tteVtV
o
o
when 1)(
when 1)(
/)(
/)(
Summary General equations when the magnitude of a voltage or
current source in the circuit changes at t = 0s for the:
voltage across a capacitor is
current through an inductor is
Superposition should be used if there are multiple voltage and/or current sources that change the magnitude of their output as a function of time.
/
/
)()0()()(
)()0()()(
t
LLLL
t
CCCC
eIIItI
eVVVtV
