Part 2 · Impedance Matching

Impedance Z = R + jX

One idea: at one frequency, a two-terminal network is fully described by a complex number Z.

Series R with L or C

Z(jω) = R + jωL or R − j/(ωC) · watch |Z| and phase

Type R (Ω) L (nH) f (GHz)

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 = +ωL (inductive). For series C: X = −1/(ωC) (capacitive).

Try it: raise frequency with series L — |Z| and ∠ climb. Switch to RC and raise f — |X| falls toward pure R. Same R, totally different “feel” at RF.

The one idea

At a single frequency, Ohm’s law for sinusoids is \(V = ZI\) with a complex \(Z\). The real part \(R\) dissipates; the imaginary part \(X\) stores and returns energy each cycle (magnetic for \(+X\), electric for \(-X\)).

Z = R + jX,\qquad |Z|=\sqrt{R^2+X^2},\qquad X_L=+\omega L,\quad X_C=-\frac{1}{\omega C}

Why we care for matching

Matching is not only about R. A conjugate match cancels X and sets R correctly. L and C are the knobs that paint X (and, in networks, can also transform R).