impedance
Impedance is the total opposition that a circuit element or a network presents to the flow of an alternating current, and it is one of the few quantities in electrical engineering that cannot be described by a single ordinary number. A resistor in a direct-current circuit is fully described by its resistance in ohms, but an alternating-current element such as a capacitor, an inductor, or a transmission line opposes current in two different ways at once: a real part that dissipates energy and an imaginary part that stores it and releases it one quarter of a cycle later. Engineers therefore write impedance as a complex number made of a real resistance and an imaginary reactance, and they read it as a magnitude and a phase angle on a plane. Because every signal that travels down an interconnect, through a plasma chamber, or into an antenna port encounters impedance, the concept sits underneath almost every design decision in radio-frequency and high-speed semiconductor work, and a professional reading of it is not optional.
**Impedance is a complex number, not a simple resistance.** The complex number that describes an element or a network has a real component called the resistance and an imaginary component called the reactance, and the two together determine how much current flows and when it flows relative to the driving voltage. A pure resistance such as a metal trace or a termination resistor carries only the real part and keeps voltage and current in phase, while any element that stores energy in an electric or magnetic field carries an imaginary part that shifts the current relative to the voltage. Writing impedance as a complex number is what allows the mathematics of direct-current circuits to be reused almost unchanged for alternating-current ones, which is why every formula that a student first learns for resistors extends naturally to networks of capacitors and inductors.
**The real part dissipates energy, and the imaginary part only stores and returns it.** When current flows through the real resistance of a component, electrical energy becomes heat and never comes back, which is the source of the power lost in a transmission line or a plasma matching network. When current flows through the imaginary reactance of a capacitor or an inductor, energy is stored in an electric or magnetic field and returned to the circuit one quarter of a cycle later, so over a full cycle a pure reactance absorbs no net power. This distinction is why a matched network can have a large imaginary part and still waste nothing, and why the real part alone decides the heat that a connector must survive while the imaginary part decides how the waveform is distorted.
**The magnitude of an impedance is the ratio of voltage to current, and the phase is the time shift between them.** The magnitude, read as the length of the impedance vector on the complex plane, tells an engineer how much current a given voltage drives, and the phase angle tells an engineer how much the current lags or leads the voltage. A load of 50 plus j85.2 ohms at the plasma band has a magnitude of 98.8 ohms and a phase of 59.6 degrees, meaning the current lags the voltage by nearly sixty degrees while the magnitude is close to one hundred ohms. The two numbers together are what an engineer reads off an instrument, and both are needed because a magnitude alone cannot tell a resistive from a reactive load.
**A resistor, an inductor, and a capacitor each produce a different kind of reactance.** A resistor's impedance is real and constant with frequency, an inductor's impedance grows in proportion to frequency because its reactance equals the angular frequency times the inductance, and a capacitor's impedance falls as frequency rises because its reactance is the reciprocal of the angular frequency times the capacitance. At the 13.56 MHz standard plasma excitation band, a 1 microhenry inductor presents an inductive reactance of 85.2 ohms, while a 100 picofarad capacitor presents a capacitive reactance of 117.4 ohms. Because the two reactances move in opposite directions with frequency, a series combination of the two crosses at a single resonance, where the inductive and capacitive reactances cancel and the network briefly looks purely resistive.
**Reactance is the reason a component behaves differently at every frequency.** Because inductive reactance rises with frequency and capacitive reactance falls, the same physical capacitor that looks like an open circuit at a low frequency behaves like a short circuit at a very high one, and the same inductor behaves the other way around. This frequency dependence is the entire basis of filtering, matching, and resonance, and it is why the impedance of a real interconnect or a plasma load changes as the signal spectrum changes. A series combination of 1 microhenry and 100 picofarads resonates near 15.9 MHz, close to the 13.56 MHz plasma band, where the net reactance is small and the network is easily matched to a source.
```flowchart
flowchart TD
A[Measure voltage and current at the port] --> B[Compute magnitude and phase of Z]
B --> C[Separate real R and imaginary X from the vector]
C --> D{Is the imaginary part X near zero?}
D -- yes --> E[Resistive load: match the real part to the reference]
D -- no --> F[Read X: inductive if positive, capacitive if negative]
F --> G[Add series or shunt reactance of opposite sign to cancel X]
G --> H[Re-measure and confirm the point moves toward the center]
H --> C
```
The table below turns the impedances plotted on the complex plane into the magnitude and phase that an engineer reads, and into the matching figure that tells how much power a mismatch reflects at a 50 ohm reference.
| Circuit | Impedance Z | Magnitude | Phase | Notes |
|---|---|---|---|---|
| 50 Ω resistor | 50 + j0 Ω | 50.0 Ω | 0° | pure resistance, matched |
| 1 µH inductor @13.56 MHz | 0 + j85.2 Ω | 85.2 Ω | +90° | reactance rises with f |
| 100 pF capacitor @13.56 MHz | 0 − j117.4 Ω | 117.4 Ω | −90° | reactance falls with f |
| R=50 + 1 µH | 50 + j85.2 Ω | 98.8 Ω | +59.6° | current lags voltage |
| R=50 + L + C | 50 − j32.2 Ω | 59.5 Ω | −32.8° | net capacitive |
The arithmetic of impedance is compact enough to write down, and it is the foundation for everything a matching network does. The impedance is a complex number with a real resistance and an imaginary reactance, and its magnitude and phase follow from the right triangle formed on the complex plane.
$$Z = R + jX, \qquad |Z| = \sqrt{R^2 + X^2}, \qquad \theta = \tan^{-1}\!\left(\frac{X}{R}\right)$$
The reactance of the two reactive elements moves in opposite directions with frequency, which is the mechanism behind resonance and matching.
$$X_L = \omega L, \qquad X_C = \frac{1}{\omega C}, \qquad \omega = 2\pi f$$
At the frequency where the inductive and capacitive reactances are equal in magnitude, a series network reaches resonance and its impedance collapses to the real part alone, which is the condition a matching network is tuned to produce at the operating band.
$$\text{resonance: } \omega L = \frac{1}{\omega C} \quad \Rightarrow \quad f_0 = \frac{1}{2\pi\sqrt{LC}}$$
A 1 microhenry inductor with a 100 picofarad capacitor resonates at 15.9 MHz, and a plasma system tuned at 13.56 MHz sits just below that natural frequency, so the small residual reactance is cancelled by a matching network that adds the opposite sign. In a semiconductor process chamber the matching network between the generator and the plasma is an automatic impedance transformer: it senses the changing impedance of the ignition plasma and adjusts a variable capacitor and inductor to keep the reflected power low as the plasma drifts. The characteristic impedance of a uniform line, which for a lossless line is the square root of the inductance per unit length divided by the capacitance per unit length, sets the reference that the whole system is matched to, and in a coaxial cable the ratio of the outer to inner conductor diameter fixes it, so an air-dielectric cable with a diameter ratio of 2.30 gives 50 ohms and one with 3.50 gives 75 ohms.
The instruments that measure impedance are the everyday tools of a radio-frequency laboratory. Keysight and Rohde & Schwarz impedance analyzers and vector network analyzers drive a signal into a device and recover the complex impedance as a magnitude and a phase across frequency, while Anritsu and Bird field units make the same measurement portable enough for a feed line or an antenna mast. The measurement itself is a ratio of a reflected to a forward wave, and the impedance is recovered from that ratio, which is why network analyzers draw the result as a curve across the complex plane. SMA and N-type connectors from Belden and Times Microwave are the ports these measurements are made through, and the impedance they are designed around is the near-universal 50 ohm reference that every matching figure in this treatment assumes.
The numbers that make impedance concrete are easy to remember once they are tied to hardware. A 50 ohm feed line carrying 100 W of forward power at 13.56 MHz reflects 4.0 percent of it at a 75 ohm load, because a 75 ohm load on a 50 ohm reference has a reflection coefficient magnitude of 0.200, and the same line reflects 11.1 percent at a 100 ohm load, where the coefficient is 0.333. A 1 microhenry inductor at 13.56 MHz shows 85.2 ohms of inductive reactance, and a 100 picofarad capacitor shows 117.4 ohms the other way, so the series combination cancels to leave a small net reactance that a matching network removes. A voltage standing wave ratio of 1.5 to 1 corresponds to that 4.0 percent reflection and a return loss of 14.0 dB, while a ratio of 2 to 1 corresponds to 11.1 percent and about 9.5 dB. The power factor that a reactive load presents is the cosine of its phase angle, so the 50 plus j85.2 ohm load delivers a power factor of 0.506 and the 50 minus j32.2 ohm load a power factor of 0.841, the higher figure meaning more of the apparent power actually reaches the load. A quarter-wave section placed between a 75 ohm load and a 50 ohm line needs a characteristic impedance of 61.2 ohms, the geometric mean of the two, which is why a short length of line can transform one impedance into another without adding reactance. A generator that trips a protection at 50 V of reflected amplitude is reading the real part of a complex mismatch, and a matching network that trims the reflected power from 11.1 percent down to 0.2 percent turns a 1000 W system from a waste of 111 W into a clean delivery. Across a 915 MHz industrial band or a 27 MHz plasma line the same reactance arithmetic applies at the new frequency, and only the component values change.
Read impedance through a *phasor* lens rather than a *resistance-only* lens: an impedance is a vector on the complex plane with a magnitude and a phase, and it carries more information than a resistance because it describes not only how much current flows but exactly when it flows. An engineer who treats a load as a single resistance is reading half the story, because the reactive part that stores and returns energy is what a matching network exists to cancel. The professional habit is to read the magnitude for how hard a source must work, to read the phase for which direction the match must move, and to know that the 85.2 ohms of inductive reactance, the 117.4 ohms of capacitive reactance, and the 98.8-ohm magnitude at a 59.6-degree phase are not separate facts but one complex number viewed from three angles.