Interview questions & answers
Q1. What is the characteristic impedance of a transmission line and how is it defined?
Characteristic impedance Z₀ = √((R+jωL)/(G+jωC)) is the impedance of an infinitely long transmission line, or equivalently the ratio of forward-traveling voltage to current on any lossless line terminated in Z₀. A standard 50 Ω coaxial cable like RG-58 has Z₀ = 50 Ω regardless of its length, set by the conductor geometry (inner and outer radii) and dielectric permittivity. For a lossless line, Z₀ = √(L/C), which is real and frequency-independent — this is why 50 Ω lines have flat frequency response and 75 Ω cable TV coax (optimized for minimum attenuation) looks the same on a spectrum analyzer.
Follow-up: Why are 50 Ω and 75 Ω the two most common standard transmission line impedances?
Q2. What is a reflection coefficient and how is it calculated?
The reflection coefficient Γ = (Z_L - Z₀)/(Z_L + Z₀) is the ratio of reflected to incident voltage wave amplitude at the load, ranging from -1 (short circuit) to +1 (open circuit) with |Γ| ≤ 1 for passive loads. A 75 Ω antenna connected to a 50 Ω coaxial feed has Γ = (75-50)/(75+50) = 25/125 = 0.2, meaning 4% of the incident power is reflected. Zero reflection (perfect match) requires Z_L = Z₀, which is why impedance matching is critical in RF systems — every 0.2 increase in |Γ| roughly doubles the mismatch loss.
Follow-up: What is the input impedance of a transmission line terminated with a reflection coefficient Γ_L?
Q3. What is VSWR and what does it indicate about the quality of impedance matching?
VSWR (Voltage Standing Wave Ratio) = (1+|Γ|)/(1-|Γ|) is the ratio of maximum to minimum voltage amplitude along a transmission line and indicates how much of the incident power is reflected due to mismatch. A VSWR of 2:1 corresponds to |Γ| = 1/3 and means about 11% of the power is reflected, which is a common specification limit for RF power amplifier outputs. A VSWR of 1:1 indicates perfect matching with no reflection, while VSWR = ∞ corresponds to a complete open or short circuit.
Follow-up: What VSWR value corresponds to a 3 dB mismatch loss?
Q4. What is the input impedance of a quarter-wave transmission line?
A quarter-wave (λ/4) transmission line with characteristic impedance Z₀ terminated in load impedance Z_L has input impedance Z_in = Z₀²/Z_L — the quarter-wave transformer inverts the normalized impedance. A λ/4 section of 70.7 Ω coax connecting a 50 Ω source to a 100 Ω antenna provides Z_in = 70.7²/100 = 50 Ω at the source, achieving perfect broadside matching. This is the quarter-wave transformer matching technique, fundamental to microwave circuit and antenna feed network design.
Follow-up: What happens to the input impedance of a quarter-wave line if the frequency deviates by 10%?
Q5. What is the input impedance of a short-circuited transmission line stub?
A short-circuited stub of length l has input impedance Z_in = jZ₀ tan(βl), which is purely reactive and ranges from -j∞ (open-circuit behavior at λ/4) to +j∞ continuously, making stubs useful as inductors or capacitors at microwave frequencies. A 10 mm long short-circuit stub in 50 Ω microstrip at 5 GHz (βl = 0.524 rad) gives Z_in = j50·tan(0.524) = j50·0.577 = j28.9 Ω, a 28.9 Ω inductive reactance. Shunt stubs are used as matching elements in microwave amplifier and filter design because they can be fabricated directly as PCB traces without discrete components.
Follow-up: What is the input impedance of an open-circuited stub of length λ/4?
Q6. What is the propagation constant of a transmission line?
The propagation constant γ = α + jβ = √((R+jωL)(G+jωC)) describes how a wave changes as it travels — α (nepers/m) is the attenuation constant giving amplitude decay and β (rad/m) is the phase constant giving the rate of phase shift per unit length. For a low-loss coaxial cable at 100 MHz with α = 0.03 dB/m and β = 2.09 rad/m, a signal travels 1 m experiencing 0.03 dB of attenuation and 2.09 radians (120°) of phase shift. In the lossless case (R = G = 0), γ = jβ = jω√(LC) and the wave travels undamped at phase velocity vp = ω/β = 1/√(LC).
Follow-up: How do you measure the attenuation constant of a transmission line experimentally?
Q7. What is a Smith chart and how is it used?
A Smith chart is a graphical representation of the complex reflection coefficient Γ = |Γ|e^(jθ) normalized to Z₀, with constant resistance circles and constant reactance arcs overlaid, allowing impedance transformations and matching network design to be performed graphically. An engineer designing a matching network for a 25 - j50 Ω transistor input impedance at 900 MHz plots the normalized impedance (0.5 - j1.0) on the 50 Ω Smith chart and rotates toward the center (matching point) by adding shunt and series elements. Clockwise rotation on the Smith chart corresponds to moving toward the generator along the transmission line, each full rotation covering λ/2 in length.
Follow-up: How does adding a shunt capacitor appear as a movement on the Smith chart?
Q8. What is the difference between phase velocity and group velocity on a transmission line?
Phase velocity vp = ω/β is the speed at which a single-frequency sinusoidal wave crest travels, while group velocity vg = dω/dβ is the speed at which the envelope of a modulated signal (and therefore information) travels. For a lossless TEM line where β = ω√(LC) is linear in ω, both velocities are equal and equal to 1/√(LC) — there is no dispersion. However, in a waveguide where β depends nonlinearly on ω, vp > c while vg < c, and their product equals c² — the phase velocity exceeds c but no information travels faster than c.
Follow-up: Why can phase velocity exceed the speed of light without violating relativity?
Q9. What causes reflections on a PCB trace and how are they minimized?
Reflections on a PCB trace occur whenever the trace's characteristic impedance changes — at via transitions, connector launches, impedance discontinuities from trace width changes, and unterminated ends — causing partial reflection of the signal back toward the source. A DDR4 memory bus trace running at 1600 MHz on an FR4 PCB must be controlled to 50 Ω ±10% to prevent reflections that cause intersymbol interference and timing violations; even a 5 mil width change can produce a 5 Ω impedance step with Γ = 0.047. Proper termination (series resistors at the source or parallel at the load), controlled-impedance PCB stackup design, and careful via modeling are the standard mitigation techniques.
Follow-up: What is the difference between source termination and load termination on a PCB trace?
Q10. What is the condition for maximum power transfer on a transmission line?
Maximum power transfer from a source to a load through a transmission line occurs when Z_L = Z₀ (for a matched line) or more generally when Z_L = Z_source* (complex conjugate matching), where the reactive parts cancel and resistive parts are equal. In an RF power amplifier output stage, the transistor's output impedance might be 5 + j15 Ω, so the matching network must transform the 50 Ω antenna to the conjugate 5 - j15 Ω to extract maximum power from the transistor. Mismatch between Z₀ and Z_L reduces the deliverable power by the mismatch factor (1 - |Γ|²).
Follow-up: How does mismatch loss differ from insertion loss in an RF transmission system?
Q11. What is TDR (Time Domain Reflectometry) and how is it used?
TDR sends a fast step pulse down a cable or PCB trace and measures the reflected waveform versus time, allowing the location and nature of impedance discontinuities to be determined from the time delay and shape of the reflection. A TDR instrument detecting a reflection at 5 ns after the step is launched indicates a discontinuity at a distance of d = vp × t/2 = 2×10⁸ × 5×10⁻⁹/2 = 0.5 m from the source. A positive reflection indicates an impedance higher than Z₀ (such as an open circuit or inductive discontinuity), while a negative reflection indicates lower impedance (short circuit or capacitive stub).
Follow-up: How do you distinguish between a capacitive and inductive discontinuity from a TDR waveform?
Q12. What is the telegrapher's equation?
The telegrapher's equations are the coupled partial differential equations ∂V/∂z = -(R+jωL)I and ∂I/∂z = -(G+jωC)V, derived from the distributed-element lumped-circuit model of an infinitesimal transmission line section, and their solutions give the forward and backward traveling wave solutions for voltage and current. Combining the two equations gives the wave equation ∂²V/∂z² = γ²V with γ = √((R+jωL)(G+jωC)), showing that voltage (and current) propagate as waves with propagation constant γ. For the lossless case, this reduces to ∂²V/∂z² = LC·∂²V/∂t², the standard wave equation with wave speed 1/√(LC).
Follow-up: How is the lumped-element circuit model of a transmission line section derived?
Q13. What happens to the input impedance of a transmission line as its length approaches zero?
As the line length approaches zero, a transmission line terminated in Z_L has input impedance approaching Z_L itself, because there is no phase shift or reflection from essentially zero length. At very short lengths (l << λ), the transmission line looks purely like the load impedance — a 1 mm PCB stub at 1 GHz (where λ/10 ≈ 15 mm) has negligible transmission line effect and can be treated as a lumped capacitance or inductance. This is the justification for using lumped-circuit models in low-frequency electronics and the reason distributed transmission line effects only matter when trace lengths are significant fractions of a wavelength.
Follow-up: At what frequency does a 10 cm PCB trace begin to require transmission line analysis?
Q14. What is a matched load and why is it used at the end of a transmission line?
A matched load has impedance equal to the characteristic impedance of the line (Z_L = Z₀), producing zero reflection (Γ = 0) and absorbing all incident power with no standing waves. A 50 Ω terminating resistor at the end of a coaxial cable used in a spectrum analyzer input eliminates the standing waves that would distort measurements if the cable were open-circuited. In digital PCB design, series resistors of 33–50 Ω placed at signal sources (source termination) convert an unterminated low-impedance driver to an effective matched source impedance, preventing multiple reflections on memory and bus lines.
Follow-up: What is the difference between a matched load and a conjugate-matched load?
Q15. What is the bandwidth of a quarter-wave transformer and how can it be increased?
A single-section quarter-wave transformer has a bandwidth defined by the frequency range over which |Γ| stays below a specified level, typically narrow (10–20% of center frequency) for large impedance ratios. A single λ/4 transformer matching 50 Ω to 100 Ω at 2.4 GHz using a 70.7 Ω section has about ±15% bandwidth before VSWR exceeds 1.5:1. Bandwidth is increased by using multi-section Chebyshev or Binomial transformers that cascade multiple λ/4 sections with carefully chosen intermediate impedances, trading insertion length for bandwidth — a 3-section Chebyshev transformer can achieve ±40% bandwidth for the same mismatch specification.
Follow-up: How many sections of quarter-wave transformer are needed to double the matching bandwidth?
Common misconceptions
Misconception: A longer transmission line always has more signal loss.
Correct: Signal loss (attenuation) is proportional to line length, but reflected power due to mismatch is independent of line length — a perfectly matched longer line has less total loss than a short mismatched line.
Misconception: The characteristic impedance depends on the frequency of the signal.
Correct: For a lossless line, Z₀ = √(L/C) is purely real and frequency-independent; it is determined entirely by the line geometry and dielectric properties, not by frequency.
Misconception: VSWR of 1:1 means no power is delivered to the load.
Correct: VSWR of 1:1 means perfect impedance matching with no reflections and maximum power transfer to the load — it is the ideal condition, not a zero-power condition.
Misconception: A short circuit at the end of a transmission line reflects all power back and delivers nothing useful.
Correct: A short-circuit terminated stub is a lossless reactive element used intentionally in microwave matching networks; it delivers no real power but provides reactive impedance that cancels unwanted reactance in the main circuit.