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Steady-State Response of Network to Periodic Signals

The voltage (periodic) is,
v(t) = Description: 3622.png
 
We want to find out the steady state current, i(t). Phasors corresponding to terms in right hand side are,
V0Description: 3628.png
 
Let, Z () = Impedance phasor of the network at any frequency ω.
 
So, the current phasors are,
I0Description: 3637.png
InDescription: 3643.png
 
By superposition principle, the net current phasor is,
 
i(t) = I0 + I1 + I2 + …
 
So, transforming from frequency domain to time domain,
 
i(t) = Description: 3649.png

Steady State Current in Series Circuits

It is known that when a sinusoidal voltage is applied to a single phase series circuit, the resulting current will also be a sinusoidal. But if an alternating voltage containing various harmonics is applied to such a circuit, each harmonic voltage will produce a component current independent of the others and the resulting current will be the phasor sum of all the harmonic currents. The wave-shape of the current may altogether be different from the wave-shape of the applied voltage.
 
We consider the following four series circuits:
  1. Purely Resistive Circuit We consider a voltage as given below be applied to a pure resistor R.
     
    Description: 3692.png
  2. Purely Inductive Circuit We consider a voltage as given below be applied to a pure inductor L.
     
    Description: 3706.png
  3. Purely Capacitive Circuit We consider a voltage as given below be applied to a pure capacitor C.
     
    Description: 3742.png
  4. General RLC Series Circuit We consider a voltage as given below be applied to a general RLC series circuit.
    Description: 3772.png

Steps for Application of Fourier Series to Circuit Analysis

  1. Fourier series of the given periodic excitation function is obtained.
  2. The circuit elements are transformed from time domain to frequency domain (i.e., R  RL  jωnLC  Description: 3800.png for nth harmonic).
  3. The Fourier series of the DC and AC components of the response are calculated.
  4. Using Superposition, the Fourier series of the response is obtained by summing up the individual DC and AC response components.

Power Spectrum

It is the distribution of the average power over the different frequency components.
 
Let, Pn be the average power for the nth harmonic component.




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