# Steady-State Response of Network to Periodic Signals

The voltage (periodic) is,

*v*(

*t*) =

We want to find out the steady state current,

*i*(*t*). Phasors corresponding to terms in right hand side are,**V**

_{0}=

Let,

**(***Z**jÏ‰*) = Impedance phasor of the network at any frequency*Ï‰*.So, the current phasors are,

**I**

_{0}=

**I**

*=*

_{n}By superposition principle, the net current phasor is,

*i*(

*t*) =

*I*

_{0}+

*I*

_{1}+

*I*

_{2}+ â€¦

So, transforming from frequency domain to time domain,

*i*(

*t*) =

# 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:

We consider a voltage as given below be applied to a pure resistor*Purely Resistive Circuit**R*.We consider a voltage as given below be applied to a pure inductor*Purely Inductive Circuit**L*.We consider a voltage as given below be applied to a pure capacitor*Purely Capacitive Circuit**C*.We consider a voltage as given below be applied to a general*General RLC Series Circuit**RLC*series circuit.

# Steps for Application of Fourier Series to Circuit Analysis

- Fourier series of the given periodic excitation function is obtained.
- The circuit elements are transformed from time domain to frequency domain (i.e.,
*R*â†’*R*,*L*â†’*jÏ‰nL*,*C*â†’ for*n*^{th}harmonic). - The Fourier series of the DC and AC components of the response are calculated.
- 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,

*P**be the average power for the*_{n }*n*^{th}harmonic component.