Phasors and AC Impedance

Analyzing alternating current (AC) circuits using sine waves in the time domain requires solving complex differential equations. To simplify this, electrical engineers transform time-domain sine waves into the Phasor domain, where variables are treated as complex numbers. This method allows AC circuits to be solved using ordinary algebra, exactly like DC circuits.

Basic Complex Number Notation

To understand phasors, we will use basic complex number notation. In electrical engineering, the imaginary unit is represented by the symbol $j$ (where $j^2 = -1$) instead of $i$ to prevent confusion with the variable for electrical current. A phasor $\mathbf{V}$ can be written in polar form as a magnitude and an angle as follows: $\mathbf{V} = V_m \angle \phi$

The Concept of Phasors

A time-domain sine wave $v(t) = V_m \cos(\omega t + \phi)$ can be transformed into a phasor as $\mathbf{V} = V_m \angle \phi$. (Note: By convention, we reference phasors to the cosine function. If given as a sine function, it must be converted to cosine by subtracting $90^\circ$.)

Impedance ($Z$)

Impedance ($\mathbf{Z}$) is the AC equivalent of resistance. It represents the total opposition to alternating current in a device. Impedance is measured in Ohms ($\Omega$), just like resistance. Because AC circuits involve phase shifts, impedance is a complex number.

The fundamental relationship in the phasor domain is Ohm's law for AC: $\mathbf{V} = \mathbf{I} \mathbf{Z}$

Impedance of Basic Devices

For the three ideal circuit elements, the impedance is defined as follows:

1. Resistor (R): $\mathbf{Z}_R = R$ (Voltage and current are in phase)

2. Inductor (L): $\mathbf{Z}_L = j \omega L$ (Current lags voltage by $90^\circ$)

3. Capacitor (C): $\mathbf{Z}_C = \frac{1}{j \omega C} = -j \frac{1}{\omega C}$ (Current leads voltage by $90^\circ$)

(The reciprocal of impedance is Admittance ($\mathbf{Y} = 1/\mathbf{Z}$), measured in Siemens (S).)

For example, calculating $\mathbf{Z}_L$: $\mathbf{Z}_L = j \omega L = j (314.16) (10 \times 10^{-3})$ $\mathbf{Z}_L = j 3.1416 \, \Omega$

The magnitude of the impedance is $3.1416 \, \Omega$ and the phase angle is exactly $+90^\circ$.

Engineering Check

Notice that the impedance of an inductor ($j \omega L$) increases with frequency, acting like an open circuit at very high frequencies. Conversely, the magnitude of a capacitor's impedance ($1/\omega C$) decreases as frequency rises, making it act like a short circuit at high frequencies. This fundamental behavior is the foundation of all electronic filtering (e.g., low-pass and high-pass filters).