Lecture 13 RC/RL Circuits, Time Dependent Op Amp Circuits

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Lecture 13RC/RL Circuits, Time Dependent Op Amp
Circuits
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RL Circuits
The steps involved in solving simple circuits
containing dc sources, resistances, and one
energy-storage element (inductance or
capacitance) are
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1. Apply Kirchhoffs current and voltage laws to
write the circuit equation.2. If the equation
contains integrals, differentiate each term in
the equation to produce a pure differential
equation.3. Assume a solution of the form K1
K2est.
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4. Substitute the solution into the differential
equation to determine the values of K1 and s .
(Alternatively, we can determine K1 by solving
the circuit in steady state)5. Use the initial
conditions to determine the value of K2.6.
Write the final solution.
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RL Transient Analysis
Find i(t) and the voltage v(t)
i(t) 0 for t lt 0 since the switch is open prior
to t 0
Apply KVL around the loop
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RL Transient Analysis
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RL Transient Analysis
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RL Transient Analysis
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RL Transient Analysis
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RL Transient Analysis
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RL Transient Analysis
Transient starts by opening switch
Prior to t 0 inductor acts as a short circuit
so that v(t) 0 for t lt 0 i(t) VS/R1 for t lt 0
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RL Transient Analysis
After t 0 current circulates through L and R,
dissipating energy in the resistance R
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RL Transient Analysis
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RL Transient Analysis
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RL Transient Analysiswith a Current Source
After the switch is opened, iR(0) 2A, IL(0)
0
Find v(t), iR(t), iL(t)
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RL Transient Analysiswith a Current Source
After the switch is closed, iR(0) 2A, IL(0)
0
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RL Transient Analysiswith a Current Source
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RL Transient Analysiswith a Current Source
iL(0) 0 ? K1 -K2
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RL Transient Analysiswith a Current Source
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RL Transient Analysiswith a Current Source
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RL Transient Analysis
Prior to t 0 i(0) 100V/100? 1A
Find i(t), v(t)
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RL Transient Analysis
Prior to t 0 i(0) 100V/100? 1A
After t 0
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RL Transient Analysis
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RL Transient Analysis
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RL Transient Analysis
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RC and RL Circuits with General Sources
First order differential equation with constant
coefficients
Forcing function
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RC and RL Circuits with General Sources
The general solution consists of two parts.
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The particular solution (also called the forced
response) is any expression that satisfies the
equation. In order to have a solution that
satisfies the initial conditions, we must add the
complementary solution to the particular solution.
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The homogeneous equation is obtained by setting
the forcing function to zero. The
complementary solution (also called the natural
response) is obtained by solving the homogeneous
equation.
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Step-by-Step Solution
Circuits containing a resistance, a source, and
an inductance (or a capacitance) 1. Write the
circuit equation and reduce it to a first-order
differential equation.
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2. Find a particular solution. The details of
this step depend on the form of the forcing
function. 3. Obtain the complete solution by
adding the particular solution to the
complementary solution xcKe-t/? which contains
the arbitrary constant K. 4. Use initial
conditions to find the value of K.
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
Take the derivative
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
The complementary solution is given by
The complete solution is given by the sum of the
particular solution and the complementary
solution
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
Initial conditions 2sin(0) 0 vC(0)
1V vR(0) vC(0) 0 ? vR(0) -1V i(0)
vR/R -1V/5000? -200?A 200cos(0)200sin(0)K
e0 200 K ? K -400?A
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
What happens if we replace the source with
2cos(200t) and the capacitor initially uncharged
vc(0)0?
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
What happens if we replace the source with
2cos(200t) and the capacitor initially uncharged
vc(0)0?
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
Take the derivative
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
The complementary solution is given by
The complete solution is given by the sum of the
particular solution and the complementary
solution
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
Initial conditions 2cos(0) 2 vC(0)
0V vR(0) vC(0) 2 ? vR(0) 2 i(0) vR/R
2V/5000? 400?A 200cos(0)-200sin(0)Ke0
200 K ? K 200?A
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Transient Analysis of an RC Circuit with a
Sinusoidal Source
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Transient Analysis of an RC Circuit with an
Exponential Source
What happens if we replace the source with 10e-t
and the capacitor is initially charged to vc(0)5?
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Transient Analysis of an RC Circuit with an
Exponential Source
Take the derivative
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Transient Analysis of an RC Circuit with an
Exponential Source
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Transient Analysis of an RC Circuit with an
Exponential Source
The complementary solution is given by
The complete solution is given by the sum of the
particular solution and the complementary
solution
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Transient Analysis of an RC Circuit with an
Exponential Source
Initial conditions 10e0 10 vC(0) 5V vR(0)
vC(0) 10 ? vR(0) 5 i(0) vR/R 5V/1M?
5?A 20e0 Ke0 20 K ? K -15?A
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Transient Analysis of an RC Circuit with an
Exponential Source
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Integrators and Differentiators
Integrators produce output voltages that are
proportional to the running time integral of the
input voltages. In a running time integral, the
upper limit of integration is t .
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If R 10 k?, C 0.1?F ? RC 0.1 ms
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Differentiator Circuit
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Differentiator Circuit