Why volts and amps quit at high frequency
At low frequencies a wire is a wire: clip a voltmeter on each end and you read the same voltage, because the signal fills the whole conductor instantaneously. But a 10 GHz signal has a wavelength of just 3 cm in air — shorter in a cable. Now the voltage at one end of a connector can be a peak while the other end, a centimetre away, sits at a null. 'The' voltage no longer exists; it depends on *where* you look. Worse, to measure current you must break the line, and any probe you insert adds its own stray inductance and capacitance, detuning the circuit you came to inspect. The classic low-frequency two-port parameters — Z, Y, h — all demand that you create a perfect open or short at a port. At microwave frequencies a 'short' is a few millimetres of inductance and an 'open' radiates like a tiny antenna. The lab conditions those parameters assume simply cannot be built.
So RF engineers changed the question. Instead of asking 'what is the voltage and current?', they ask 'when I send a travelling wave into a port, how much bounces back, and how much comes out the other ports?'. Waves are the natural currency of high frequency because they happen to be what *actually* propagates down a transmission line. You never have to break the line or create a short — you just terminate every port in a clean, well-known reference impedance (almost always 50 Ω), launch a wave, and measure the reflections and transmissions with a directional coupler. That measurement is non-invasive, repeatable, and works from DC to hundreds of gigahertz. The numbers it produces are the scattering parameters, or S-parameters.
Reading S-parameters: reflection, transmission, and a matrix
Picture each port as a tube with two waves inside it: an incident wave *a* flowing in, and a reflected wave *b* flowing out. For a two-port device — an amplifier, a filter, a length of cable — that's four waves: a₁, b₁ at the input and a₂, b₂ at the output. The S-parameters are simply the ratios that tell you which outgoing wave each incoming wave produces: b = S·a. Each S-parameter S_ij answers one tidy question — *the wave coming OUT of port i, caused by the wave going IN to port j, with every other port terminated in 50 Ω so nothing reflects back to confuse the measurement.*
Two-port, every port terminated in 50 Ω:
a1 --> <-- a2
+---------------------------+
port 1 | DEVICE | port 2
+---------------------------+
<-- b1 b2 -->
Definition: b = S · a
[ b1 ] [ S11 S12 ] [ a1 ]
[ b2 ] = [ S21 S22 ] [ a2 ]
S11 = b1/a1 (a2=0): INPUT reflection -> how well port 1 is matched
S21 = b2/a1 (a2=0): FORWARD transmission-> gain / insertion loss
S12 = b1/a2 (a1=0): REVERSE transmission-> isolation / leakage backward
S22 = b2/a2 (a1=0): OUTPUT reflection -> how well port 2 is matched
'a2 = 0' just means: port 2 is terminated in 50 Ω, so no wave
is launched back into the device from that side.Two of these dominate everyday RF life. S11 is the input reflection coefficient: it tells you what fraction of the wave you sent in came straight back because the input wasn't perfectly matched. A small |S11| means a good match — power went *in* instead of bouncing off the door. S21 is the forward transmission: for an amplifier it's the gain, for a filter or cable it's the insertion loss. S22 is the output match (the mirror of S11), and S12 is the reverse transmission — usually tiny, and you *want* it tiny, because it measures how much a signal at the output leaks backward toward the source. In an amplifier, small S12 means good isolation: your antenna can't shove noise back into your sensitive front end.
Because S-parameters span an enormous range — a tiny reflection here, a 1000× gain there — they're almost always quoted in decibels: S in dB = 20·log₁₀|S|. The handy intuition: every 20 dB is a factor of 10 in voltage wave amplitude, and every 6 dB is roughly a factor of 2. A return loss of −20 dB means only 1/10 of the wave amplitude (1/100 of the power) bounced back — an excellent match. An S21 of +20 dB means the output wave is 10× the input in amplitude, i.e. 100× the power: 20 dB of gain.
From reflection coefficient to VSWR and return loss
S11 is really the same animal as the reflection coefficient Γ (gamma) you met in the matching rung — they are equal when the reference impedance is your 50 Ω system. Γ is a *complex* number: its magnitude says how much of the wave bounces, and its phase says with what timing. When the load equals the line impedance, Γ = 0 and nothing reflects — a perfect match. When the load is an open or a short, |Γ| = 1 and *everything* reflects. Every real load sits somewhere in between, as a point inside the unit circle |Γ| ≤ 1.
The reflected and incident waves don't just coexist — they interfere, creating a fixed pattern of peaks and troughs along the line called a standing wave. The ratio of the biggest voltage to the smallest is the voltage standing-wave ratio, VSWR. A perfect match gives a flat line: VSWR = 1:1. A total mismatch gives VSWR = ∞:1. VSWR, return loss and |Γ| are three views of one fact — *how much power is bouncing back* — and you should be able to slide between them on sight.
The three faces of one mismatch:
Reflection coefficient: Γ = (Z_L - Z0) / (Z_L + Z0) (= S11)
Return loss (dB): RL = -20·log10|Γ| (positive = good)
VSWR: VSWR = (1 + |Γ|) / (1 - |Γ|)
|Γ| Return loss VSWR Power reflected
---- ----------- ------- ---------------
0.00 infinite 1.00:1 0% (perfect)
0.10 20 dB 1.22:1 1% (excellent)
0.20 14 dB 1.50:1 4% (good)
0.33 9.5 dB 2.00:1 11% (marginal)
0.50 6 dB 3.00:1 25% (poor)
1.00 0 dB infinite 100% (open/short)
Worked: a load Z_L = 75 Ω on a 50 Ω line:
Γ = (75-50)/(75+50) = 25/125 = 0.20
RL = -20·log10(0.20) = 14 dB
VSWR = 1.2/0.8 = 1.5:1 -> 4% of the power reflectsThe Smith chart: the whole impedance plane on one circle
In 1939 Phillip Smith, an engineer at Bell Labs, was tired of grinding through the reflection-coefficient formula by hand. He realised that if you plot Γ as a point on a polar plane — magnitude as distance from the centre, phase as angle — then *every possible impedance in the universe* maps to a point inside the unit circle. Impedances of zero to infinity, which would run off the edge of an ordinary graph, all fold neatly inside one bounded disc. The Smith chart is that disc, with the impedance grid pre-drawn on top of the Γ-plane so you can read both at once. It is, in essence, a slide rule for the equation Γ = (Z−Z₀)/(Z+Z₀).
Learn the landmarks and the chart starts to speak. The centre is Γ = 0, the perfect 50 Ω match — every design wants to land here. The horizontal axis is pure resistance: the centre is 50 Ω, the far-left point is a dead short (0 Ω), the far-right point is an open circuit (∞ Ω). The upper half is inductive (positive reactance), the lower half capacitive (negative reactance). Distance from the centre is |Γ|, so a circle drawn around the centre is a constant-VSWR circle: ride around it and your match quality stays the same while the phase changes. One glance at where a point sits tells you simultaneously the impedance, the reflection coefficient, the return loss and the VSWR.
Smith chart landmarks (impedance chart):
inductive (upper half, +jX)
_____________
/ | \
/ o <- inductive \
short o---+---------(o)---------+---o open
(0 Ω) \ matched CENTRE / (inf Ω)
\ Z0 = 50 Ω, Γ=0 /
\ o <- capacitive /
\_____|_________/
capacitive (lower half, -jX)
Distance from centre = |Γ| -> constant-VSWR circles
Angle around centre = phase of Γ
Constant-RESISTANCE circles: arcs touching the OPEN point
Constant-REACTANCE arcs: arcs fanning from the OPEN point
Normalize first! z = Z / Z0 (so 50 Ω -> 1.0 at the centre)
e.g. 25 + j50 Ω -> z = 0.5 + j1.0 (upper half, left of centre)Moving on the chart: matching as a journey to the centre
Here is where the Smith chart stops being a lookup table and becomes a *design surface*. Every component you add to a network moves your point along a predictable path, and matching becomes a literal navigation problem: get from wherever your stubborn load sits to the centre. Adding a series element slides you along a constant-resistance circle (it changes reactance, not resistance). Adding a shunt element slides you along a constant-conductance circle on the admittance view. A series inductor walks you clockwise *up*; a series capacitor walks you counter-clockwise *down*; shunt elements do the mirror moves. And walking down a length of transmission line simply *rotates* your point clockwise around the centre — a half-wavelength brings you all the way back.
- Plot the load. Normalize the measured impedance (or read S11 straight off the network analyzer) and mark the point. If it's not at the centre, you have a mismatch to fix.
- Pick your path. Decide whether the first element is series or shunt; that tells you which family of circles you slide along toward the centre.
- Add reactance, not resistance. L-match networks use two reactive elements (an inductor and a capacitor) — lossless, so they move you *to* the match without burning power in a resistor.
- Land at the centre. When your point reaches z = 1.0, |Γ| = 0: the source now delivers maximum power into the load. Read off the component values from the arcs you traversed.
This is the deep reason the chart has survived the computer age. Modern tools — Keysight ADS, scikit-rf, your network analyzer's screen — all still draw the Smith chart, because it makes *intuition* visible. A designer can see at a glance that a load is 'too inductive, slightly low resistance' and know instantly that a shunt capacitor followed by a series inductor will spiral it home. The numbers a solver spits out don't build that intuition; the picture does.
A worked example: reading an amplifier's datasheet
Let's put it all together on a real-flavoured part: a 2.4 GHz Wi-Fi gain block, the kind of low-noise amplifier sitting right behind an antenna. Its datasheet hands you four complex S-parameters at the design frequency. Reading them is the daily literacy of an RF engineer — each number is a verdict on one aspect of the device, and together they tell you whether the part will play nicely in your 50 Ω system.
Amplifier S-parameters at 2.4 GHz (Z0 = 50 Ω):
Parameter |Value| in dB Verdict
--------- ------- ---------- --------------------------------
S11 0.18 -15 dB input return loss 15 dB -> good match
S21 5.6 +15 dB forward gain = +15 dB (×31 power)
S12 0.02 -34 dB reverse isolation 34 dB -> excellent
S22 0.25 -12 dB output return loss 12 dB -> OK, tune-able
Convert S11 to VSWR:
|Γ| = 0.18
VSWR = (1+0.18)/(1-0.18) = 1.18/0.82 = 1.44:1 -> good input match
Convert S22 to VSWR:
|Γ| = 0.25
VSWR = 1.25/0.75 = 1.67:1 -> usable, but the output could be matched
better with an L-network nudging it toward the chart centre.
Power check on S21: +15 dB = 10^(15/10) = ~31.6× power gain.
Reverse leakage S12 = -34 dB means a signal at the output comes back to
the input 34 dB weaker (~1/2500 in power): your front end is well isolated.Now you can *act*. The input is already a respectable 1.44:1, so leave it. The output at 1.67:1 is fine but not great; plot S22 on the Smith chart and you'll find it sitting a little below and right of centre — slightly capacitive, resistance a touch high. A shunt inductor and a series capacitor would walk it back to the middle, lifting the output return loss above 20 dB and squeezing a bit more power into the next stage. The forward gain of 15 dB and the 34 dB of reverse isolation you simply *use*: they're what the part promises to deliver, and S-parameters are how that promise is written down, measured, and checked.