FPGA Based Digital World - Sinusoidal Signal and Phasor

Sinusoidal Signal and Phasor

AC circuit is driven by a source voltage or current which is time-varying, usually in sinusoidal waveform.

Siusoidal Signal

Sinusoid is a signal in the form of sine or cosine function.

Sine wave and cosine wave can be transformed to each other, as listed below.

sin(x) = cos(x-90°), cos(x)=sin(x+90°)

sin_cos_relation

In addition, a sine wave and cosine wave can be added together to form another sinusoidal wave, and an example relationship is listed below.

Acos(x)+Bsin(x)= C cos(x-θ)

where, C=sqrt(A*A+B*b), θ=arctg(B/A)

And it could be also be estimated graphically below.

sinu_wave_add

Phasor

A sine or cosine wave is determined by it below parameters

- Frequency

- Amplitude

- Initial Phase

Once the above parameters are fixed, the wave is uniquely identified.

sine_wave_parameters

The circuit element usually changes the amplitude and phase when a voltage or current passes it. So, a term of phasor is used to describe both the amplitude and phase of a sinusoidal wave.

A phasor is defined as a combination of the amplitude Vm and phase Ф of a sinusoid.

V=Vm*exp(jФ) ,

where

V is the phasor representation of the sisusoid,

Vm is the amplitude and Ф is the phase.

Basically a rotating vector, simply called a “Phasor” is a scaled line whose length represents an AC quantity that has both magnitude (“peak amplitude”) and direction (“phase”) which is “frozen” at some point in time.

Relationship between Phasor and Sinusoid

The relation between a sinusoidal wave to a phasor is shown below.

v1(t)= Vm*cos(ωt+Ф)= Re(Vm*exp(jωt+Ф)) = Re( V*exp(jωt) )

v2(t)= Vm*sin(ωt+Ф)= Im(Vm*exp(jωt+Ф)) = Im( V*exp(jωt) )

So, a sinusoid can be regarded as a phasor multiplied by the time factor of exp(jωt), and then projected to the real or imagniary axis.

Since a phasor has magnitude and phase("direction"), it behaves like a vector, and can be represented in a phsor diagram.

phasor_diagram

The relationship between a phasor and sinusoid can be expressed graphically below(projection on the imaginary axis, as sine wave. if the projection is on the real axis, then the wave is cosine wave).

Below is some example relationship between a time domain sinusoidal signal to a phasor or frequency domain representation.

Vm*cos(ωt+Ф) <--> Vm*exp(Ф)

Vm*sin(ωt+Ф) <--> Vm*exp(Ф-90o)

dv/dt <--> jωV

∫vdt <--> V/jω or -jV/ω

As you know, a complex number multiplied by j is to rotate the complext number counterclockwisely by 90o, while by -j is to rotate the complext number clockwisely by 90o. So, the dv/dt leads v by 90°,while ∫vdt lags v by 90°.

Please note:

1) Phasor V is not time-dependant, while v(t) is;

2) Phasor V is generally complex, while v(t) is real.

v(t) is the real or imaginary part of V, and is real.

Phasor Analysis

Phasor analysis can be used to convert calculus(integration and differentiation) operations into algebraic operations,as mentioned above.

Please note that

1)Phasor analysis only applies to sinusoidal signal;

2)Phasor analysis only applies to constant frequency;

3)Phasor analysis applies in manupulating two or more sinnusoidal signals only if they are of the same frequency;

In a linear, time invariant(LTI)system, there are four basic elements: arithmetic operators(+-*/), scale operation, differentiators, and integrators. Since sinusoidal frequency is not altered by these operations, if the input(s) to a system are sinusoids, then all signals in the system will be sinusoidal at the same frequency.

So phasor analysis can be used in LTI circuit analysis.

Extension Part - Waveforms in Electric Circuit

Sinusoidal wave is the basis of periodic waves, and is the important wave in AC circuit analysis.

Other frequently encountered curves in electric circuit are square wave,triangular wave and saw-tooth wave.

The harmanic series of these waves are to be discussed in Fourier series.

waves_in_circuit


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