S-Parameters

S-parameters (scattering parameters) are the standard way to describe how RF devices behave. They tell you what happens to signals when they “scatter” off your antenna, pass through a filter, or interact with any RF component. The NanoVNA-H measures S11 (reflection) and S21 (transmission).

The Two-Port Model

Think of your device under test as a “black box” with two ports:

flowchart LR
  subgraph DUT[Device Under Test]
      direction LR
      P1((Port 1)) --- Box[Black Box] --- P2((Port 2))
  end

  A[Signal In<br/>a1] --> P1
  P1 --> B[Reflected<br/>b1]
  P2 --> C[Transmitted<br/>b2]

• a1: Signal going INTO Port 1 (from the NanoVNA)
• b1: Signal coming OUT of Port 1 (reflected back)
• b2: Signal coming OUT of Port 2 (transmitted through)

S-parameters are simply ratios of these waves:

Parameter	Definition	What It Tells You
S11	b1 / a1	Reflection coefficient at Port 1
S21	b2 / a1	Transmission from Port 1 to Port 2

S12 and S22

A full 2-port VNA also measures S12 (reverse transmission) and S22 (reflection at Port 2). The NanoVNA-H measures S11 and S21, which covers most ham radio needs. For symmetric devices like filters, S12 = S21 and S22 = S11.

S11: Reflection Coefficient

S11 measures how much signal bounces back when it hits your device. It is a complex number with magnitude and phase:

S11 = |S11| * e^(j*phase)

The NanoVNA displays this in several useful formats:

Return Loss (dB)

Return loss converts the reflection magnitude to decibels:

Return Loss = -20 * log10(|S11|) dB

| Return Loss | |S11| | Power Reflected | Interpretation | |-------------|-------|-----------------|----------------| | 0 dB | 1.00 | 100% | Total reflection (open or short) | | 6 dB | 0.50 | 25% | Poor match | | 10 dB | 0.32 | 10% | Acceptable for many uses | | 14 dB | 0.20 | 4% | Good match | | 20 dB | 0.10 | 1% | Excellent match | | 30 dB | 0.03 | 0.1% | Near perfect |

Rule of Thumb

For HF antennas, aim for at least 10 dB return loss (SWR under 2:1). For VHF/UHF repeater feedlines, aim for 20 dB or better.

SWR (Standing Wave Ratio)

SWR is the classic ham radio metric, derived from S11 magnitude:

SWR = (1 + |S11|) / (1 - |S11|)

The firmware calculates this in plot.c:

static float swr(int i, const float *v) {
  float x = linear(i, v);  // |S11| = sqrt(re^2 + im^2)
  if (x > 0.99f)
    return infinityf();
  return (1 + x)/(1 - x);
}

| SWR | Return Loss | |S11| | Power Reflected | |-----|-------------|-------|-----------------| | 1.0:1 | infinity | 0.00 | 0% | | 1.5:1 | 14 dB | 0.20 | 4% | | 2.0:1 | 10 dB | 0.33 | 11% | | 3.0:1 | 6 dB | 0.50 | 25% | | infinity | 0 dB | 1.00 | 100% |

Phase

S11 phase tells you whether the impedance is inductive or capacitive:

static float phase(int i, const float *v) {
  return (180.0f / VNA_PI) * vna_atan2f(v[1], v[0]);
}

• Positive phase (0 to +180): Inductive reactance (antenna too long, add capacitance)
• Negative phase (0 to -180): Capacitive reactance (antenna too short, add inductance)
• Zero phase: Purely resistive (resonance point)

S21: Transmission Coefficient

S21 measures how much signal passes from Port 1 to Port 2. For passive devices, |S21| is always less than or equal to 1 (signal cannot magically increase).

Insertion Loss (dB)

For filters and cables, we express S21 as insertion loss:

Insertion Loss = -20 * log10(|S21|) dB

| Insertion Loss | |S21| | Power Transmitted | Typical Device | |----------------|-------|-------------------|----------------| | 0 dB | 1.00 | 100% | Perfect thru | | 0.5 dB | 0.94 | 89% | Good filter passband | | 3 dB | 0.71 | 50% | 3 dB attenuator, filter cutoff | | 6 dB | 0.50 | 25% | 6 dB attenuator | | 20 dB | 0.10 | 1% | Filter stopband | | 40 dB | 0.01 | 0.01% | Good filter rejection |

Gain (for Active Devices)

If |S21| > 1, we have gain (amplifiers):

Gain = 20 * log10(|S21|) dB

Power Limits

The NanoVNA outputs about -10 dBm (0.1 mW). Do not connect amplifiers or transmitters that could send power back into the VNA. The input circuits can be damaged by signals above +10 dBm.

From S11 to Impedance

The magic of a VNA is converting S11 (which the hardware measures) to impedance (which you actually want to know). The relationship is:

Z = Z0 * (1 + S11) / (1 - S11)

Where Z0 is the reference impedance (50 ohms). In the firmware (plot.c):

// Resistance: Real part of impedance
static float resistance(int i, const float *v) {
  return get_s11_r(1.0f - v[0], -v[1], PORT_Z);
}

// Reactance: Imaginary part of impedance
static float reactance(int i, const float *v) {
  return get_s11_x(1.0f - v[0], -v[1], PORT_Z);
}

// Impedance magnitude
static float mod_z(int i, const float *v) {
  const float z0 = PORT_Z;
  return z0 * vna_sqrtf(get_l(1.0f + v[0], v[1]) / get_l(1.0f - v[0], v[1]));
}

Understanding the Traces

The NanoVNA-H supports many trace types, all derived from S11 and S21:

S11 Traces

Trace Type	What It Shows	Use Case
LOGMAG	Return loss in dB	Quick SWR check
PHASE	S11 phase angle	Inductive vs capacitive
SWR	Standing wave ratio	Classic ham metric
SMITH	Impedance on Smith chart	Matching network design
R	Resistance (ohms)	Antenna resistance
X	Reactance (ohms)	Antenna reactance
	Z
Q	Quality factor	Component quality

S21 Traces

Trace Type	What It Shows	Use Case
LOGMAG	Insertion loss in dB	Filter response
PHASE	S21 phase angle	Phase shift through device
DELAY	Group delay	Signal delay vs frequency
LINEAR		S21

Group Delay

Group delay measures how long signals take to pass through your device, and whether different frequencies are delayed differently:

Group Delay = -d(phase) / d(frequency)

The firmware calculates this by comparing phase at adjacent frequency points:

float groupdelay_from_array(int i, const float *v) {
  int bottom = (i == 0) ? 0 : -1;
  int top = (i == sweep_points-1) ? 0 : 1;
  freq_t deltaf = get_sweep_frequency(ST_SPAN) / ((sweep_points - 1) / (top - bottom));
  return groupdelay(&v[2*bottom], &v[2*top], deltaf);
}

Practical Examples

Example 1: Dipole Antenna

You measure a 40m dipole at 7.15 MHz:

• S11 magnitude: 0.33 (-9.6 dB return loss)
• S11 phase: +45 degrees
• Calculated: SWR = 2.0:1, Z = 35 + j35 ohms

Interpretation: The antenna is slightly short (positive reactance = inductive), and the resistance is lower than 50 ohms. Lengthen the antenna slightly to reduce reactance, or use a matching network.

Example 2: Bandpass Filter

You measure a 2m bandpass filter:

• At 145 MHz: S21 = -0.5 dB (good passband)
• At 144 MHz: S21 = -1.0 dB (band edge)
• At 130 MHz: S21 = -35 dB (good rejection)

Interpretation: The filter passes the 2m band with minimal loss and provides 35 dB of rejection for out-of-band signals.

Summary

Term	Formula	Range	Good Values
Return Loss	-20*log(	S11	)
SWR	(1+	S11	)/(1-
Insertion Loss	-20*log(	S21	)
Impedance	Z0*(1+S11)/(1-S11)	Complex ohms	50+j0 ohms

Next Steps

Now that you understand S-parameters, learn to visualize them on the Smith Chart---the most powerful tool for impedance matching.