Natural and Step
Responses of RLC Circuits
Qi Xuan
Zhejiang University of Technology
Nov 2015
Electric Circuits
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Structure
•  Introduc)on to the Natural Response of a Parallel RLC Circuit •  The Forms of the Natural Response of a Parallel RLC Circuit •  The Step Response of a Parallel RLC Circuit •  The Natural and Step Response of a Series RLC Circuit •  A Circuit with Two Integra)ng Amplifiers Electric Circuits
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An Igni)on Circuit The rapidly changing current in the
primary winding induces via magne)c
coupling (mutual inductance) a very
high voltage in the secondary winding.
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Introduc)on to the Natural Response of a Parallel RLC Circuit The ini)al voltage on the capacitor, V0
represents the ini)al energy stored in
the capacitor. The ini)al current
through the inductor, I0 represents the
ini)al energy stored in the inductor. Second-ordered circuit
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General Solu.on for the Second-­‐
Order Differen.al Equa.on
Two roots:
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Some Useful Nota.ons
Deno)ng the two corresponding solu)ons v1 and v2, respec)vely,
we can show that their sum also is a solu)on. Denoting
(Proof from Eq. 8.10 to 8.12)
Only when s1 ≠ s2
Unit: radians per second (rad/s)
We have
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Over damped: ω02 < α2 , both roots will be real and dis)nct Under damped: ω02 > α2, two roots will be complex and
conjungates Cri.cal damped: ω02 = α2, two roots will be real and equal Electric Circuits
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Example #1
a)  Find the roots of the characteris)c equa)on that governs the transient behavior of the voltage if R = 200 Ω, L=50 mH,and C = 0.2 µF. b)  Will the response be overdamped, underdamped, or cri)cally damped? c)  Repeat (a) and (b) for R = 312.5 Ω. d)  What value of R causes the response to be cri)cally damped? Electric Circuits
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Solu.on for Example #1
a) For the given values of R, L, and C, we have
= 5000 rad/s
= 20000 rad/s
b) In this case, α2 > ω02, thus the voltage response is overdamped.
c) Increase of R will result in decrease of α, thus the voltage
response will be from overdamped to underdamped.
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d) For cri)cal damping, α2 = ω02, so Electric Circuits
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The Forms of the Natural Response of a parallel RLC Circuit
•  Calculate the two roots s1 and s2, based on the given R, L and C; •  Determine whether the response is over-­‐, under-­‐, or cri)cally damped; •  Find the unknown coefficients, such as A1 and A2
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Overdamped Voltage Response
= V0
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We summarize the process for finding the overdamped
response, v(t), as follows: 1.  Find the roots of the characteris)c equa)on, s1 and
s2, using the values of R, L, and C. 2.  Find v(0+) and dv(0+)/dt using circuit analysis. 3.  Find the values of A1 and A2 by solving the following
Equa)ons simultaneously: v(0+) = A1 + A2 dv(0+)/dt = iC(0+)/C = s1A1 + s2A2
4.  Subs)tute the values for s1, s2, A1, and A2 into Eq.
8.18 to determine the expression for v(t) for t > 0. Electric Circuits
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Underdamped Voltage Response
B1
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Real!
B2
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Because α determines how quickly the oscilla)ons subside, it is also
referred to as the damping factor (or damping coefficient).
The oscillatory behavior is possible because of the two types of energy-­‐
storage elements in the circuit: the inductor and the capacitor. Rè∞，αè0
The oscilla)on at ωd = ω0
is sustained! No Dissipa)on on R.
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Overdamped or Underdamped?
•  When specifying the desired response of a second order system, you may want to reach the final value in the shortest .me possible, and you may not be concerned with small oscilla.ons about that final value. If so, you would design the system components to achieve an underdamped response. •  On the other hand, you may be concerned that the response not exceed its final value, perhaps to ensure that components are not damaged. In such a case, you would design the system components to achieve an overdamped response, and you would have to accept a rela.vely slow rise to the final value. Electric Circuits
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Cri.cally Damped Voltage Response
In the case of a repeated
root, the solu)on involves
a simple exponen)al term
plus the product of a linear
and an exponen)al term. You will rarely encounter cri)cally damped systems in prac)ce, largely
because ω0 must equal α exactly. Both of these quan))es depend on
circuit parameters, and in a real circuit it is very difficult to choose
component values that sa)sfy an exact equality rela)onship.
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Example #2
In the given circuit, V0 = 0 and I0 = -12.25 mA. a)  Calculate the roots of the characteris)c equa)on. b)  Calculate v and dv/dt at t = 0. c)  Calculate the voltage response for t ≥ 0. d)  Plot v(t) versus t for the )me interval 0 ≤ t ≤ 11 ms. Electric Circuits
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a)
Solu.on for Example #2
Underdamped
b)
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=0
= 98,000 V/s
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Step Response of a Parallel RLC Circuit Electric Circuits
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The Indirect Approach
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The Direct Approach
=I
=0
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Example #3
The ini)al energy stored in the given circuit is zero. At t = 0, a dc current source of 24 mA is applied to the circuit. The value of the resistor is 400 Ω. a)  What is the ini)al value of iL? b)  What is the ini)al value of diL/dt? c)  What are the roots of the characteris)c equa)on? d)  What is the numerical expression for iL(t) when t > 0? Electric Circuits
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Solu.on for Example #3
a)  No energy is stored in the circuit prior to the applica)on of the dc current source, so the ini.al current in the inductor is zero. The inductor prohibits an instantaneous change in inductor current; therefore iL(0) = 0 immediately a`er the switch has been opened. b)  The ini.al voltage on the capacitor is zero before the switch has been opened; therefore it will be zero immediately a`er. Now, because v = LdiL/dt, we have Electric Circuits
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c)  From the circuit elements, we obtain Overdamped
d)  If = I = 24 mA
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The Natural Response of a Series RLC Circuit
Differen)ate Rearrange
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The Step Response of a Series RLC Circuit
Compare Eq. 8.66 and Eq. 8.41:
the procedure for finding vC
parallels that for finding iL Electric Circuits
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Example #4
No energy is stored in the 100 mH inductor or the 0.4
µF capacitor when the switch in the given circuit is closed. Find vC(t) for t > 0. Electric Circuits
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Solu.on for Example #4
Calculate the two roots:
The roots are complex, so the voltage response is underdamped. Thus, we have
No energy is stored in the circuit ini)ally, so both vC(0) and dvC(0+)/dt are zero, then
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A Circuit with Two Integra)ng Amplifier
Two integra)ng amplifiers connected
in cascade: the output signal of the
first amplifier is the input signal for
the second amplifier. For the first integrator:
For the second integrator:
Second-­‐order
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Example #5
No energy is stored in the given circuit when the input voltage vg jumps instantaneously from 0 to 25 mV. a)  Derive the expression for vo(t) for 0 < t < tsat. b)  How long is it before the circuit saturates? Electric Circuits
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Solu.on for Example #5
a)
Integrate
b) When t = 3 s, the second amplifier
saturates, at that time we get:
= -3 V > -5 V
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The first amplifier doesn’t saturates!
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Summary
•  characteris)c equa)on •  Second-­‐order differen)al equa)on •  Overdamped, underdamped, and cri)cal damped •  Natural Step responses
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