14.2 Antenna Analysers, Vector Network Analysers, and Smith Charts

14.2 Antenna Analysers, Vector Network Analysers, and Smith Charts

Optimising an antenna system requires understanding the complex impedance presented at its feedpoint. While a simple inline SWR meter provides a basic indication of forward versus reflected power under transmit conditions, it offers zero insight into whether an antenna is electrically too long or too short, nor can it identify reactive components required for matching networks. Modern radio amateurs utilize scalar antenna analysers and Vector Network Analysers (VNAs) to inspect impedance across wide frequency bands, mapping complex impedance onto the Smith Chart and locating transmission-line discontinuities from their reflection delay.

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Operating Principles of RF Impedance Analysers

             Scalar Bridge                         Vector Measurement

 Source --> Bridge --> DUT                   Source --> DUT --> Port2 receiver
              |                                |        |
      Multiple amplitude samples          Reference  Reflected
              |                             channel    channel
       Scalar processing                       \        /
              |                           Complex receiver ratios
       VSWR; R, |X| or |Z|                       |
       when supported                     S11 and S21, magnitude/phase

 Scalar amplitude data can leave +X/-X ambiguous. The vector system needs
 incident, reflected and transmitted channels for forward two-port tests.

1. Scalar Antenna Analysers

A scalar antenna analyser integrates a low-power variable RF source (such as a Direct Digital Synthesis, DDS, generator) with an internal Wheatstone bridge or directional coupler and diode detectors. The instrument excites the unknown load with a known RF voltage and samples the reflected voltage wave:

• The detector measures the magnitude of the voltage reflection coefficient ($|\Gamma|$):

$|\Gamma| = \frac{|V_{\text{reflected}}|}{|V_{\text{forward}}|}$

• From $|\Gamma|$, the instrument computes Voltage Standing Wave Ratio (VSWR):

$\text{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}$

• Additional bridge-voltage measurements can provide $R$, $|X|$ and $|Z|=\sqrt{R^2+X^2}$ in instruments equipped for them. These do not follow from $|\Gamma|$ alone, which specifies only a constant-VSWR circle.

The Critical Blindness of Scalar Analysers:

Without a phase-sensitive reference, the scalar bridge does not directly determine the sign of the voltage/current phase angle. Even where several amplitude samples allow $R$ and $|X|$ to be calculated, the two reactance signs remain ambiguous:

• A scalar analyser cannot determine whether reactance is inductive ($+jX$) or capacitive ($-jX$).
• If an antenna presents $50 + j25\ \Omega$ (electrically too long) or $50 - j25\ \Omega$ (electrically too short), a scalar analyser reports an identical VSWR of $1.6:1$ and $|Z| = 55.9\ \Omega$.
• The sign can often be inferred by sweeping frequency around a simple antenna resonance. A small known change in antenna length and its effect on the reading is another test.

2. Vector Network Analysers (VNAs)

A Vector Network Analyser (VNA)—exemplified in amateur radio by the compact, dual-port NanoVNA and high-end laboratory instruments—measures both magnitude and phase of reflected and transmitted waves across a designated frequency sweep. A VNA treats the device under test (DUT) as an electrical network described by Scattering Parameters (S-parameters):

                          Two-Port Network (S-Parameters)

          Port 1 (Input)                             Port 2 (Output)
          Incident a_1 --->  +-----------------+  ---> Transmitted b_2
                             |    Device Under | 
          Reflected b_1 <--- |     Test (DUT)  |  <--- Incident a_2
                             +-----------------+

            S_11 = b_1 / a_1, S_21 = b_2 / a_1, both with a_2 = 0
            (Port 2 terminated in the reference impedance.)

• $S_{11}$: With Port 2 matched to $Z_0$ ($a_2=0$), $b_1/a_1$ is the input reflection coefficient. For a one-port antenna, this gives feedpoint impedance, return loss and VSWR at the calibrated reference plane.
• $S_{21}$: Under the same matched-output condition, $b_2/a_1$ is the forward transmission coefficient, used for filter response, insertion loss, amplifier gain and cable attenuation.

Quadrature I/Q Down-Conversion Architecture

Inside the VNA, a reference channel samples the incident signal, while measurement channels sample reflected ($S_{11}$) and transmitted ($S_{21}$) signals. Each channel feeds an I/Q (In-phase and Quadrature) mixer driven by a local oscillator. By multiplying the incoming RF by $\cos(\omega t)$ and $\sin(\omega t)$, the VNA resolves the response into real ($I$) and imaginary ($Q$) components:

$\Gamma = \Gamma_r + j\Gamma_i = |\Gamma| e^{j\theta} = |\Gamma| \cos(\theta) + j |\Gamma| \sin(\theta)$

Resolving Complex Impedance ($R \pm jX$)

Knowing both magnitude $|\Gamma|$ and phase $\theta$, the VNA computes complex load impedance directly via the bilinear conformal transformation:

$Z_L = Z_0 \cdot \frac{1 + \Gamma}{1 - \Gamma} = R + jX$

where $Z_0$ is the system characteristic impedance ($50.0\ \Omega$).

• If $\theta > 0^\circ$, the reflected wave leads the incident reference, revealing inductive reactance ($+jX$) (voltage leads current; the antenna is electrically too long or contains series inductance).
• If $\theta < 0^\circ$, the reflected wave lags the incident reference, revealing capacitive reactance ($-jX$) (current leads voltage; the antenna is electrically too short or contains series capacitance).

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The Smith Chart in Amateur Radio Practice

The Smith Chart, introduced by Philip H. Smith in 1939, is drawn in the complex reflection-coefficient plane. A grid labelled in normalized impedance is overlaid using $\Gamma=(z-1)/(z+1)$. It therefore lets the same point show both reflection coefficient and load impedance.

                                  The Smith Chart

                               Inductive Reactance (+jx)
                                    Upper Hemisphere
                                        /  |  \
                                      /    |    \
                                    /   r=1.0     \
                                  /        |        \
          Short Circuit          +---------O---------+         Open Circuit
          r = 0, x = 0           |      Prime        |         Z tends to infinity
          (Γ = -1.0)             |      Center       |         (Γ = +1.0)
                                  \    (50 + j0 Ω)  /
                                    \      |      /
                                      \    |    /
                                        \  |  /
                               Capacitive Reactance (-jx)
                                    Lower Hemisphere

1. Conformal Mapping and Normalized Impedance

The boundary of the Smith Chart corresponds to the unit circle of the reflection coefficient ($|\Gamma| = 1.0$), encompassing all passive impedances with non-negative resistance ($R \ge 0$). To make the chart universal across any characteristic impedance, impedances are normalized by dividing by $Z_0$ ($50\ \Omega$):

$z = \frac{Z_L}{Z_0} = \frac{R + jX}{Z_0} = r + jx$

• Example: A physical impedance of $Z_L = 25 + j50\ \Omega$ normalizes in a $50\ \Omega$ system to:

$z = \frac{25 + j50}{50} = 0.5 + j1.0$

2. Geometry of the Chart

The Smith Chart consists of orthogonal circles and arcs:

1. The Horizontal Equator (Real / Resistance Axis): The horizontal diameter represents purely resistive impedance ($x = 0$, zero reactance).

   • Far Left Point ($0.0$): Zero resistance and zero reactance ($z = 0 + j0\ \Omega$). Represents an ideal Short Circuit ($\Gamma = -1.0, \theta = 180^\circ$).
   • Far Right Point ($\infty$): Infinite impedance ($z = \infty$). Represents an ideal Open Circuit ($\Gamma = +1.0, \theta = 0^\circ$).
   • Prime Center Point ($1.0$): Normalized impedance $z = 1.0 + j0$, representing a perfect $50 + j0\ \Omega$ match. At this point, $\Gamma = 0$ and $\text{VSWR} = 1.0:1$.

2. Circles of Constant Resistance ($r$ circles): All constant resistance circles are centered on the horizontal axis and are mutually tangent to each other at the open-circuit point on the far right ($r = \infty$). Every point on the $r = 1.0$ circle has a real resistive component of exactly $50\ \Omega$ (e.g., $50 + j25\ \Omega$, $50 - j50\ \Omega$).

3. Arcs of Constant Reactance ($x$ arcs): Curves of constant reactance branch outward from the open-circuit point and cross the resistance circles perpendicularly:

   • Upper Hemisphere: Represents Inductive Reactance ($+jx$). Any impedance trajectory lying in the upper half of the chart has positive reactance ($+jX_L$).
   • Lower Hemisphere: Represents Capacitive Reactance ($-jx$). Any trajectory lying in the lower half has negative reactance ($-jX_C$).

4. Circles of Constant VSWR: Because reflection coefficient magnitude ($|\Gamma|$) is directly related to VSWR, any contour of constant VSWR forms a perfect circle centered on the prime center ($1.0$).

   • To read the VSWR of any point on the chart, draw a circle centered at the prime center through that point. The value where the right-hand edge of this circle crosses the horizontal resistance axis equals the VSWR directly!
   • A point at $z = 2.0 + j0$ lies on the $\text{VSWR} = 2.0:1$ circle. A point at $z = 0.5 + j0$ also lies on the $\text{VSWR} = 2.0:1$ circle ($50 / 25 = 2.0$).

3. Transmission Line Behavior on the Smith Chart

When traversing a transmission line, impedance transforms continuously:

• Direction of Travel: Moving along a line toward the transmitter/generator corresponds to moving clockwise around the Smith Chart ("Toward Generator"). Moving toward the load corresponds to moving counter-clockwise ("Toward Load").
• Electrical Length: One complete $360^\circ$ rotation around the Smith Chart corresponds to an electrical distance of half a wavelength ($\lambda / 2$).
• Quarter-Wave Inversion: A rotation of $180^\circ$ corresponds to a quarter-wavelength ($\lambda / 4$), inverting low impedance to high impedance (transforming a short circuit into an open circuit).
• Transmission Line Loss: In a lossless line, the impedance trajectory follows a perfect circle of constant radius ($|\Gamma| = \text{constant}$). In a real, lossy coaxial cable, attenuation absorbs power in both directions, causing the trajectory to spiral inward toward the prime center ($50\ \Omega$). This illustrates why a long, lossy coaxial line makes a severely mismatched antenna appear to have a "good" SWR at the shack end!

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Calibration and Transmission Line Diagnostics

1. SOLT Calibration and Shifting the Reference Plane

A VNA cannot deliver accurate measurements without systematic calibration. The internal directional couplers, detector diodes, internal cables, and external test leads possess stray inductance, shunt capacitance, and phase delay. Calibration measures these systematic errors and mathematically subtracts (de-embeds) them from measurements.

                               SOLT Calibration Architecture

     [ VNA Port 1 ] =====( Test Cable / Jumper )=====[ Reference Plane ]---->[ Device Under Test ]
                                                            |
                                                  Calibration Standards:
                                                  - SHORT (Zero Ω, Phase = 180°)
                                                  - OPEN  (Infinite Ω, Phase = 0°)
                                                  - LOAD  (Precision 50.0 Ω, Γ = 0)
                                                  - THRU  (Direct connection Port 1 to 2)

The standard calibration protocol is SOLT:

1. SHORT: A precision short circuit presenting $0\ \Omega$ (reflection coefficient $\Gamma = 1.0 \angle 180^\circ$).
2. OPEN: A precision open circuit presenting infinite impedance ($\Gamma = 1.0 \angle 0^\circ$).
3. LOAD: A precision non-inductive $50.0\ \Omega$ termination ($\Gamma = 0.0$, absorbs all incident power).
4. THRU: A direct zero-length connection between Port 1 and Port 2 (for two-port transmission measurements).

2. Time Domain Reflectometry (TDR) via IFFT

A VNA measures complex $S_{11}$ over frequency and can transform those data into a time-domain response. Lowpass impulse/step modes require suitable harmonic frequency spacing and a DC estimate; a bandpass impulse mode can use a band-limited sweep. The display mode matters when interpreting a fault.

           Ideal Lowpass Step Responses: Alternative Cable Ends

   Gamma
   +1 |                        +---------------- Open
      |                        |
    0 +------------------------+----------------------> Delay
      |                        |
   -1 |                        +---------------- Short
                            round-trip delay

A signed lowpass-step response distinguishes an impedance increase from a decrease. A bandpass magnitude peak locates a discontinuity but does not by itself give its reactive sign. Time/distance resolution depends on frequency span and windowing; display zoom cannot create millimetre measurement resolution.

By knowing the coaxial cable's Velocity Factor ($v_f$), the time delay ($t$) of any reflected pulse converts directly into physical distance ($d$):

$d = \frac{v_f \cdot c \cdot t}{2}$

where $c$ is the speed of light ($3.00 \times 10^8\text{ m/s}$) and the factor of 2 accounts for the two-way round trip.

Diagnostic Interpretation (Lowpass Step Mode):

• Positive change: A higher impedance; an ideal lossless open termination has $\Gamma=+1$.
• Negative change: A lower impedance; an ideal short termination has $\Gamma=-1$.
• Distributed lower-impedance region: This can indicate a dielectric or geometry change. Water ingress can increase effective permittivity and lower impedance, but the trace alone does not uniquely identify the cause. Cable loss and calibration affect the returned amplitude.

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Step-by-Step Worked Calculations

Worked Example 1: Reflection Coefficient and VSWR Calculation

A VNA measures the complex feedpoint impedance of an off-center fed dipole at $14.15\text{ MHz}$ as $Z_L = 30 - j40\ \Omega$. In a $50.0\ \Omega$ system, calculate:

1. The complex reflection coefficient ($\Gamma$).
2. The reflection coefficient magnitude ($|\Gamma|$).
3. The resulting VSWR.

Calculation:

1. Compute $\Gamma$ using the reflection coefficient formula:

$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} = \frac{(30 - j40) - 50}{(30 - j40) + 50} = \frac{-20 - j40}{80 - j40}$

Simplify by dividing numerator and denominator by $20$:

$\Gamma = \frac{-1 - j2}{4 - j2}$

Multiply numerator and denominator by the complex conjugate of the denominator ($4 + j2$):

$\Gamma = \frac{(-1 - j2)(4 + j2)}{(4 - j2)(4 + j2)} = \frac{-4 - j2 - j8 - j^2(4)}{16 - j^2(4)} = \frac{-4 - j10 + 4}{16 + 4} = \frac{0 - j10}{20} = 0 - j0.50$

1. Compute the magnitude $|\Gamma|$:

$|\Gamma| = \sqrt{0^2 + (-0.50)^2} = 0.50$

The phase angle is $\theta = -90^\circ$ (purely capacitive reflection).

1. Compute VSWR:

$\text{VSWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|} = \frac{1 + 0.50}{1 - 0.50} = \frac{1.50}{0.50} = 3.0:1$

Result: The reflection coefficient is $\Gamma = 0.50 \angle -90^\circ$, producing a $\text{VSWR}$ of $3.0:1$.

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Worked Example 2: Normalizing Coordinates on the Smith Chart

An antenna tuning network presents an input impedance of $Z = 75 + j100\ \Omega$. Determine its normalized coordinates on a standard $50\ \Omega$ Smith Chart, identify which hemisphere it occupies, and state the type of reactance present.

Calculation:

$z = \frac{Z}{Z_0} = \frac{75 + j100\ \Omega}{50\ \Omega} = \frac{75}{50} + j\left(\frac{100}{50}\right) = 1.5 + j2.0$

Analysis:

• Normalized resistance $r = 1.5$. The impedance lies along the $r = 1.5$ constant resistance circle.
• Normalized reactance $x = +2.0$. Because the reactance sign is positive, the point lies in the upper hemisphere of the Smith Chart, indicating inductive reactance.

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Worked Example 3: Locating a Coaxial Cable Fault

An $80\text{-meter}$ dipole develops very high SWR. A VNA sweep from $1\text{ MHz}$ to $50\text{ MHz}$ shows that the unwrapped $S_{11}$ phase of its open-ended feedline changes by $360^\circ$ every $\Delta f=2.45\text{ MHz}$. Assume one dominant end reflection and solid-polyethylene RG-213 with constant velocity factor $v_f=0.66$. Estimate the distance to that end reflection.

Calculation:

For a nondispersive line, the reflected wave accumulates phase over the round-trip path $2d$. A further $360^\circ$ phase change occurs for a frequency increment $\Delta f=v/(2d)$. An ideal open or short has $|\Gamma|=1$; the method uses phase rotation, not reflection-magnitude nulls:

$\Delta f = \frac{v}{2d} = \frac{v_f \cdot c}{2d}$

Rearranging to solve for distance $d$:

$d = \frac{v_f \cdot c}{2 \cdot \Delta f}$

Substitute the known values:

$d = \frac{0.66 \times (3.00 \times 10^8\text{ m/s})}{2 \times (2.45 \times 10^6\text{ Hz})} = \frac{1.98 \times 10^8}{4.90 \times 10^6} \approx 40.41\text{ meters}$

Result: The dominant end reflection is approximately $40.4\text{ meters}$ from the reference plane, using the stated velocity factor. Velocity-factor uncertainty, dispersion and measurement errors limit the physical accuracy.

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Comparison Table: Impedance Diagnostic Instruments