Phasor of Sinusoidal Steady State Analysis
A rotating vector that represents a Sinusoidal -varying quality is called a phasor. Vector will rotate with an angular velocity equal to the angular frequency of that quantity. Its length represents the amplitude of the quantity, and its projection upon a fixed axis gives the instantaneous value of that quantity.
[Tex]V=Vm<\phi [/Tex]
- where V is the phasor representing the voltage
- Vm is the amplitude of the sinusoidal waveform
- [Tex]<\phi [/Tex] is the phase angle.
Derivation of the Phasor
We know that the phasor representation in sinusoidal steady state analysis is for a linear time-invariant system.
So, Assuming the input is a sinusoidal signal,
[Tex]V(t) = V_0 \sin(\omega t + \phi) [/Tex]
[Tex]cosine formate- V(t) = V_0 \cos(\omega t + \phi_o-\pi/2) [/Tex]
using euler’s identities,
[Tex]e^{j\theta} = \cos(\theta) + j\sin(\theta) [/Tex]
[Tex]V_oe^{j(wt + \phi_o – \frac{\pi}{2})} = V_o\cos(wt + \phi_o – \frac{\pi}{2}) + jV_o\sin(wt + \phi_o – \frac{\pi}{2}) [/Tex]
[Tex]V_oe^{j( \phi_o – \frac{\pi}{2})}e^{j(wt)} = V_o\cos(wt + \phi_o – \frac{\pi}{2}) + jV_o\sin(wt + \phi_o – \frac{\pi}{2}) [/Tex]
[Tex]\text{Re}\{V_o e^{j(\phi_o – \frac{\pi}{2})}e^{j\omega t}\} = V_o \cos(wt + \phi_o – \frac{\pi}{2}) =V(t) [/Tex]
where [Tex]V_s=V_o e^{j(\phi_o – \frac{\pi}{2})}
[/Tex] = phasor.
Now angle Notation:-
[Tex]1 \angle \phi^\circ = 1e^{j\phi}[/Tex]
Sinusoidal Steady State Analysis – Electric circuits
In steady state (the fully charged state of the cap), current through the capacitor becomes zero. The sinusoidal steady-state analysis is a key technique in electrical engineering, specifically used to investigate how electric circuits respond to sinusoidal AC (alternating current) signals. This method simplifies the intricate details involved in time-varying AC circuits by representing voltages and currents as phasors—complex quantities that succinctly convey both amplitude and phase information.
Table of Content
- Sinusoidal Steady State Analysis
- Sinusoidal Source
- Derivation
- V-I Relation for an Inductor
- V-I Relationship for a Capacitor
- Frequency Response
- Bode Plots
- Examples