8.3 Resonant Line Sections, Quarter-Wave Transformers, and Stubs

8.3 Resonant Line Sections, Quarter-Wave Transformers, and Stubs

In conventional low-frequency electronics, reactive impedance is realized exclusively through physical inductors and capacitors. At radio frequencies—particularly across the VHF, UHF, and microwave spectra—physical lumped components suffer from severe parasitic effects: inductors exhibit distributed inter-turn capacitance, and capacitors exhibit lead inductance and dielectric heating. To circumvent these limitations, RF engineers exploit a remarkable property of transmission lines: a section of transmission line transforms impedances along its length.

By carefully choosing the line's characteristic impedance ($Z_0$), electrical length ($l$), and terminal condition (shorted, open, or loaded), a simple section of coaxial cable or parallel open-wire line can function as an impedance inverter, an impedance repeater, an ultra-low-loss tuned resonant tank, an inductive choke, a capacitive trimmer, or a notch filter.

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The General Transmission Line Input Impedance Equation

Consider a lossless uniform transmission line of characteristic impedance $Z_0$, terminated at $x = 0$ with an arbitrary complex load impedance $Z_L$. At any distance $l$ from the load (looking back into the line toward the load), the input impedance ($Z_{\text{in}}$) is the ratio of total line voltage to total line current:

$$Z_{\text{in}}(l) = \frac{V(l)}{I(l)} = \frac{V^+ e^{+j\beta l} + V^- e^{-j\beta l}}{\frac{V^+}{Z_0} e^{+j\beta l} - \frac{V^-}{Z_0} e^{-j\beta l}}$$

Substituting $V^- = \Gamma V^+$ and $\Gamma = (Z_L - Z_0) / (Z_L + Z_0)$, and expressing the complex exponentials in terms of Euler's trigonometric identities ($\cos \beta l$ and $j \sin \beta l$), yields the classical General Transmission Line Input Impedance Equation:

$$Z_{\text{in}}(l) = Z_0 \left[ \frac{Z_L + j Z_0 \tan(\beta l)}{Z_0 + j Z_L \tan(\beta l)} \right]$$

where:

• $\beta = \frac{2\pi}{\lambda}$ is the phase constant in radians per meter.
• $l$ is the physical length of the line section.
• $\beta l$ is the electrical length of the line section in radians (where $2\pi\text{ rad} = 360^\circ = 1\lambda$).

Because the tangent function is periodic with a period of $\pi$ radians ($180^\circ$, or $\lambda/2$), the input impedance transforms cyclically along the line, repeating its exact value every electrical half-wavelength.

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Quarter-Wave ($\lambda/4$) Line Sections and Impedance Inversion

A transmission line whose electrical length is an odd multiple of a quarter-wavelength ($l = \lambda/4, 3\lambda/4, 5\lambda/4, \dots$) exhibits unique and powerful impedance transformation properties.

When $l = \lambda/4$, the electrical angle is $\beta l = (2\pi / \lambda)(\lambda / 4) = \pi / 2$ radians ($90^\circ$). As $\beta l$ approaches $\pi / 2$, $\tan(\beta l)$ approaches infinity ($\tan(\pi/2) \to \infty$). Dividing the numerator and denominator of the general impedance equation by $\tan(\beta l)$:

$$Z_{\text{in}} = Z_0 \lim_{\beta l \to \pi/2} \left[ \frac{\frac{Z_L}{\tan(\beta l)} + j Z_0}{\frac{Z_0}{\tan(\beta l)} + j Z_L} \right] = Z_0 \left[ \frac{0 + j Z_0}{0 + j Z_L} \right] = Z_0 \left( \frac{Z_0}{Z_L} \right)$$ $$Z_{\text{in}} = \frac{Z_0^2}{Z_L}$$

This is the celebrated Quarter-Wave Impedance Inversion Formula.

Input upper  o-------- line length lambda/4 -------+
                                                   |
                                                 [ Z_L ]
                                                   |
Input return o-------------------------------------+

Z_in = Z_0^2 / Z_L
Short load --> open input; open load --> short input.
Low resistive load --> high input resistance.
Capacitive load --> inductive input, and conversely.

Extreme Resonant Cases of Quarter-Wave Sections

1. Short-Circuited Quarter-Wave Section ($Z_L = 0$): $Z_{\text{in}} = \frac{Z_0^2}{0} \to \infty$ A short-circuited quarter-wave line appears at its input terminals as an infinite impedance (open circuit). It behaves identically to an ideal parallel-resonant LC tank circuit at anti-resonance.

   • Practical Application: Metallic Insulator / Standoff Support. In VHF/UHF antenna arrays and power amplifier cavities, a shorted quarter-wave copper rod or stub can be connected directly between the live RF conductor and a grounded metal chassis. At the operating frequency, it presents infinite impedance and draws zero RF current, acting as a rigid, mechanically robust, weatherproof insulator made entirely of solid metal!
   • Practical Application: DC Grounding and Surge Protection. A grounded shorted stub provides a DC path for static charge while presenting high impedance at its design frequency. It is not a zero-impedance path for every lightning component: surge current can divide into equipment and on-frequency energy can pass. Use coordinated lightning protection and the device’s specified RF insertion loss and surge ratings.

2. Open-Circuited Quarter-Wave Section ($Z_L = \infty$): $Z_{\text{in}} = \frac{Z_0^2}{\infty} = 0$ An open-circuited quarter-wave line appears at its input terminals as a dead short circuit ($Z_{\text{in}} = 0$). It behaves identically to a series-resonant LC circuit at resonance.

   • Practical Application: Harmonic Notch Stubs. An open quarter-wave stub in shunt presents low impedance at its design frequency. If cut for $2f_0$, it rejects that harmonic but is a capacitive load at $f_0$. A shorted quarter-wave stub cut for $f_0$ instead presents high impedance at $f_0$ and low impedance at $2f_0$.

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Quarter-Wave Matching Transformers (Q-Sections)

The impedance inverting property of quarter-wave sections makes them ideal for matching two disparate, purely resistive impedances, such as transforming an antenna's radiation resistance ($R_L$) to match a standard feedline ($R_{\text{source}}$).

From the inversion formula, $R_{\text{in}} = Z_0^2 / R_L$. Setting $R_{\text{in}} = R_{\text{source}}$ and solving for the required transformer characteristic impedance ($Z_0$):

$$Z_0 = \sqrt{R_{\text{source}} \times R_L} = \sqrt{R_1 R_2}$$

Step-by-Step Worked Example: Matching Two Paralleled Yagi Antennas

An amateur radio operator stacks two identical 2-meter Yagi antennas. Each antenna has a feedpoint impedance of $50\ \Omega$. To combine them in parallel, each antenna must first be transformed to $100\ \Omega$ so that when connected in parallel at a central coaxial tee junction, the total combined impedance is $100\ \Omega \parallel 100\ \Omega = 50\ \Omega$, matching the main station feedline.

                    Dual-Antenna Quarter-Wave Matching Stacking

     Antenna 1 (50 Ω) o----[ λ/4 Q-Section (Z_0 = 70.7 Ω) ]----+
                                                               |
                                                           (T-Junction)
                                                               |  Combined = 50 Ω
                                                               +--- To Transmitter (50 Ω Coax)
                                                               |
     Antenna 2 (50 Ω) o----[ λ/4 Q-Section (Z_0 = 70.7 Ω) ]----+

Step 1: Calculate the required transformer impedance ($Z_0$):

$$Z_0 = \sqrt{R_{\text{ant}} \times R_{\text{target}}} = \sqrt{50\ \Omega \times 100\ \Omega} = \sqrt{5000} \approx 70.71\ \Omega$$

Step 2: Practical Implementation: Readily available $75\ \Omega$ low-loss coaxial cable (such as RG-11/U or RG-6/U) is remarkably close to $70.71\ \Omega$. Substituting $Z_0 = 75\ \Omega$ back into the formula:

$$R_{\text{in}} = \frac{Z_0^2}{R_L} = \frac{75^2}{50} = \frac{5625}{50} = 112.5\ \Omega$$

Paralleling two $112.5\ \Omega$ legs yields $112.5 / 2 = 56.25\ \Omega$. The resulting SWR on a $50\ \Omega$ main feedline is:

$$\text{SWR} = \frac{56.25}{50} = 1.125:1$$

This produces an imperceptible return loss of $24.6\text{ dB}$ (only $0.35\%$ reflected power), demonstrating why $75\ \Omega$ cable is universally used by amateurs to build stacking transformers.

Step 3: Calculate the physical length at $f = 144.30\text{ MHz}$: If using RG-11/U foam polyethylene cable with a velocity factor $VF = 0.84$:

$$\lambda_0 = \frac{2.9979 \times 10^8\text{ m/s}}{144.30 \times 10^6\text{ Hz}} \approx 2.0775\text{ m}$$ $$l_{\text{phys}} = \frac{\lambda_0}{4} \times VF = \frac{2.0775\text{ m}}{4} \times 0.84 \approx 0.5194\text{ m} \times 0.84 = 0.4363\text{ m} = 43.63\text{ cm}$$

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Half-Wave ($\lambda/2$) Line Sections: The 1:1 Impedance Repeater

When a transmission line's electrical length is an exact half-wavelength ($l = \lambda/2$) or an integer multiple of a half-wavelength ($l = n \times \lambda/2$, where $n = 1, 2, 3, \dots$):

The electrical angle is $\beta l = (2\pi / \lambda)(\lambda / 2) = \pi$ radians ($180^\circ$). Since $\tan(\pi) = 0$, substituting $\tan(\beta l) = 0$ into the general impedance equation produces a dramatic simplification:

$$Z_{\text{in}} = Z_0 \left[ \frac{Z_L + j Z_0 (0)}{Z_0 + j Z_L (0)} \right] = Z_0 \left( \frac{Z_L}{Z_0} \right) = Z_L$$

Practical Utility of Half-Wave Repeaters

1. Remote Antenna Measurements: An analyser measures impedance at its calibrated reference plane. A random feedline transforms the antenna's impedance; this is its input impedance, rather than a false reading. A lossless electrical half-wave repeats the load exactly at the design frequency. Real cable attenuation still changes the result, so accurate remote measurements require calibration at the antenna or compensation for the intervening line.
2. Equal-Phase Feed Networks: Half-wave lines introduce an exact $180^\circ$ phase shift with unity impedance scaling, making them essential for feeding phased vertical arrays, collinear arrays, and balun balance loops.

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Non-Resonant Line Sections: Reactive Stubs

An ideal lossless line terminated in an open or short circuit dissipates no real power. Away from its resonant open/short input conditions, it presents reactance and stores electric and magnetic energy. Real stubs also have loss resistance.

1. Short-Circuited Stubs ($Z_L = 0$)

Substituting $Z_L = 0$ into the general impedance equation:

$$Z_{\text{in}} = Z_0 \left[ \frac{0 + j Z_0 \tan(\beta l)}{Z_0 + j (0) \tan(\beta l)} \right] = +j Z_0 \tan(\beta l)$$

• Length $0 < l < \lambda/4$ ($0^\circ < \beta l < 90^\circ$): Because $\tan(\beta l) > 0$, the input impedance is positive imaginary: $+jX_L$ (Pure Inductor). The effective inductive reactance is: $X_L = Z_0 \tan(\beta l)$
• Length $\lambda/4 < l < \lambda/2$ ($90^\circ < \beta l < 180^\circ$): Because $\tan(\beta l) < 0$, the input impedance is negative imaginary: $-jX_C$ (Pure Capacitor). The effective capacitive reactance is: $X_C = -Z_0 \tan(\beta l)$

2. Open-Circuited Stubs ($Z_L = \infty$)

Substituting $Z_L = \infty$ into the general impedance equation (dividing numerator and denominator by $Z_L$):

$$Z_{\text{in}} = Z_0 \left[ \frac{1 + j(Z_0 / Z_L) \tan(\beta l)}{(Z_0 / Z_L) + j \tan(\beta l)} \right] = \frac{Z_0}{j \tan(\beta l)} = -j Z_0 \cot(\beta l)$$

• Length $0 < l < \lambda/4$ ($0^\circ < \beta l < 90^\circ$): Because $\cot(\beta l) > 0$, the input impedance is negative imaginary: $-jX_C$ (Pure Capacitor). The effective capacitive reactance is: $X_C = Z_0 \cot(\beta l)$
• Length $\lambda/4 < l < \lambda/2$ ($90^\circ < \beta l < 180^\circ$): Because $\cot(\beta l) < 0$, the input impedance is positive imaginary: $+jX_L$ (Pure Inductor). The effective inductive reactance is: $X_L = -Z_0 \cot(\beta l)$

Master Summary Table of Stub Impedances

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Stub Matching Techniques: Single-Stub Tuners

When an antenna feedpoint presents an impedance mismatch containing both resistance and reactance, a single-stub matching network can establish a perfect $1:1$ match to a $50\ \Omega$ feedline without using lumped coils or capacitors.

                Shunt-stub input       Distance d to load
Signal from TX o-------+-------------------------+
                       |                         |
                    [ Z_stub ]                 [ Z_L ]
                       |                         |
Return to TX   o-------+-------------------------+

Z_stub is the input impedance of a complete open- or
short-terminated two-conductor stub of length l.

Matching Principle

1. Normalize admittance to the main line: $y=Y/Y_0=g+jb$, where $Y_0=1/Z_0$. The normalized quantities are dimensionless.
2. Move a distance $d$ from the load toward the transmitter to a point where $g=1$, so $y(d)=1+jb$.
3. Connect a shunt stub of length $l$ whose normalized input admittance is $y_{\text{stub}}=-jb$.
4. The susceptances cancel: $y_{\text{total}}=(1+jb)-jb=1$. Thus the physical admittance is $Y_0$, and the input impedance is $Z_0$.
5. At the design frequency, the lossless main line between the junction and transmitter is matched. The load-side section still has its original standing waves.

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Practical Amateur Applications of Stubs

1. Coaxial Harmonic Notch Stubs

A shunt stub can suppress a harmonic by presenting low impedance across the feedline at that frequency. Choose its termination and length from the impedance equations:

• Shorted stub, quarter-wave at $f_0$: At $f_0$, $Z_{\text{in}}=jZ_0\tan90^\circ\to\infty$, so the ideal stub leaves the wanted carrier unloaded. At $2f_0$, it is a half-wave long and repeats the short circuit. This rejects the second harmonic by reflection.
• Open stub, quarter-wave at $2f_0$: It presents a short at $2f_0$, but at $f_0$ its length is $\lambda/8$ and $Z_{\text{in}}=-jZ_0\cot45^\circ=-jZ_0$. For a $50\ \Omega$ stub this is $-j50\ \Omega$ in shunt, so the fundamental match must also be assessed.
• Open stub, half-wave at $f_0$: It repeats an open circuit at both $f_0$ and $2f_0$, so it does not provide this second-harmonic notch.

Real notch depth and carrier insertion loss depend on cable loss, connections, physical length and parasitic reactance. Verify the assembled filter's pass and rejection responses at the operating frequencies.

2. Multi-Transmitter Repeater Site Isolation

Closely spaced repeater channels require selective filter or duplexer networks, commonly using high-Q cavities. A shorted quarter-wave resonator has high input impedance at resonance; its coupling and connection determine the network's pass or rejection response. Select and measure isolation at the interfering transmitter frequency together with insertion loss and matching at the wanted receiver frequency.