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Electric Circuits IThe Source-Free Parallel RLC
Circuit
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Dr. Firas Obeidat
Dr. Firas Obeidat – Philadelphia University 2
Second Order Differential equations Circuits containing two storage elements are
known as second-order circuits because theirresponses are described by differential equationsthat contain second derivatives.
A second-order circuit may have two storageelements of different type or the same type.
A second-order circuit is characterized by a second-order differential equation. It consists of resistorsand the equivalent of two energy storage elements.
Dr. Firas Obeidat – Philadelphia University 3
The Source-Free Parallel RLC Circuit
Assume initial inductor current Io and initialcapacitor voltage Vo
Our experience with first-order equations might suggest that we at least try the exponential form once more. Thus, we assume
Dr. Firas Obeidat – Philadelphia University 4
The Source-Free Parallel RLC CircuitLet us assume that we replace s by s1
The above equations are satisfythe differential equation
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The Source-Free Parallel RLC Circuit
The roots of the characteristicequation are real and negative.
The roots are real and equal.
In this case the roots are complexand may be expressed as
ωd is called the damped natural frequency
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The Source-Free Parallel RLC CircuitThe constants A1 and A2 in each case can bedetermined from the initial conditions. We needv(o) and dv(0)/dt
(a) Overdamped response,(b) critically damped response,(c) underdamped response.
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The Source-Free Parallel RLC CircuitExample: in the parallel circuit in the shownfigure, find v(t) for t>0, assuming v(o)=5V, i(0)=0,L=1H and C=10mF. Consider these cases:R=1.923Ω, R=5Ω, R=6.25Ω,
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The Source-Free Parallel RLC Circuit
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The Source-Free Parallel RLC Circuit
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The Source-Free Parallel RLC Circuit
Notice that by increasing thevalue of R, the degree ofdamping decreases and theresponses differ.
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The Source-Free Parallel RLC CircuitExample: Find v(t) for t>0 in theRLC circuit of shown figure.
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The Source-Free Parallel RLC Circuit
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