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core30 min studyLesson 290 in Physics

Scattering parameters and network measurements

Define RMS power-normalized port waves, interpret two-port scattering coefficients, and connect calibration, reference planes, and power balance.

Before you begin
On this page

The question for today

What you’ll understand

  • Define incident and outgoing waves using real reference resistances and RMS phasors.
  • Interpret S-parameter indices with other ports matched.
  • Check passive power balance and explain reference-plane and calibration effects.

At microwave frequencies, directly probing an internal voltage can significantly disturb the circuit. Network measurements instead infer incident and outgoing waves at specified ports. Scattering parameters describe their complex ratios, including phase, for a linear device at each frequency. They become physically interpretable only after the reference impedance, wave normalization, port orientation, and reference planes are declared. A numerical S21 without that context is incomplete, even when its magnitude looks like a familiar gain.

Normalize waves so squared magnitude is power

At each port choose current positive into the device and a positive real reference resistance R. Use RMS voltage and current phasors. Defining incident and outgoing amplitudes by the sum and difference of voltage and resistance times current gives amplitudes in square root watts. Their squared magnitudes subtract to give net average power into the device. If the port transmission line is lossless with characteristic impedance equal to R, these quantities coincide with its power-normalized traveling waves.

The definition can also be used as an algebraic port transformation when reference resistance differs from the actual line impedance. In that case the normalized quantities must not automatically be treated as literal physical traveling fields. Complex reference impedances require a carefully specified generalized definition, with conjugation and normalization chosen for the desired power convention. This lesson restricts the calculation to real positive reference resistances, avoiding the false extension of real-line formulas to every lossy-waveguide normalization.

ak=Vk+RkIk2Rk,bk=Vk−RkIk2Rk,Pk=∣ak∣2−∣bk∣2a_k=\frac{V_k+R_kI_k}{2\sqrt{R_k}},\quad b_k=\frac{V_k-R_kI_k}{2\sqrt{R_k}},\quad P_k=|a_k|^2-|b_k|^2
Vk and Ik are RMS phasors in volts and amperes with current into the device, Rk > 0 is a real reference resistance in ohms, and ak,bk have units √W. Pk is net average power entering port k in watts.↔ Scroll wide equations horizontally.

Indices identify output and excitation ports

For a linear time-invariant two-port at a fixed frequency, outgoing waves are a matrix times incident waves. The first index names the observed outgoing port and the second names the excited incoming port. Measuring one matrix column requires zero incident waves at the other port, usually established by a matched termination and accounted for by calibration. S21 is forward transmission from port one to port two; S11 is input reflection under that matched-output condition.

If the output load is mismatched, it returns an incident wave into port two, which can travel back through the device and alter input reflection. The resulting loaded input reflection is generally not equal to bare S11. A linear small-signal scattering matrix also does not fully specify an amplifier at large signal, a mixer translating frequencies, or a nonlinear resonator whose response changes with drive power. Frequency, bias, operating power, and termination conditions belong with the reported data.

(b1b2)=(S11S12S21S22)(a1a2),Sij=biaj∣ak≠j=0\begin{pmatrix}b_1\\b_2\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}a_1\\a_2\end{pmatrix},\qquad S_{ij}=\left.\frac{b_i}{a_j}\right|_{a_{k\ne j}=0}
Sij is dimensionless and complex. The definition assumes fixed frequency, linear response and declared port references. For a two-port, the unexcited port is matched so its incident wave is zero; outgoing power may still leave it.↔ Scroll wide equations horizontally.

Power checks and reference planes constrain interpretation

With only port one excited, the reflected power fraction is the squared magnitude of S11 and the transmitted fraction is the squared magnitude of S21 under this normalization. Their sum cannot exceed one for a passive two-port if all relevant outgoing channels are counted. The missing fraction can be dissipated or escape through unrepresented radiation or modes. A lossless complete network has a unitary scattering matrix, which imposes phase relationships as well as column power sums.

Moving a reference plane through a matched lossless line changes propagation phase but not the relevant coefficient magnitudes. Reflection acquires a round-trip phase, while transmission acquires the sum of the input and output line phases. Calibration estimates systematic instrument and fixture effects so the reported planes correspond to the intended device boundary. It does not eliminate random noise, connector variability, or uncertainty in the standards. Keep cable delay, device dispersion, and genuine absorption distinct when interpreting a resonance measurement.

PreflPinc=∣S11∣2,PtransPinc=∣S21∣2,S†S=I(complete lossless network)\frac{P_{\mathrm{refl}}}{P_{\mathrm{inc}}}=|S_{11}|^2,\quad \frac{P_{\mathrm{trans}}}{P_{\mathrm{inc}}}=|S_{21}|^2,\quad S^\dagger S=I\quad\text{(complete lossless network)}
The power ratios assume port two matched and RMS power-normalized waves with real references. For a passive network S†S ≤ I; equality requires no dissipative or omitted outgoing channels. Reference planes and frequency remain part of the definition.↔ Scroll wide equations horizontally.

Work it through

Read a two-port power budget

A passive two-port with matched port two has |S11| = 0.2 and |S21| = 0.8 at the measurement frequency. Port one receives 10 mW incident power. Find reflected, transmitted, and unaccounted power.

  1. The reflected fraction is 0.22=0.040.2^2=0.04. Reflected power is therefore 0.04 × 10 mW = 0.4 mW.

  2. The transmitted fraction is 0.82=0.640.8^2=0.64. The matched output receives 0.64 × 10 mW = 6.4 mW.

  3. The remaining fraction is 1 − 0.04 − 0.64 = 0.32, or 3.2 mW. If the two ports exhaust all propagating channels, this is internal dissipation; otherwise omitted radiation or modes can also account for it.

The result

Reflected power is 0.4 mW, transmitted power is 6.4 mW, and 3.2 mW is dissipated or escapes through channels absent from this two-port description.

The living notebook

Explore the model

Interactive lab

Interactive lab · vectors

Reference-plane phase is a rotation

Reset and set BOTH Angle to 0 degrees and Magnitude to 1; note the Cartesian components. Then Reset and set BOTH Angle to 90 degrees and Magnitude to 1; predict the same magnitude but rotated components. This static vector illustrates a complex coefficient’s phase change. It does not implement network calibration, multiple ports, S-matrix multiplication, or power-wave normalization.

Reference-plane phase is a rotationPlayback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Current readings: x component: 1.393; y component: 0.975; Magnitude: 1.7. All parameters are adjustable with the labeled controls below.−7.46−3.7303.737.46−3.5−1.7501.753.5xyvvₓ = 1.39
Vectorx / y projectionsTrace marker
x component
1.393
y component
0.975
Magnitude
1.7
t = 0 s

Playback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA changed phase reference can rotate a coefficient without changing its magnitude or implying gain or loss.

Pause & reason

Check your understanding

Under real-reference power normalization, with port two matched, what does |S21|² represent?

Your turn

Try it yourself

At a port with real reference R = 50 Ω, RMS phasors are V = 10 V and I = 0.10 A, in phase, with current into the device. Calculate incident and outgoing wave powers and net accepted power.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Form V + RI and V − RI before dividing by 2√R, then square the amplitudes.

Compare with the solution

Here RI = 5 V. Thus ∣a∣2=(10+5)2/(4×50)=225/200=1.125 W|a|^2=(10+5)^2/(4\times50)=225/200=1.125\,\mathrm W and ∣b∣2=(10−5)2/200=0.125 W|b|^2=(10-5)^2/200=0.125\,\mathrm W. Net power is 1.000 W, equal to Re⁡(VI∗)=10(0.10)\operatorname{Re}(VI^*)=10(0.10). The voltage/current ratio is 100 Ω, so reflection relative to 50 Ω is 1/3 and its power fraction is 1/9, consistent with 0.125/1.125.

Keep the idea

Take it with you

  • Scattering coefficients need a declared wave normalization, frequency, and reference plane.
  • Each matrix column is defined with the other incident port waves set to zero.
  • Power deficits and phase shifts require distinct checks for dissipation, omitted channels, and reference-plane delay.

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