Impedance Z = R + jX — Impedance Matching

Along the way

1
One frequency, one \(Z\). Sinusoidal Ohm’s law is \(V = ZI\). The phasor is the whole story at that \(f\) — nothing more is hiding.
2
\(R\) is heat (emerald). Real part, in phase with current. Same \(R\) we matched in Part 1.
3
\(X\) stores and returns (rose). \(+X\) is inductive (\(X_L = \omega L\)); \(-X\) is capacitive (\(X_C = -1/\omega C\)). No net energy per cycle, but it counts in \(|Z|\) and phase.
4
Sweep \(f\) on the charts below. Series L: \(|Z|\) and \(\angle Z\) climb. Series C: \(|X|\) falls toward pure \(R\). Same \(R\), different RF feel.
5
Matching will cancel \(X\). A conjugate match sets \(X_L = -X_S\) and then gets \(R\) right. L and C are the knobs that paint \(X\).

Series RL or RC only — enough to see \(R\) vs \(X\). Networks that transform \(R\) come in Parts 4–6.

Next: Part 3 — the same \(Z\) as a point \(\Gamma\) on the unit disk.

Series R with L or C

\(Z(j\omega) = R + j\omega L\) or \(R - j/(\omega C)\) · phasor + sweep

Type R (Ω) L (nH) f (GHz)

R (real) X (imag) Z phasor this f

Phasor · Re = R · Im = X

R—

X—

|Z|—

∠Z—

\[ \omega = 2\pi f \]

= —

\[ X = +\omega L \]

= —

\[ Z = R + jX \]

= —

\[ |Z| = \sqrt{R^2 + X^2} \]

= —

\[ \angle Z = \mathrm{atan2}(X,R) \]

= —

For series L: \(X = +\omega L\) (inductive). For series C: \(X = -1/(\omega C)\) (capacitive).

\(|Z|(f)\) · amber line = this frequency

\(\angle Z(f)\) · + = inductive · − = capacitive

Try it: “50 + j25 @ 1 GHz”, then raise \(f\) — the phasor climbs and \(|Z|(f)\) keeps rising. Switch to series RC and raise \(f\) — \(|X|\) collapses toward \(R\). Same resistor, opposite frequency story.

Analog Intuition

Impedance \(Z = R + jX\)

Series R with L or C