Transmission lines and waveguides

What to remember

• A line matched to its characteristic impedance (load ZL = Z0) has no reflection, Γ = 0 and VSWR = 1; all power goes to the load.
• A rectangular waveguide cannot carry frequencies below its cut-off; it supports TE and TM modes but no TEM mode, and the dominant mode is TE10.
• A quarter-wave line transforms impedance as Zin = Z0²/ZL; a Smith chart is the graphical tool for line calculations.

Line parameters

A transmission line is described by four distributed parameters per unit length: series resistance R (Ω/m), series inductance L (H/m), shunt conductance G (S/m) and shunt capacitance C (F/m). Line equations (telegraph equations) give voltage and current as waves travelling in both directions.

Characteristic impedance: Z0 = √((R + jωL)/(G + jωC)). Propagation constant: γ = √((R + jωL)(G + jωC)) = α + jβ, where α is the attenuation constant (Np/m) and β is the phase constant (rad/m). Wave velocity v = ω/β; wavelength λ = 2π/β.

Lossless line (R = G = 0): Z0 = √(L/C), purely resistive; α = 0; β = ω√(LC); v = 1/√(LC). Low-loss line at high frequency: Z0 ≈ √(L/C) and α ≈ R/(2Z0) + G Z0/2.

Distortionless line (Heaviside condition): R/L = G/C. Then α = √(RG) and β = ω√(LC); all frequencies travel at the same velocity and attenuate equally. Loading coils (adding inductance) are used on telephone lines to approach this condition.

Typical values: coaxial cable 50 Ω or 75 Ω; twin-wire line 300 Ω; 50 Ω for radio equipment and 75 Ω for television reception. For a coaxial line, Z0 = (60/√εr) ln(b/a) ohms. Example: for an air-filled coaxial line with b/a = 2.3, Z0 ≈ 60 × ln 2.3 ≈ 50 Ω.

Reflection, standing waves and matching

Reflection coefficient at the load: Γ = (ZL − Z0)/(ZL + Z0). Its magnitude is between 0 and 1.

VSWR S = (1 + |Γ|)/(1 − |Γ|), so |Γ| = (S − 1)/(S + 1). For a purely resistive load, S = ZL/Z0 or Z0/ZL, whichever is greater than 1. Return loss = −20 log|Γ| dB. Reflected power fraction = |Γ|². Example: ZL = 100 Ω on a 50 Ω line gives Γ = 50/150 = 1/3 and S = 2; reflected power = 11.1 %.

At a voltage maximum, the impedance is maximum (Z0 × S); at a voltage minimum, the impedance is minimum (Z0/S). Voltage maxima and minima are separated by λ/4, and successive maxima by λ/2.

Input impedance of a lossless line of length l: Zin = Z0 (ZL + jZ0 tan βl)/(Z0 + jZL tan βl). A shorted line of length l: Zin = jZ0 tan βl. An open line: Zin = −jZ0 cot βl. A shorted quarter-wave stub looks like an open circuit at its input (very high impedance); an open quarter-wave line looks like a short. A half-wave line repeats the load impedance: Zin = ZL. A quarter-wave transformer matches two resistances: Z0′ = √(Z0 ZL). Example: match 100 Ω to 50 Ω with a quarter-wave section of √(50 × 100) = 70.7 Ω.

Matching methods: quarter-wave transformer, single-stub and double-stub matching (a short-circuited stub of suitable length and position), and tapered lines. Stubs are used because a short reactive stub can cancel the load reactance.

Smith chart

A Smith chart is a polar plot of the reflection coefficient with normalised impedance circles (constant resistance and constant reactance). The centre is the matched point (z = 1). The outer circle has |Γ| = 1. Moving one full turn around the chart corresponds to a half-wavelength (λ/2). Movement toward the generator is clockwise. The left end of the real axis is a short circuit (z = 0); the right end is an open circuit (z = ∞). The same chart can be used as an admittance chart by moving a half-turn (λ/4). It gives VSWR (read at the real-axis crossing, right half), impedance at any distance, and stub matching positions.

Waveguides

A waveguide is a hollow metallic tube that guides electromagnetic waves at microwave frequencies (typically a few GHz and above). Compared with coaxial cable, a waveguide has lower loss, higher power handling and no centre conductor, but it is bulky and has a cut-off frequency.

Modes: TE (transverse electric, no E in the direction of propagation), TM (transverse magnetic), and TEM (not possible in a single-conductor hollow guide). Rectangular guide of broad dimension a and narrow dimension b: cut-off frequency of TEmn mode:

fc = (c/2) √((m/a)² + (n/b)²)

The dominant mode is TE10 (lowest cut-off): fc = c/(2a), cut-off wavelength λc = 2a. Example: a = 2.29 cm gives fc ≈ 6.55 GHz. For a = 4 cm, fc = 3 × 10⁸/(0.08) = 3.75 GHz. The next modes are TE20 and TE01, followed by TE11 and TM11 (the lowest TM mode). Single-mode operation lies between the TE10 cut-off and the next higher cut-off.

Guide wavelength: λg = λ0/√(1 − (fc/f)²), always longer than the free-space wavelength. Phase velocity vp = c/√(1 − (fc/f)²) is greater than c; group velocity vg = c√(1 − (fc/f)²) is less than c; vp × vg = c². Wave impedance of TE mode: ZTE = η/√(1 − (fc/f)²), which is greater than 377 Ω; for TM, ZTM = η√(1 − (fc/f)²).

Example: for fc = 3.75 GHz and f = 5 GHz, √(1 − 0.5625) = 0.661, so vg = 0.661c ≈ 1.98 × 10⁸ m/s and vp ≈ 4.54 × 10⁸ m/s (check: vp × vg = 9 × 10¹⁶ = c²).

Circular waveguide: dominant mode TE11. Resonant cavity: a closed metallic box with high Q, used as a tuned circuit at microwave frequencies; resonant modes depend on the dimensions. Excitation of a waveguide is done by a probe (for E field) or a loop (for H field). A waveguide cannot propagate below cut-off; the field decays exponentially (evanescent).

Other microwave components: directional coupler, circulator, isolator, attenuator, E-plane tee, H-plane tee and magic tee. Striplines and microstrip lines are planar lines for microwave circuits; microstrip supports a quasi-TEM mode. Optical fibre is a dielectric waveguide that works by total internal reflection.

Losses, power and practical lines

Attenuation on a line is usually given in dB per unit length: dB = 8.686 × α (for α in Np/m), since 1 Np = 8.686 dB. Conductor loss rises with frequency because skin effect raises the resistance; dielectric loss also rises with frequency. Power on a line: the power delivered to the load is the incident power times (1 − |Γ|²). The maximum voltage on a line with standing waves is |V+|(1 + |Γ|) and the minimum is |V+|(1 − |Γ|). An open-wire line is used at lower frequencies; coaxial cable is used from audio to several GHz; a waveguide is used above a few GHz where coaxial cable loss becomes large. A balun connects a balanced line (such as twin-wire) to an unbalanced line (such as coaxial cable). Characteristic impedance of a two-wire line of spacing D and wire diameter d in air: Z0 ≈ 276 log10(2D/d) ohms. Slotted-line measurements find VSWR and the position of the voltage minimum to determine an unknown load.

Waveguide excitation and attenuation

Waveguide attenuation is low in the middle of the band and rises near cut-off, where the wave becomes evanescent. Standard waveguides are used over about 1.25 to 1.9 times the cut-off frequency of the dominant mode, to avoid both high loss and the second mode. Waveguide walls are made of copper, brass or aluminium, sometimes silver-plated inside. A matched waveguide termination absorbs all the power; a movable short is used for tuning. A cavity resonator's resonant frequency rises when its dimensions are made smaller.

Worked examples

• 1. Line with L = 0.25 µH/m and C = 100 pF/m (lossless): Z0 = √(0.25 × 10⁻⁶/10⁻¹⁰) = √2500 = 50 Ω; v = 1/√(0.25 × 10⁻⁶ × 10⁻¹⁰) = 2 × 10⁸ m/s.
• 2. A 75 Ω line terminated in 150 Ω: Γ = 75/225 = 1/3; VSWR = 2.
• 3. A line of Z0 = 50 Ω with load 25 Ω: S = 2; Γ = −1/3.
• 4. A shorted stub of length λ/8: Zin = jZ0 tan 45° = jZ0.

Exam traps

• Short circuit gives Γ = −1; open circuit gives Γ = +1; both give VSWR = ∞.
• Quarter-wave line inverts impedance; half-wave line repeats it.
• Phase velocity in a waveguide is greater than c, but group velocity is less than c.
• TE10, not TM11, is the dominant mode in a rectangular guide; TM00 and TM10 do not exist.
• Guide wavelength is longer than free-space wavelength.
• Waveguide wave impedance for TE mode is above 377 Ω.
• Heaviside condition is R/L = G/C, which is the same as RC = LG, and not R = G.
• A matched line has VSWR 1, not 0.

One-liners

• 1. Z0 = √(L/C) for a lossless line.
• 2. v = 1/√(LC) on a lossless line.
• 3. Γ = (ZL − Z0)/(ZL + Z0).
• 4. VSWR = (1 + |Γ|)/(1 − |Γ|).
• 5. Quarter-wave transformer impedance is √(Z0 ZL).
• 6. Distortionless: R/L = G/C.
• 7. TE10 is the dominant rectangular-guide mode.
• 8. fc (TE10) = c/(2a).
• 9. vp × vg = c².
• 10. TEM mode cannot exist in a hollow single-conductor waveguide.
• 11. Smith chart centre is the matched point.
• 12. A resonant cavity is a high-Q microwave resonator.

Practice questions

1. The characteristic impedance of a lossless transmission line is

   1. √(C/L)
   2. 1/√(LC)
   3. √(LC)
   4. √(L/C)
   Answer

   D. √(L/C)

   With R = G = 0, Z0 = √(L/C).

2. A lossless line has L = 0.25 µH/m and C = 100 pF/m. Its characteristic impedance is

   1. 100 Ω
   2. 50 Ω
   3. 25 Ω
   4. 75 Ω
   Answer

   B. 50 Ω

   √(0.25 × 10⁻⁶/10⁻¹⁰) = √2500 = 50 Ω.

3. The velocity of a wave on the line with L = 0.25 µH/m and C = 100 pF/m is

   1. 3 × 10⁸ m/s
   2. 4 × 10⁸ m/s
   3. 1 × 10⁸ m/s
   4. 2 × 10⁸ m/s
   Answer

   D. 2 × 10⁸ m/s

   v = 1/√(LC) = 1/√(2.5 × 10⁻¹⁷) = 2 × 10⁸ m/s.

4. The Heaviside condition for a distortionless line is

   1. R = G
   2. R/L = G/C
   3. RL = GC
   4. R/G = L/C
   Answer

   B. R/L = G/C

   R/L = G/C gives frequency-independent α and velocity.

5. For a distortionless line, the attenuation constant α equals

   1. √(RG)
   2. ω√(LC)
   3. √(LC)
   4. R/L
   Answer

   A. √(RG)

   Under the Heaviside condition, α = √(RG).

6. The unit of the attenuation constant α is

   1. henry per metre
   2. radian per metre
   3. ohm
   4. neper per metre
   Answer

   D. neper per metre

   β is in rad/m; α in Np/m.

7. The reflection coefficient of a line with load ZL on a line with Z0 is

   1. (Z0 − ZL)/Z0
   2. (ZL + Z0)/(ZL − Z0)
   3. (ZL − Z0)/(ZL + Z0)
   4. ZL/Z0
   Answer

   C. (ZL − Z0)/(ZL + Z0)

   Standard definition.

8. A short-circuited line has reflection coefficient

   1. +0.5
   2. −1
   3. +1
   4. 0
   Answer

   B. −1

   ZL = 0 gives Γ = −1.

9. An open-circuited line has VSWR equal to

   1. 1
   2. 0
   3. infinity
   4. 2
   Answer

   C. infinity

   |Γ| = 1, so S = ∞.

10. A 50 Ω line is terminated in a 100 Ω resistor. The VSWR is

    1. 2
    2. 1
    3. 3
    4. 0.5
    Answer

    A. 2

    S = 100/50 = 2.

11. A 50 Ω line has a load of 100 Ω. The reflection coefficient magnitude is

    1. 1/2
    2. 2/3
    3. 1
    4. 1/3
    Answer

    D. 1/3

    (100 − 50)/(150) = 1/3.

12. A line has VSWR of 3. The magnitude of the reflection coefficient is

    1. 0.67
    2. 0.5
    3. 0.25
    4. 0.33
    Answer

    B. 0.5

    (3 − 1)/(3 + 1) = 0.5.

13. With |Γ| = 0.5, the fraction of incident power reflected is

    1. 12.5 %
    2. 25 %
    3. 75 %
    4. 50 %
    Answer

    B. 25 %

    |Γ|² = 0.25.

14. The return loss of a load with |Γ| = 0.1 is

    1. 20 dB
    2. 10 dB
    3. 0.1 dB
    4. 1 dB
    Answer

    A. 20 dB

    −20 log 0.1 = 20 dB.

15. The VSWR of a matched line is

    1. 0
    2. 1
    3. 2
    4. infinity
    Answer

    B. 1

    No reflection gives S = 1.

16. A quarter-wave transformer matching a 100 Ω load to a 50 Ω line needs a section of impedance about

    1. 25 Ω
    2. 150 Ω
    3. 75 Ω
    4. 70.7 Ω
    Answer

    D. 70.7 Ω

    √(50 × 100) = 70.7 Ω.

17. A half-wave line terminated in ZL has input impedance equal to

    1. Z0²/ZL
    2. Z0
    3. ZL
    4. zero
    Answer

    C. ZL

    A λ/2 line repeats the load impedance.

18. A short-circuited quarter-wave line at the input looks like

    1. a short circuit
    2. a capacitor
    3. an open circuit
    4. a pure resistance of Z0
    Answer

    C. an open circuit

    Zin = Z0²/0 → ∞.

19. The distance between successive voltage maxima on a standing-wave line is

    1. λ
    2. λ/4
    3. 3λ/4
    4. λ/2
    Answer

    D. λ/2

    Maxima repeat every half wavelength.

20. The input impedance of a short-circuited line of length l is

    1. jZ0 tan βl
    2. −jZ0 cot βl
    3. Z0 tan βl
    4. Z0
    Answer

    A. jZ0 tan βl

    Standard stub formula.

21. A short-circuited stub of length λ/8 has input impedance

    1. 0
    2. Z0
    3. −jZ0
    4. jZ0
    Answer

    D. jZ0

    tan 45° = 1.

22. On the Smith chart, one full revolution corresponds to a line length of

    1. λ
    2. 2λ
    3. λ/2
    4. λ/4
    Answer

    C. λ/2

    Impedance repeats every half wavelength.

23. The centre of the Smith chart represents

    1. short circuit
    2. the matched condition
    3. infinite VSWR
    4. open circuit
    Answer

    B. the matched condition

    Normalised z = 1, Γ = 0.

24. Moving toward the generator on a Smith chart is

    1. clockwise
    2. anticlockwise
    3. radially outward
    4. along the real axis
    Answer

    A. clockwise

    Standard Smith chart convention.

25. A hollow rectangular waveguide cannot support

    1. TM11 mode
    2. TEM mode
    3. TE10 mode
    4. TE11 mode
    Answer

    B. TEM mode

    A single hollow conductor cannot support TEM.

26. The dominant mode in a rectangular waveguide is

    1. TE11
    2. TM11
    3. TE10
    4. TM01
    Answer

    C. TE10

    It has the lowest cut-off frequency.

27. The cut-off frequency of the TE10 mode in a guide of broad dimension a = 4 cm is

    1. 1.9 GHz
    2. 7.5 GHz
    3. 15 GHz
    4. 3.75 GHz
    Answer

    D. 3.75 GHz

    fc = c/2a = 3 × 10⁸/0.08.

28. The cut-off wavelength of the TE10 mode in a rectangular guide of broad dimension a is

    1. 2a
    2. a
    3. 4a
    4. a/2
    Answer

    A. 2a

    λc = 2a for TE10.

29. For a wave above cut-off in a rectangular guide, the guide wavelength is

    1. longer than free-space wavelength
    2. equal to free-space wavelength
    3. shorter than free-space wavelength
    4. independent of frequency
    Answer

    A. longer than free-space wavelength

    λg = λ0/√(1 − (fc/f)²).

30. In a waveguide, the phase velocity and group velocity satisfy

    1. vp/vg = c
    2. vp + vg = c
    3. vp = vg
    4. vp × vg = c²
    Answer

    D. vp × vg = c²

    For hollow guide.

31. In a waveguide above cut-off, the phase velocity is

    1. less than velocity of light
    2. greater than the velocity of light
    3. equal to the group velocity
    4. zero
    Answer

    B. greater than the velocity of light

    vp = c/√(1 − (fc/f)²) > c.

32. In a waveguide with fc = 3 GHz at an operating frequency of 5 GHz, the group velocity is

    1. 0.5c
    2. 0.6c
    3. 0.8c
    4. 1.25c
    Answer

    C. 0.8c

    vg = c√(1 − 0.36) = 0.8c.

33. The wave impedance of a TE mode in a waveguide above cut-off is

    1. greater than 377 Ω
    2. exactly 377 Ω
    3. less than 377 Ω
    4. zero
    Answer

    A. greater than 377 Ω

    ZTE = η/√(1 − (fc/f)²).

34. The dominant mode in a circular waveguide is

    1. TE10
    2. TEM
    3. TE11
    4. TM01
    Answer

    C. TE11

    Standard result.

35. Consider the statements. 1. A quarter-wave line inverts impedance. 2. A half-wave line gives input impedance equal to Z0 for any load. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    A half-wave line repeats ZL, not Z0.

36. Consider the statements. 1. A short circuit gives |Γ| = 1. 2. An open circuit gives |Γ| = 1. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both give total reflection.

37. Consider the statements. 1. Below cut-off a waveguide wave is evanescent. 2. Group velocity in a waveguide exceeds c. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Group velocity is less than c.

38. Consider the statements. 1. A rectangular waveguide supports TE and TM modes. 2. TE10 has the lowest cut-off frequency when a > b. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are standard.

39. Consider the statements. 1. Voltage maxima occur at points of maximum impedance. 2. Voltage minima occur at points of minimum impedance. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Z = V/I; at the voltage maximum current is minimum.

40. Match the load with the reflection coefficient on a 50 Ω line: P. 50 Ω Q. Short R. Open 1. +1 2. 0 3. −1

    1. P-3, Q-2, R-1
    2. P-2, Q-3, R-1
    3. P-2, Q-1, R-3
    4. P-1, Q-2, R-3
    Answer

    B. P-2, Q-3, R-1

    Matched 0, short −1, open +1.

41. Match the line with its input impedance: P. Shorted λ/4 Q. Open λ/4 R. Any load λ/2 1. Zero 2. ZL 3. Infinity

    1. P-1, Q-3, R-2
    2. P-3, Q-2, R-1
    3. P-2, Q-1, R-3
    4. P-3, Q-1, R-2
    Answer

    D. P-3, Q-1, R-2

    λ/4 inverts: short → open, open → short; λ/2 repeats the load.

42. Match the quantity with the expression: P. Guide wavelength Q. Group velocity R. Phase velocity 1. c√(1 − (fc/f)²) 2. λ0/√(1 − (fc/f)²) 3. c/√(1 − (fc/f)²)

    1. P-3, Q-1, R-2
    2. P-2, Q-1, R-3
    3. P-2, Q-3, R-1
    4. P-1, Q-2, R-3
    Answer

    B. P-2, Q-1, R-3

    Standard waveguide relations.

43. Match the item with its use: P. Stub Q. Smith chart R. Cavity resonator 1. Graphical line calculation 2. Impedance matching 3. Microwave tuned circuit

    1. P-3, Q-1, R-2
    2. P-1, Q-2, R-3
    3. P-2, Q-3, R-1
    4. P-2, Q-1, R-3
    Answer

    D. P-2, Q-1, R-3

    Stubs match, the chart computes, cavities resonate.

44. Consider the statements. 1. A matched line has VSWR of 1. 2. A matched line has reflection coefficient of 1. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Matched means Γ = 0.

45. Consider the statements about a lossless line. 1. Its characteristic impedance is real. 2. Its attenuation constant is non-zero. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    α = 0 on a lossless line.