Impedance
Resistance opposes current the same way at every frequency. Reactance opposes current in a frequency-dependent way, but introduces a 90-degree phase shift. In real circuits you almost always have both present at the same time — a resistor in series with a capacitor, or an inductor coil with both reactance and resistance. Impedance is the unified concept that combines resistance and reactance into a single quantity describing the total opposition to AC current at a specific frequency. It is measured in ohms, just like resistance, but it carries both a magnitude and a phase angle.
The impedance triangle. Resistance (R) and net reactance (X = XL − XC) are at right angles to each other. The impedance magnitude (Z) is the hypotenuse: Z = √(R² + X²). The phase angle θ is tan-1(X/R).
View LargerWhy You Cannot Simply Add R and X
When you have two resistors in series, you simply add them: Rtotal = R1 + R2. Why can you not do the same with resistance and reactance?
The reason is phase. Resistance causes voltage and current to be in phase (0° between them). Reactance causes a 90° phase shift between voltage and current. Two voltages that are 90° out of phase cannot be added arithmetically — their instantaneous values do not simply combine the way two in-phase voltages do.
Think of it geometrically: if two forces act at right angles to each other on an object, you cannot find the total force by adding their magnitudes. You use the Pythagorean theorem: total force = √(F₁² + F₂²). Impedance works the same way. Resistance and reactance are perpendicular quantities. Their combination (impedance) is found using the Pythagorean theorem.
The Impedance Triangle
Draw a right triangle with resistance (R) along the horizontal axis and reactance (X) along the vertical axis. The hypotenuse of this right triangle is the impedance magnitude (Z). By the Pythagorean theorem:
Z = √(R² + X²)
where X is the net reactance = XL − XC
Z is measured in ohms (Ω)
The angle between the impedance (hypotenuse) and the resistance (horizontal side) is the phase angle θ (theta). It tells you the phase difference between the total voltage across the circuit and the current through it.
θ = arctan(X / R) = tan-1(X / R)
Positive θ means the circuit is net inductive (voltage leads current)
Negative θ means the circuit is net capacitive (current leads voltage)
θ = 0° means the circuit is purely resistive (voltage and current in phase)
The impedance triangle gives two complete pieces of information about the circuit: its magnitude (how much total opposition to current) and its angle (the phase relationship between voltage and current). Together these constitute the full impedance of the circuit.
Complex Notation: R + jX
Electrical engineers write impedance using complex numbers: Z = R + jX, where j = √(−1) is the imaginary unit (engineers use j instead of the mathematician's i to avoid confusion with current i). This notation keeps track of both the resistive and reactive parts simultaneously.
- The real part (R) is the resistance — the part that dissipates power
- The imaginary part (jX) is the reactance — inductive reactance is positive (+jXL), capacitive reactance is negative (−jXC)
For a circuit with a resistor, inductor, and capacitor in series:
Z = R + j(XL − XC) = R + jX
where X = XL − XC (net reactance)
|Z| = √(R² + X²) (impedance magnitude, in ohms)
θ = tan-1(X/R) (phase angle, in degrees)
You do not need to work extensively with complex number arithmetic for most practical applications — the magnitude formula and phase angle formula handle the calculations you will encounter. But understanding that Z = R + jX is the complete representation of impedance helps make sense of impedance specifications and Smith chart readings.
Series RLC Impedance
In a series RLC circuit, a resistor, inductor, and capacitor are connected end-to-end so the same current flows through all three. The impedance of this circuit is found in three steps:
- Calculate XL = 2πfL
- Calculate XC = 1/(2πfC)
- Find net reactance X = XL − XC, then Z = √(R² + X²)
A series RLC circuit. The same current I flows through all three components. The voltages across individual components (VR, VL, VC) are phase-shifted relative to each other and cannot be added arithmetically — they are added as phasors.
View LargerA series circuit contains R = 50 Ω, L = 2.0 µH, and C = 25 pF. Find the impedance at 7 MHz.
Step 1: XL = 2π × 7 × 106 × 2.0 × 10-6 = 87.96 Ω ≈ 88.0 Ω
Step 2: XC = 1 / (2π × 7 × 106 × 25 × 10-12) = 1 / (1.0996 × 10-3) = 909.5 Ω
Step 3: X = XL − XC = 88.0 − 909.5 = −821.5 Ω (net capacitive)
Step 4: Z = √(50² + 821.5²) = √(2500 + 674,862) = √677,362 = 823.0 Ω
Phase angle: θ = arctan(−821.5 / 50) = arctan(−16.43) = −86.5°
This circuit is strongly capacitive at 7 MHz (C dominates), presenting about 823 ohms of impedance with the current leading the voltage by 86.5°.
For the same circuit (R = 50 Ω, L = 2.0 µH, C = 25 pF), find the frequency at which XL = XC (resonance).
Set 2πfL = 1/(2πfC): f = 1/(2π√LC)
f = 1/(2π√(2.0 × 10-6 × 25 × 10-12))
f = 1/(2π√(5 × 10-17)) = 1/(2π × 2.236 × 10-8) = 22.5 MHz
At 22.5 MHz, XL = XC, so X = 0 and Z = √(50² + 0²) = 50 Ω purely resistive. This is resonance, covered in depth in the Resonance lesson.
An 8-ohm loudspeaker measured at 3 kHz presents an impedance of 7.5 + j2.5 Ω (its voice coil adds inductive reactance as frequency rises). What is the magnitude of the impedance?
Z = √(7.5² + 2.5²) = √(56.25 + 6.25) = √62.5 = 7.91 Ω
The speaker is close to its nominal 8 ohms here, but its impedance is partly inductive. The phase angle is θ = arctan(2.5/7.5) = +18.4° (slightly inductive). At higher frequencies the voice coil's reactance grows further, which is why a speaker's impedance rises at the top of the audio range.
Series RLC Impedance Calculator
Series RLC Impedance Calculator
Enter R, L, C, and frequency. The calculator computes XL, XC, net reactance X, impedance magnitude Z, and phase angle θ.
Ohm's Law for AC Circuits
Once you have impedance Z, Ohm's Law for AC works exactly like DC Ohm's Law, but using impedance instead of resistance:
V = I × Z I = V / Z Z = V / I
Use RMS values for V and I, and the result gives the correct RMS relationship.
A 14.2 MHz signal with an amplitude of 10 V RMS is applied across a series circuit with R = 50 Ω, XL = 75 Ω, and XC = 25 Ω. Find the current.
X = XL − XC = 75 − 25 = 50 Ω
Z = √(50² + 50²) = √5000 = 70.71 Ω
I = V / Z = 10 / 70.71 = 141.4 mA RMS
Phase angle: θ = arctan(50/50) = arctan(1) = 45° (inductive, voltage leads current by 45°)
Impedance in Practice
Standard impedances. Many systems standardize on one impedance so that sources, cables and loads match: 50 ohms for test equipment and high-frequency coaxial cable, 75 ohms for video and TV cable, 4–8 ohms for loudspeakers, 600 ohms for traditional audio lines. When source, cable and load share the same purely resistive impedance, signals transfer without reflections and readings from different instruments agree.
Loudspeaker impedance. A loudspeaker's "8 ohms" is a nominal value. Its real impedance is complex and changes with frequency: it rises sharply at the cone's resonance, dips near its nominal value in the midrange, and climbs again as the voice coil's inductance takes over at high frequencies. Crossover networks and amplifiers must cope with this varying load.
Input impedance of an amplifier or instrument. An oscilloscope input is typically 1 MΩ in parallel with about 15 pF; a microphone preamplifier might present 2 kΩ. The source and the input impedance form a divider, so a high-impedance source connected to a low-impedance input loses much of its signal — the same loading effect you met with meters, now with a frequency-dependent part.
Impedance of an LC network. Filters and matching networks combine inductors and capacitors so that, together with the source and load, they present a chosen impedance at a chosen frequency — for example, transforming the low impedance of a transducer to the impedance its driver expects. The Pi, L and T networks lesson shows how.
Frequently Asked Questions
What does it mean when a load has a purely resistive impedance?
A purely resistive impedance means Z = R + j0 — the reactive part is zero. For a circuit containing inductance and capacitance, this happens at resonance: the reactances cancel exactly, and the load is neither capacitive nor inductive at that frequency. All the power delivered to it is converted to useful work or heat, with no energy sloshing back and forth. A purely resistive load equal to the source impedance also gives a perfect match — for example, 50 + j0 Ω connected to a 50-ohm generator and cable.
Is impedance the same as resistance?
No. Resistance is a special case of impedance where the reactive part is zero (Z = R + j0). Impedance is the more general concept that includes both resistive and reactive components. Resistance is always positive and frequency-independent. Impedance can have any magnitude and any phase angle from −90° to +90°, and changes with frequency whenever reactive components are present. In DC circuits, impedance equals resistance because there is no reactance at zero frequency.
Why does 50 ohms appear so often in electronics?
50 ohms was chosen as a compromise between two competing considerations for coaxial cable: minimum loss (which occurs at about 77 ohms for a standard air-dielectric coax) and maximum power handling capacity (which peaks at about 30 ohms). 50 ohms splits the difference. It was standardized for military and industrial equipment during World War II and has been the default for test equipment and high-frequency cable ever since. Television and video equipment uses 75-ohm coax (closer to the minimum-loss optimum), which is why matching adapters are needed between 75-ohm and 50-ohm equipment.
Test Your Knowledge
Answer the questions below to check your understanding of this lesson. Every answer can be found in the lesson above.