BLM1612 Circuit Theory
Basic RL and RC Circuits
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reproduction or display.
Capacitors and Inductors
C→L
v→i
i →v
The Source-Free RL Circuit
Applying KVL:
di R
+ i =0
dt L
We can solve for the natural response
if we know the initial condition i(0)=I0:
i(t)=I0e-Rt/L for t>0
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Example 8.2: RL with a Switch
Show that the voltage v(t) will be -12.99 volts at
t=200 ms.
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The Exponential Response
The time constant
τ=L/R
determines the rate of
decay.
i(t) = I0e
−t /τ
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The Source-Free RC Circuit
Applying KCL:
dv
1
+
v =0
dt RC
We can solve for the natural response
if we know the initial condition v(0)=V0
v(t)=V0e-t/RC for t>0
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RC Natural Response
The time constant is
τ=RC
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Example 8.3: The Source Free RC Circuit
Show that the voltage v(t) is 321 mV at t=200 µs.
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General RL Circuits
The time constant of a single-inductor circuit
will be τ=L/Req where Req is the resistance
seen by the inductor.
Example: Req=R3+R4+R1R2 / (R1+R2)
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General RC Circuits
The time constant of a single-capacitor circuit will
be τ=ReqC where Req is the resistance seen by
the capacitor.
Example 8.5: Req=R2+R1R3 / (R1+R3)
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10for
reproduction or display.
1st Order Response Observations
The voltage on a capacitor or the current through
a inductor is the same prior to and after a switch
at t=0.
Resistor voltage (or current) prior to the switch
v(0-) can be different from the voltage after the
switch v(0+).
All voltages and currents in an RC or RL circuit
follow the same natural response e-t/τ.
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Example 8.4: L and R Current
Find i1(t) and iL(t) for t>0.
Answer: τ=20 µs; i1=-0.24e-t/τ,iL=0.36e-t/τ for t>0
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The Unit Step Function
The unit-step function u(t) is a convenient notation
to respresent change:
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Switches and Steps
The unit step models a double-throw switch.
A single-throw switch is open circuit for t<0, not
short circuit.
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Modeling Pulses using u(t)
Rectangular
pulse
Pulsed
sinewave
:
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Driven RL Circuits
The two circuits shown both
have i(t)=0 for t<0 and are
also the same for t>0.
We now have to find both the
natural response and and the
forced response due to the
source V0
– i=in+if
– in=Ae-Rt/L
– if=V0/R
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Driven RL Circuits
The total response is the
combination of the
transient/natural response and
the forced response:
V0
−Rt / L
i(t) = (1 − e
u(t)
)
R
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Driven RL Circuits
V0
−Rt / L
i(t) = (1 − e
u(t)
)
R
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Example 8.7
For the circuit , find i(t) for t=∞, 3−, 3+, and 100 µs
after the source changes value.
Example 8.8: RL Circuit with Step
Show that
i(t)=25+25(1-e-t/2)u(t) A
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Example 8.9: Voltage Pulse
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Example 8.10: Driven RC Circuits (part 1 of 2)
vC=20 + 80e-t/1.2 V and i=0.1 + 0.4e−t/1.2 A
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Example 8.10: Driven RC Circuits (part 2 of 2)
vC=20 + 80e-t/1.2 V
i=0.1 + 0.4e−t/1.2 A
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Chapter 8 Summary