Second-Order Circuits (6.3)

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Second-Order Circuits (6.3)

• Prof. Phillips
• April 7, 2003

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2nd Order Circuits

• Any circuit with a single capacitor, a single
  inductor, an arbitrary number of sources, and an
  arbitrary number of resistors is a circuit of
  order 2.
• Any voltage or current in such a circuit is the
  solution to a 2nd order differential equation.

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Important Concepts

• The differential equation
• Forced and homogeneous solutions
• The natural frequency and the damping ratio

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A 2nd Order RLC Circuit
i (t)
R

C
vs(t)
L

• The source and resistor may be equivalent to a
  circuit with many resistors and sources.

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Applications Modeled by a 2nd Order RLC Circuit

• Filters
• A lowpass filter with a sharper cutoff than can
  be obtained with an RC circuit.

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The Differential Equation
vr(t)

i (t)

R


vc(t)
C
vs(t)

vl(t)


L

• KVL around the loop
• vr(t) vc(t) vl(t) vs(t)

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Differential Equation
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The Differential Equation

• Most circuits with one capacitor and inductor are
  not as easy to analyze as the previous circuit.
  However, every voltage and current in such a
  circuit is the solution to a differential
  equation of the following form

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Important Concepts

• The differential equation
• Forced and homogeneous solutions
• The natural frequency and the damping ratio

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The Particular Solution

• The particular (or forced) solution ip(t) is
  usually a weighted sum of f(t) and its first and
  second derivatives.
• If f(t) is constant, then ip(t) is constant.
• If f(t) is sinusoidal, then ip(t) is sinusoidal.

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The Complementary Solution

• The complementary (homogeneous) solution has the
  following form
• K is a constant determined by initial conditions.
• s is a constant determined by the coefficients of
  the differential equation.

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Complementary Solution
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Characteristic Equation

• To find the complementary solution, we need to
  solve the characteristic equation
• The characteristic equation has two roots-call
  them s1 and s2.

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Complementary Solution

• Each root (s1 and s2) contributes a term to the
  complementary solution.
• The complementary solution is (usually)

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Important Concepts

• The differential equation
• Forced and homogeneous solutions
• The natural frequency and the damping ratio

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Damping Ratio (?) andNatural Frequency (?0)

• The damping ratio is ?.
• The damping ratio determines what type of
  solution we will get
• Exponentially decreasing (? gt1)
• Exponentially decreasing sinusoid (? lt 1)
• The natural frequency is w0
• It determines how fast sinusoids wiggle.

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Roots of the Characteristic Equation

• The roots of the characteristic equation
  determine whether the complementary solution
  wiggles.

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Real Unequal Roots

• If ? gt 1, s1 and s2 are real and not equal.
• This solution is overdamped.

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Overdamped
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Complex Roots

• If ? lt 1, s1 and s2 are complex.
• Define the following constants
• This solution is underdamped.

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Underdamped
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Real Equal Roots

• If ? 1, s1 and s2 are real and equal.
• This solution is critically damped.

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Example
i (t)
10W

769pF
vs(t)
159mH

• This is one possible implementation of the filter
  portion of the IF amplifier.

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More of the Example

• For the example, what are z and w0?

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Example continued

• z 0.011
• w0 2p455000
• Is this system over damped, under damped, or
  critically damped?
• What will the current look like?

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Example (cont.)

• The shape of the current depends on the initial
  capacitor voltage and inductor current.

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Slightly Different Example
i (t)
1kW

769pF
vs(t)
159mH

• Increase the resistor to 1kW
• What are z and w0?

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Example cont.

• z 2.2
• w0 2p455000
• Is this system over damped, under damped, or
  critically damped?
• What will the current look like?

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Example (cont.)

• The shape of the current depends on the initial
  capacitor voltage and inductor current.

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Damping Summary
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Class Examples

• Learning Extension E6.9
• Learning Extension E6.10