Impedance and Q Factor

Understanding impedance and Q factor is essential for antenna work, filter design, and component measurement. The NanoVNA-H can measure these parameters directly, but knowing what they mean helps you interpret the results and make better design decisions.

Impedance: R + jX

Impedance (Z) is the AC equivalent of resistance. It has two components:

Z = R + jX

• R (Resistance): The real part - dissipates power as heat
• X (Reactance): The imaginary part - stores and releases energy
• j: The imaginary unit (engineers use j instead of i)

Resistance (R)

Resistance is what you measure with an ohmmeter, but at RF frequencies, additional effects appear:

• Skin effect: Current flows only near the conductor surface, increasing R at high frequencies
• Radiation resistance: In antennas, the “resistance” that represents radiated power
• Loss resistance: Unwanted heating in conductors and materials

The NanoVNA calculates resistance from S11:

static float resistance(int i, const float *v) {
  // Z = Z0 * (1 + S11) / (1 - S11), extract real part
  return get_s11_r(1.0f - v[0], -v[1], PORT_Z);
}

Reactance (X)

Reactance is the opposition to current flow from energy storage elements:

Inductive Reactance (+jX)

XL = 2 * pi * f * L = omega * L

• Increases with frequency
• Positive sign (above the horizontal axis on Smith chart)
• Current lags voltage by 90 degrees

Capacitive Reactance (-jX)

XC = -1 / (2 * pi * f * C) = -1 / (omega * C)

• Decreases with frequency (becomes less negative)
• Negative sign (below the horizontal axis on Smith chart)
• Current leads voltage by 90 degrees

Sign Convention

Engineers use +jX for inductors and -jX for capacitors. This matches the Smith chart where inductive loads appear in the upper half and capacitive loads in the lower half.

Series vs Parallel Impedance

Components can be connected in series or parallel, and the NanoVNA can display both representations.

Series Model (Default)

When components are in series, impedances add directly:

Z_series = R + jX

The NanoVNA displays:

• R: Series resistance
• X: Series reactance
• sL: Series inductance (calculated from +X)
• sC: Series capacitance (calculated from -X)

static float series_l(int i, const float *v) {
  const float zi = reactance(i, v);  // Get X
  const float w = get_w(i);          // omega = 2*pi*f
  return zi / w;                     // L = X / omega
}

static float series_c(int i, const float *v) {
  const float zi = reactance(i, v);
  const float w = get_w(i);
  return -1.0f / (w * zi);           // C = -1 / (omega * X)
}

Parallel Model

For parallel circuits, use admittance (Y = 1/Z = G + jB):

Y_parallel = G + jB

Where:

• G (Conductance): 1/Rp (inverse of parallel resistance)
• B (Susceptance): Parallel reactive component

The NanoVNA displays:

• Rp: Parallel resistance
• Xp: Parallel reactance
• pL: Parallel inductance
• pC: Parallel capacitance
• G: Conductance
• B: Susceptance

static float parallel_r(int i, const float *v) {
  return 1.0f / conductance(i, v);
}

static float parallel_x(int i, const float *v) {
  return -1.0f / susceptance(i, v);
}

When to Use Each Model

Series Model

Use the series model when:

• Measuring components in series with the transmission line
• Analyzing series resonant circuits
• The component is between the signal path and ground
• Reactance dominates over resistance

Parallel Model

Use the parallel model when:

• Measuring components across (shunting) the line
• Analyzing parallel resonant circuits
• Designing shunt matching networks
• Working with tank circuits

Resonance

Resonance occurs when inductive and capacitive reactances cancel:

XL + XC = 0
omega * L = 1 / (omega * C)
f_resonance = 1 / (2 * pi * sqrt(L * C))

At resonance:

• Total reactance is zero (X = 0)
• Impedance is purely resistive
• On the Smith chart, the point lies on the horizontal axis
• For antennas, this is often the target operating frequency

Finding Resonance

To find an antenna’s resonant frequency, look for where the X trace crosses zero, or where the Smith chart trace crosses the horizontal axis. The marker search function can locate this automatically.

Q Factor (Quality Factor)

The Q factor measures how “sharp” or “selective” a resonant circuit is. Higher Q means:

• Narrower bandwidth
• More energy stored vs. dissipated
• Sharper frequency response

Definition

Q = |X| / R = (Energy stored per cycle) / (Energy dissipated per cycle)

The NanoVNA calculates Q directly from S11:

static float qualityfactor(int i, const float *v) {
  const float r = 1.0f - get_l(v[0], v[1]);  // Related to resistance
  const float x = 2.0f * v[1];               // Related to reactance
  return vna_fabsf(x / r);
}

Interpreting Q Values

Q Value	Interpretation	Typical Application
< 10	Low Q	Wideband antennas, lossy components
10-50	Moderate Q	General-purpose inductors
50-200	High Q	Quality RF inductors, crystals
> 200	Very High Q	Precision crystals, cavity resonators

Q and Bandwidth

For a resonant circuit, Q relates directly to bandwidth:

BW = f_center / Q

Example: A 7 MHz antenna with Q = 20 has a bandwidth of 350 kHz (7000/20).

Q at Different Frequencies

Q varies with frequency. A component may have Q = 100 at 1 MHz but Q = 30 at 100 MHz due to skin effect and parasitic elements. Always measure at your operating frequency.

Practical Examples

Example 1: Measuring an Inductor

You measure an unknown inductor at 10 MHz:

• R: 5 ohms (series resistance)
• X: +150 ohms (inductive)

Calculations:

L = X / (2 * pi * f) = 150 / (2 * pi * 10e6) = 2.39 uH
Q = X / R = 150 / 5 = 30

Example 2: Antenna at Resonance

You measure your 40m dipole at 7.15 MHz:

• R: 72 ohms
• X: 0 ohms (resonant)

The antenna is resonant at this frequency. The 72 ohms represents radiation resistance plus loss resistance. No matching network is needed if your feedline can tolerate slight mismatch (SWR = 1.44:1).

Example 3: Crystal Measurement

You measure a 10 MHz crystal:

• At series resonance: R = 15 ohms, X = 0
• Just above series resonance: X increases rapidly

Calculations:

Q = f_s / BW_3dB

Crystals typically have Q > 10,000, giving extremely narrow bandwidth.

Converting Between Series and Parallel

Sometimes you need to convert between representations:

Series to Parallel:

Rp = Rs * (1 + Q^2)
Xp = Xs * (1 + 1/Q^2)

Parallel to Series:

Rs = Rp / (1 + Q^2)
Xs = Xp / (1 + Q^2)

Where Q = |X|/R for either representation.

High-Q Approximation

When Q > 10, the formulas simplify considerably:

• Rp = Q^2 * Rs
• Xs = Xp

Summary

Parameter	Symbol	Unit	What It Tells You
Resistance	R	Ohms	Power dissipation
Reactance	X	Ohms	Energy storage (+L, -C)
Impedance	Z	Ohms	Total opposition to current
Conductance	G	Siemens	1/R parallel
Susceptance	B	Siemens	Parallel reactive component
Q Factor	Q	Dimensionless	Selectivity, sharpness

Next Steps

Now that you understand impedance and Q, learn how the NanoVNA corrects measurement errors in Calibration Error Model.