FPGA Based Digital World - AC Circuit Analysis

AC Circuit Analysis

An AC circuit is driven by a source voltage or current which is time-varying, usually in sinusoidal waveform. A sinusoid is a signal that has the form of sine or cosine function.

A sinusoidal signal is expressed in parameters

- Amplitude

- Phase

- Freqency

Phasor, as a complex number, is used to represent the amplitude and pahse of the sinusoidal signal. With phasor, the AC circuit(with sinusoidal source) could be analyzed using the same laws applied to DC circuit, for example, KVC, KVL, superposition theorem, Thevenin's and Norton's theorem, etc.In phasor analysis, the frequency of the voltage or current source is assumed to be constant, and focuses on the amplitude with phase.

If the amplitude of the sinusoidal source remains constant and the frequency is varied, the circuit's frequency response is obtained. The freqency response of a circuit is the variation in its behavior with change in signal frequency.

Phasor method transforms the circuit analysis from time domain to frequency domain or phasor domain.The phasor result can be transformed back to time domain.

For periodic but nonsinusoidal source, Fourier Series is a useful tool for analysis. The basic idea is to express a periodic signal in terms of sinusoids, then phasor method can be used to analyze the circuit.

The Fourier series provides the spectrum of a signal, which consists of the amplitudes and phases of the harmonics versus frequency. It helps us to identify the pertinent features of a signal.

To extend the concept of Fourier series from periodic signal to noperiodic signal, Fourier transform is introduced. In Fourier transform, a nonperiodic signal is assumed to be a periodic signal with its period to be infite, thus integral tranform is used.

Phasor method, Fourier series and Fourier transform are suitable to analyze the circuit steady-state. To analyze circuit with intial condition, Laplace transform is introduced. In addtion, the Laplace transform is also capable of providing the total response of the circuit comprising both the natural and forced responses.

In phasor method, Fourier series or Fourier tranform method, the transfer function H(ω) is a useful analytical tool to find the frequency response of a circuit. In fact, the freqency response of a circuit is the plot of the circuit's transfer function versus ω with w varying from ω=0 to ω=∞. A transfer function is defined as the ratio of output response y(ω) to the input excitation x(ω) of a circuit, varing with freqency. Bode plot is a useful tool to plot transfer function in an easier way.

In s domain, the transfer function has a similar definition as

H(S)=Y(s)/X(s),

where Y(s) and X(s) are the response and excitation in s domain, assuming all initial conditions to be zero.

The circuit laws in DC also apply to frequency or s domain.

Laplace transform can only handle circuit with inputs for t>0 with intial conditions, while Forier transform can handle circuit with t<0 as well as those for t>0.


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