Question

How do you add phasors in AC circuit analysis?

Answer 1

In AC (alternating current) circuit analysis, phasors are used to represent sinusoidal voltage and current quantities as complex numbers, which simplifies calculations and analysis. Adding phasors in AC circuit analysis is essentially a process of algebraic addition, similar to adding vectors. Here's a step-by-step guide on how to add phasors in AC circuit analysis:

Convert to Phasor Representation:
Start by representing the sinusoidal quantities (voltages or currents) as phasors. A phasor is a complex number with a magnitude (amplitude) and a phase angle.

Add the Phasors Algebraically:
To add phasors, simply add the corresponding complex numbers algebraically. Add the real parts together and the imaginary parts together.

Magnitude and Phase Angle:
After adding the phasors algebraically, the resultant phasor can be expressed in polar form as well. Find the magnitude by taking the absolute value of the complex sum and find the phase angle using trigonometric functions.

Convert Back to Time Domain:
Once you've performed the addition in the phasor domain, you might need to convert the result back to the time domain to obtain the time-varying quantity. This can be done using Euler's formula:

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V(t) = Re{V_phasor * e^(jωt)}


Where:

V(t) is the time-varying quantity.
V_phasor is the phasor representation of the quantity.
j is the imaginary unit.
ω is the angular frequency (equal to 2π times the frequency of the AC signal).
e is the base of the natural logarithm.

Here's a simple example of adding two phasors:

Let's say you have two phasors:

Phasor A: 100∠30° (magnitude of 100 units at a phase angle of 30 degrees)
Phasor B: 80∠-45° (magnitude of 80 units at a phase angle of -45 degrees)

To add these two phasors:

Convert them to rectangular form:

Phasor A: 100 * cos(30°) + j * 100 * sin(30°)
Phasor B: 80 * cos(-45°) + j * 80 * sin(-45°)

Add the real and imaginary parts separately:

Real part: 100 * cos(30°) + 80 * cos(-45°)
Imaginary part: 100 * sin(30°) + 80 * sin(-45°)

Calculate the magnitude and phase angle of the resultant phasor:

Magnitude: √(Real^2 + Imaginary^2)
Phase angle: atan(Imaginary / Real)

Convert back to polar form if needed:

Resultant Phasor: Magnitude∠PhaseAngle

Remember that phasor addition assumes that the frequencies of the sinusoidal quantities are the same. If the frequencies are different, additional considerations are necessary.