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Higher Order Circuits

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Time Waveforms. If si is a real root, it corresponds to an exponential term ... Initial conditions will determine the values of the constants. Lecture 17. 13. Example ... – PowerPoint PPT presentation

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Title: Higher Order Circuits


1
Higher Order Circuits
2
Higher Order Circuits
  • The text has a chapter on 1st order circuits and
    a chapter on 2nd order circuits.
  • The text has no chapter on 3rd order circuits.
  • Why?

3
Higher Order Circuits are Boring!
  • The behavior of a higher order (3rd or higher
    order) circuit is not qualitatively different
    than that of a 1st or 2nd order circuit.
  • Particular solutions are similar, especially for
    constant and sinusoidal sources.

4
More on Higher Order Circuits
  • The natural response is a sum of decaying
    exponentials and/or exponentially decaying
    sinusoids.
  • The responses of higher order circuits have the
    same sort of characteristics as 1st and 2nd order
    circuits
  • There are more terms in the solution.

5
Mathematical Justification
  • Any voltage or current in an nth order circuit is
    the solution to a differential equation of the
    form

6
Particular Solution
  • The particular solution vp(t) is typically a
    weighted sum of f(t) and its first n derivatives.
  • If f(t) is constant, then vp(t) is constant.
  • If f(t) is sinusoidal, then vp(t) is sinusoidal.

7
Complementary Solution
  • The complementary solution is the solution to
  • Complementary solution has the form

8
Characteristic Equation
  • s1 through sn are the roots of the characteristic
    equation

9
Time Waveforms
  • If si is a real root, it corresponds to an
    exponential term
  • If si is a complex root, there is another complex
    root that is its complex conjugate, and together
    they correspond to an exponentially decaying
    sinusoidal term

10
Example
  • A 3rd order circuit has the following
    characteristic equation
  • s3 6s2 11s 6 0
  • What terms would we expect in the complementary
    solution?

11
Finding Roots of Polynomials with MATLAB
  • We have a polynomial with coefficients 1, 6, 11,
    and 6.
  • The following MATLAB command finds its roots
  • roots(1 6 11 6)

12
Answer
  • The roots of the characteristic equation are
  • -1, -2, and -3
  • The complementary solution is
  • Initial conditions will determine the values of
    the constants.

13
Example
  • A 4th order circuit has the following
    characteristic equation
  • s4 s3 - 2s2 2s 4 0
  • What terms would we expect in the complementary
    solution?

14
Answer
  • The roots of the characteristic equation are
  • -1, -2, 1j, and 1-j
  • The complementary solution is
  • Initial conditions will determine the values of
    the constants.

15
Summary
  • In an nth order linear circuit, any voltage or
    current is the solution to an nth order linear
    constant coefficient differential equation.
  • The particular solution is usually
  • Constant for constant sources (DC SS)
  • Sinusoidal for sinusoidal sources (AC SS)

16
Summary (cont.)
  • The complementary solution is usually a sum of
    decaying exponentials and exponentially decaying
    sinusoids.
  • Time constant
  • Damping ratio and natural frequency

17
Summary (cont.)
  • Transients usually are associated with the
    complementary solution.
  • The actual form of transients usually depends on
    initial capacitor voltages and inductor currents.
  • Steady state responses usually are associated
    with the particular solution.

18
Summary (cont.)
  • You should be able to work problems in which
    capacitors in first order circuits charge and
    discharge.
  • You should be able to find the damping ratio and
    natural frequency in second order circuits and
    determine if they are under damped, over damped,
    or critically damped.

19
Summary (cont.)
  • You should be able to identify the transient and
    steady state portions of a waveform.
  • You should be able to describe the role of
    initial conditions in the transient portion of
    the waveform.
  • You should be able to explain why 3rd and higher
    order circuits dont act qualitatively different
    than 1st and 2nd order circuits.
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