Phasor Representation and Complex Numbers in AC Circuits

In alternating current (AC) circuit analysis, the behavior of sinusoidal voltages and currents can be complex due to their time-varying nature. To simplify analysis, engineers commonly use phasors and complex numbers—mathematical tools that transform sinusoidal waveforms into manageable, steady-state quantities.

What Is a Phasor?

A phasor is a complex number that represents a sinusoidal waveform—like voltage or current—as a rotating vector in the complex plane.

A typical sinusoidal voltage is written as:

$$v(t) = V_{\text{peak}} \cdot \sin(\omega t + \phi)$$

Where:

• $V_{\text{peak}}$ = Peak voltage

• $\omega$ = Angular frequency in radians/second

• $\phi$ = Phase angle

• $t$ = Time

In phasor form, this becomes:

$$V = V_{\text{RMS}} \angle \phi$$

Where:

• $V_{\text{RMS}}$ = Root Mean Square value

• $\angle \phi$= Phase angle

Phasor notation transforms a time-varying signal into a constant vector, simplifying circuit analysis.

Why Use Phasors?

Phasors convert differential equations into algebraic equations, allowing easier manipulation of sinusoidal AC signals. Key advantages include:

• Simplified operations: Addition, subtraction, multiplication, and division become straightforward.

• Vector visualization: Phasors show both magnitude and phase.

• Efficient analysis: Engineers can analyze circuits using basic algebra instead of calculus.

Complex Numbers in AC Circuit Analysis

Phasors are expressed using complex numbers, which have both real and imaginary components:

Rectangular Form:

$$Z = x + jy$$Where:

• $x$ = Real part

• $y$ = Imaginary part

• $j$ = Imaginary unit ($\sqrt{-1}$; $j$ is used in electrical engineering to avoid confusion with current, $i$)

Polar Form:

$$Z = |Z| \angle \theta$$

Where:

• $|Z|$ = Magnitude

• $\theta$ = Phase angle (in degrees or radians)

This representation is key to describing voltages, currents, and impedances in AC systems.

Application: Complex Impedance of Circuit Elements

Every passive component in an AC circuit has a complex impedance $Z$, which relates voltage and current phasors using Ohm’s Law:

$$V = I \cdot Z$$Impedances:

• Resistor: $Z_R = R$ (purely real)

• Inductor: $Z_L = j\omega L$ (positive imaginary)

• Capacitor:

$$Z_C = \frac{1}{j\omega C} = -j \cdot \frac{1}{\omega C}$$

(negative imaginary)

Where:

• $\omega = 2\pi f$: Angular frequency

• $f$: Frequency in Hz

These expressions allow for uniform treatment of different components using phasor algebra.

Benefits of Phasor and Complex Number Analysis

Summary Table

Conclusion

Phasor representation and complex numbers are essential tools in AC circuit analysis. They turn time-domain problems into steady-state solutions, making the design, analysis, and troubleshooting of AC systems significantly more efficient and intuitive.