In AC (alternating current) circuit analysis, phasors are commonly used to represent sinusoidal voltages and currents. Phasors simplify the analysis of AC circuits by converting sinusoidal functions into complex numbers, making calculations involving multiplication and division much easier. Here's how multiplication and division of phasors work:
Multiplication of Phasors:
When you multiply two phasors, you simply multiply their magnitudes and add their angles (or phases). Mathematically, if you have two phasors A and B:
Phasor A: A = A_mag ∠ θ_A
Phasor B: B = B_mag ∠ θ_B
Then the product of A and B, denoted as C = A * B, is:
Phasor C: C = (A_mag * B_mag) ∠ (θ_A + θ_B)
This reflects the fact that the amplitude of the resulting waveform is the product of the amplitudes of the individual waveforms, and the phase shift is the sum of the individual phase shifts.
Division of Phasors:
When you divide one phasor by another, you divide their magnitudes and subtract the divisor's angle (or phase) from the dividend's angle. Mathematically, if you have two phasors A and B:
Phasor A: A = A_mag ∠ θ_A
Phasor B: B = B_mag ∠ θ_B
Then the quotient of A divided by B, denoted as C = A / B, is:
Phasor C: C = (A_mag / B_mag) ∠ (θ_A - θ_B)
This reflects the fact that the amplitude of the resulting waveform is the ratio of the amplitudes of the individual waveforms, and the phase shift is the difference between the individual phase shifts.
It's important to note that these operations assume that the phasors are at the same frequency. If the frequencies are different, you would need to consider additional factors.
These operations are particularly useful when analyzing AC circuits with multiple components like resistors, capacitors, and inductors. By converting sinusoidal functions into phasors, complex calculations involving AC circuit elements become straightforward algebraic operations.