Total circuit impedance is represented as a complex number, written in the form Z = R + jX, where R is resistance and X is reactance. This notation captures both how much a circuit resists current flow and how much the voltage and current fall out of sync with each other. Impedance is measured in ohms (Ω) and follows the same basic relationship as Ohm’s law: Z = V/I, where V is voltage and I is current.
The Basic Formula: Ohm’s Law for AC
In a DC circuit, resistance is simply voltage divided by current. Impedance extends that idea to AC circuits, where both voltage and current vary over time. The relationship Z = V/I still holds, but V and I are now complex values that encode not just size but timing. This makes Z itself a complex quantity, which is why representing it requires more than a single number.
Rectangular Form: R + jX
The most common way to express impedance is in rectangular form: Z = R + jX. Here, R is the resistance (the real part) and X is the reactance (the imaginary part). The letter “j” serves as the imaginary unit, used instead of the mathematician’s “i” to avoid confusion with current.
Resistance represents energy that the circuit actually consumes, mostly as heat. Reactance represents energy that gets temporarily stored and returned by inductors and capacitors. A purely resistive circuit like a 15-ohm heating element would be written as Z = 15 + j0. A circuit with only a capacitor providing 30 ohms of capacitive reactance would be Z = 0 − j30. A circuit that combines both, say 8.66 ohms of resistance with 5 ohms of inductive reactance, would be Z = 8.66 + j5 Ω.
The sign of X tells you the circuit’s character. Positive X means the circuit is inductive (coils, motors, transformers). Negative X means it’s capacitive.
Polar Form: Magnitude and Phase Angle
The same impedance can be written in polar form as Z = |Z|e^(jφ), or more practically as a magnitude paired with an angle. The magnitude |Z| tells you the overall opposition to current, calculated as the square root of R² + X². The phase angle φ is found using φ = arctan(X/R).
That phase angle carries real physical meaning. It tells you how far the voltage waveform is shifted in time relative to the current waveform. A positive phase angle means voltage leads current, which happens in inductive circuits. A negative phase angle means current leads voltage, which happens in capacitive circuits. The mnemonic “ELI the ICE man” is a common way to remember this: voltage (E) leads current (I) in an inductor (L), and current (I) leads voltage (E) in a capacitor (C).
Polar form is especially useful when you need to quickly see the overall size of the impedance and the phase relationship. Rectangular form is more convenient when you’re adding impedances together.
How Reactance Depends on Frequency
What makes impedance more dynamic than simple resistance is that reactance changes with frequency. Inductive reactance is calculated as X_L = 2πfL, where f is frequency in hertz and L is inductance. As frequency increases, an inductor opposes current more strongly. Capacitive reactance works in reverse: X_C = 1/(2πfC). A capacitor opposes current less as frequency rises.
This frequency dependence means the total impedance of any circuit containing inductors or capacitors shifts as the signal frequency changes. At one frequency, a circuit might be predominantly inductive. At another, it could be capacitive. At a specific frequency where inductive and capacitive reactances cancel each other out, the impedance becomes purely resistive. This is the basis of resonance.
The Impedance Triangle
A simple geometric tool for visualizing impedance is the impedance triangle. Resistance forms the horizontal side, reactance forms the vertical side, and impedance is the hypotenuse. Inductive reactance points upward (positive direction), while capacitive reactance points downward (negative direction). The net reactance, X_L minus X_C, determines the vertical side’s length and direction.
The angle between the horizontal resistance side and the hypotenuse is the phase angle φ. You can read the triangle to extract both the rectangular and polar forms at a glance: the two legs give you R and X directly, while the hypotenuse length and angle give you the polar representation.
Combining Impedances in a Circuit
How you calculate total impedance depends on whether components are wired in series or parallel. For series circuits, you simply add the impedances together:
Z_total = Z_1 + Z_2
If Z_1 = R_1 + jX_1 and Z_2 = R_2 + jX_2, the result is (R_1 + R_2) + j(X_1 + X_2). You add resistances together and reactances together, then convert to polar form if needed using the magnitude and arctan formulas.
For parallel circuits, the math is more involved:
Z_total = (Z_1 × Z_2) / (Z_1 + Z_2)
This is why polar form exists alongside rectangular form. Multiplication and division of complex numbers are easier in polar form (you multiply magnitudes and add angles), while addition and subtraction are easier in rectangular form. Most real circuit calculations require switching between the two representations depending on the step.
Visualizing Impedance on the Complex Plane
Beyond the impedance triangle, engineers plot impedance directly on the complex plane. The horizontal axis represents resistance (always positive, since circuits don’t generate energy on their own), and the vertical axis represents reactance. Any impedance value occupies a single point on this plane. A series RLC circuit, for example, has its total impedance at the point R + j(Lω − 1/Cω), where ω is the angular frequency.
On this plane, phasors (rotating vectors) represent the time-varying voltages and currents. The voltage across an inductor sits 90 degrees ahead of the current, the voltage across a capacitor sits 90 degrees behind the current, and the voltage across a resistor stays perfectly in phase with the current. These relationships all fall out naturally from the complex impedance representation, which is why the notation exists in the first place. It turns difficult differential equations about oscillating waveforms into straightforward algebra.