Complex Impedance, Reactance & Admittance
At DC, resistance is the only opposition to current flow. At RF, inductors and capacitors also oppose current — but in a frequency-dependent and phase-shifting way. The complete description requires complex numbers: impedance Z = R + jX, where the imaginary part X is the reactance.
Phasors and Sinusoidal Steady State
In a circuit driven by a sinusoidal source at frequency f, all voltages and currents are sinusoids at the same frequency. We represent them as phasors — complex numbers whose magnitude gives amplitude and whose angle gives phase. Multiplying a phasor by j rotates it 90° (a quarter cycle), which is exactly what an inductor does: it makes current lag voltage by 90°.
Reactance
Reactance X is the imaginary part of impedance. It stores and releases energy without dissipating it:
| Component | Impedance Z | Reactance X | Phase |
|---|---|---|---|
| Resistor R | R | 0 | V and I in phase |
| Inductor L | jωL | +ωL = +2πfL | I lags V by 90° |
| Capacitor C | 1/(jωC) | −1/(ωC) = −1/(2πfC) | I leads V by 90° |
At resonance (ωL = 1/ωC), the inductive and capacitive reactances cancel and the impedance is purely resistive.
Admittance Y = 1/Z
For parallel circuit analysis, admittance Y = G + jB is more convenient than impedance. G is the conductance (1/R) and B is the susceptance (−1/X for an inductor, +1/X for a capacitor — note the sign flip). On a Smith Chart, the admittance chart is the impedance chart rotated 180°.
Series vs Parallel Equivalents
A real inductor has both inductance L and series resistance R_s. It can be represented equivalently as a parallel combination of L_p and R_p, where the two representations are equivalent at one frequency. This conversion matters when designing matching networks:
Quality Factor Q
The Q factor of a reactive element describes how "ideal" it is — the ratio of energy stored to energy lost per cycle:
High Q means low loss. A typical SMD inductor at 100 MHz has Q = 30–80. A quartz crystal resonator has Q = 10,000–100,000. Q also sets bandwidth: \(BW = f_0/Q\).
Frequency Dependence
At higher frequencies, parasitics dominate. A real capacitor has series inductance (ESL) that causes it to become inductive above its self-resonant frequency (SRF). A real inductor has inter-winding capacitance that causes it to self-resonate. Always check the SRF of a component against your operating frequency:
| Component value | Typical SRF | Notes |
|---|---|---|
| 10 µF electrolytic | 500 kHz | Useless for RF bypassing |
| 100 nF ceramic (0402) | 50 MHz | Good for audio, limited at RF |
| 10 nF ceramic (0402) | 200 MHz | OK for HF/VHF bypassing |
| 100 pF ceramic (0402) | 2 GHz | Good RF bypass to ~1 GHz |
| 10 pF ceramic (0402) | 8 GHz | Good for microwave matching |
| 1 pF ceramic (0402) | 20+ GHz | Usable at mmWave |
Impedance vs Frequency: What to Expect
A 50 Ω transmission line port expects 50 + j0 Ω. Impedance that deviates from this causes reflections. The Smith Chart provides an intuitive visual representation of complex impedance across frequency — a single-frequency impedance is a point, and an impedance varying with frequency traces a curve. Understanding complex impedance is the prerequisite for reading and using the Smith Chart.