ELE1110A Tutorial Notes

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ELE1110A Tutorial Notes

• Second-Order Circuits
• by Tian Feng

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1.Typical Examples of Second-order Circuits

• Typical examples of second-order circuits
  includes
• (a) Series RLC circuit, (b) Parallel RLC circuit,
  (c) RL circuit, (d) RC circuit

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2. How to get v(0), i(0), dv(0)/dt, di(0)/dt,
v(8), i(8)

• Main points that should be take care of
• We must carefully handle the polarity of voltage
  v (t) across the capacitor and the direction of
  the current i (t) through the inductor.
• Keep in mind that v and i are defined strictly
  according to the passive sign convention. One
  should carefully observe how these are defined
  and apply them accordingly.
• Keep in mind that the capacitor voltage is always
  continuous so that V(0)V(0-) where t 0-
  denotes the time just before the switching event,
  and t 0 is the time after the switching event
  and the inductor current is always continuous so
  that i(0)i(0-)

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Classwork 1

• Determine (a) i(0) and v(0)(b) di(0)/dt
  and dv(0)/dt (c) i(8) and v(8).

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3. The Source-free Series RCL Circuit

• A source-free series RCL circuit occurs when
  its dc source is suddenly disconnected. The
  energy initially stored in the capacitor and
  inductor is released to the resistor. A
  source-free RCL circuit is typically constructed
  as depicted in Fig. 3, comprising only three
  elements R, C and L.

The final equation is
The initial conditions are v(0)v(0-)v0,
i(0)i(0-)i0, and
Therefore the final solution is
Where s1 and s2 are solutions of
Where A1 and A2 are to be determined.
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Discussions
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Conclusions

• The behavior is characterized by the idea of
  damping, which is the gradual loss of the initial
  stored energy. The damping is due to the presence
  of the resistance R. If R 0, we have an LC
  circuit with as the oscillation frequency that is
  lossless.
• Oscillatory response is possible due to the
  presence of the two types of storage elements.
  Having both L and C allows the flow of energy
  back and forth between the two elements. The
  damped oscillation exhibited by the underdamped
  response is known as ringing.
• The waveforms of the responses differ. The
  critical damped response decays faster than the
  overdamped response.

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4. The Source-free Parallel RCL Circuit
The equations are
where
and A1 and A2 are to be determined.
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Discussions

• Overdamped case(agt?0)
• Critically damped case (a?0)
• Underdamped case (alt?0)

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5. Step Response of a Series RCL Circuit
The final equation is
1)vn is the natural (or transient) response and
dependent on time. And has been discussed
before, for three different cases. 2) vf is the
forced (or steady-state) response, independent of
time. Because the voltages drop across the
inductor and resistor will be zero after
switching for long time.
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Classwork 2

• For the circuit in following figure, find v(t)
  and i(t) for tgt0. Consider the cases R5ohm,
  R4ohm and R1ohm.

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6. Step Response of a Parallel RCL Circuit
For the above circuit, the equations
1)in is the natural (or transient) response and
dependent on time. And in has been discussed
before, for three different cases. 2) if is the
forced (or steady-state) response, independent of
time Because the currents through the
capacitor and resistor will be zero after
switching for long time.
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Classwork 3
In the circuit, find i(t) and iR(t) for tgt0
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7. General Second-Order Circuits

• We first determine the initial conditions x(0)
  and dx(0)/dt and the final value of x(8).
• We find the natural response xn(t) by switching
  off independent sources and applying KCL and KVL
  and let icCdVc./dt, VLLdiL/dt. Once a
  second-order differential equation is obtained,
  we can find the standard solutions for xn(t) by
  examining a and ?0.
• We obtain the forced (steady-state) response as
• The total response is the sum of the natural
  response and forced response

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Classwork 4

• Determine v and i for tgt0 in the circuit.