Microwave Circuits Review Part I

2.5. Multipath Propagation

3. Transmission Lines

3.1. Characteristics and Reflection Coefficient

We can describe a section of TL using a distributed model:

We have the following equations:

v(z) – v(z+Δz) = rΔzi + LΔz ∂i/∂t, i(z) – i(z+Δz) = gΔzv + CΔz ∂v/∂t,

hence -∂v/∂z = ri + L ∂i/∂t, -∂i/∂t = gv + C ∂v/∂t.

For sinusoidal excitation e^jωt, replace i and v by phasors

∂V/∂z = -(r+jωL)I, ∂I/∂z = -(g+jωC)V, we obtain

∂²V/∂z² = γ²V, ∂²I/∂z² = γ²I, where γ = √[(r+jωL)(g+jωC)] = α+jβ.

γ is called propagation constant, α is called attenuation constant, and β is called phase constant.

Solving the wave equation yields

V = V⁺e^-γz + V^–e^γz, I = I⁺e^-γz – I^–e^γz, with I^± = V^±/Z₀, Z₀ = √[(r+jωL)/(g+jωC)] is called characteristic impedance.

Low-Loss TL

In many cases the losses due to r and g are small and negligible, i.e., α<<β, so γ≈jβ,

Z₀ ≈ √(L/C), β ≈ ω√(LC),

V = V⁺e^-jβz + V^–e^jβz, I = (V⁺/Z₀)e^-jβz – (V^–/Z₀)e^jβz,

restoring the harmonic-time dependence ejωt to the phasors:

V(z,t) = V⁺e^j(ωt-βz) + V^–e^j(ωt+βz), I(z,t) = (V⁺/Z₀)e^j(ωt-βz) – (V^–/Z₀)e^j(ωt+βz), where

e^j(ωt-βz) represents a +z (incident) wave, e^j(ωt+βz) represents a -z (reflected) wave.

For constant phase in +z propagation, ωt – βz = constant, ω – β ∂z/∂t = 0. ∂z/∂t = v_p is phase velocity.

v_p = ω/β = 1/√(LC) = fλ, β = ω/v_p = 2π/λ.

For a lossless line, ∂P/∂z = 0. For a lossy line, ∂P/∂z = -(1/2)(g|V|² + r|I|²).

Reflection Coefficient Γ

At the load, or z=0, Γ_L = V^–/V⁺,

V_L = V⁺+V^– = V⁺(1+Γ_L), I_L = I⁺-I^– = (V⁺/Z₀)(1-Γ_L);

Z_L = V_L/I_L = Z₀(1+Γ_L)/(1-Γ_L), Γ_L = (Z_L-Z₀)/(Z_L+Z₀);

Z_in(z) = V/I = Z₀(V⁺e^-jβz + V^–e^jβz)/(V⁺e^-jβz – V^–e^jβz) = Z0(1 + Γ_Le^2jβz)/(1 – Γ_Le^2jβz),

Γ(z) = Γ_Le^j2βz, which is periodic with period λ/2,  Γ_in(-l) = Γ_Le^-j2βl.

Note:

Γ_L = 0 if load is matched to line (Z_L = Z₀), which indicates no reflection;

Γ_L = +1 if load is open, which indicates total positive reflection;

Γ_L = -1 if load is shorted, which indicates total negative reflection.

Z_in(z) = Z₀(Z_L – jZ₀tanβz)/(Z₀ – jZ_Ltanβz).

For a matched line (Z_L = Z₀), Z_in(-l) = Z₀;

For a λ/4 line, βl = π/2, Z_in(-l) = Z₀²/Z_L, this line is called Z-transformer;

For a shorted line Z_L = 0, Z_in(-l) = jZ₀tanβl;

For an open line Z_L = ∞, Z_in(-l) = -jZ₀cotβl.

Standing Waves Ratio (SWR)

V = V⁺e^-jβz + V^–e^jβz, |V| = |V⁺+V^–e^2jβz|

At z = z₀ = nπ/β, V_max = |V⁺| + |V^–|, I_min = |V⁺/Z₀| – |V^–/Z₀|, Z_max = Z₀(1+|Γ|)/(1-|Γ|);

at z = z₀ ± λ/4, V_min = |V⁺| – |V^–|, I_max = |V⁺/Z₀| + |V^–/Z₀|, Z_min = Z₀(1-|Γ|)/(1+|Γ|).

We define the SWR as a measure of Z-mismatch:

SWR = V_max/V_min = (1+|Γ|)/(1-|Γ|).

For a matched line, SWR = 1;

For a shorted or open line, SWR = ∞.

 

For a general case of partial reflection, V(z) = V⁺(1-Γ_L)e^-jβz+2Γ_LV⁺cosβz

V(z,t) = V⁺(1-Γ_L)e^j(ωt-βz) + 2Γ_LV⁺cos(βz)e^jωt, which consists of a propagating wave and a standing wave, and the standing wave is proportional to Γ_L.

|Γ| = (SWR – 1)/(SWR + 1)

Power Flow in a TL

Maximum Power Transfer MPT: Z-Matching at Both Ends

Input: for max P_in, Z_in = Z_G*, where Z_in = Z₀(Z_L+jZ₀tanβl)/(Z₀+jZ_Ltanβl);

Output: for max P_L, Z_out = Z_L*, where Z_out = Z₀(Z_G+jZ₀tanβl)/(Z₀+jZ_Gtanβl).

Transmission Coefficient TL

The counterpart of the reflection coefficient is the transmission coefficient:
V(0) = V⁺ + V^– = T_L*V⁺, T_L = 1 + V^–/V⁺ = 1 + Γ_L.

Mismatch Losses

Return Loss RL

RL measures the reflection fraction of incident power,

RL(dB) = -10log(P^–/P⁺), where incident power P⁺ = |V⁺|²/(2Z₀), reflected power P^– = |V^–|²/(2Z₀),

hence RL = – 10log|Γ|² = -20log|Γ|.

Insertion Loss IL

IL measures the transmitted fraction of incident power,

IL(dB) = -10log(P_t/P⁺) = -10log(1 – P^–/P⁺),

hence IL = -10log(1-|Γ|²) is positive.

For matched load, Γ_L = 0, RL = ∞ dB, IL = 0 dB;

For open/short load, Γ_L = ±1, RL = 0 dB, IL = ∞ dB.