Derive input impedance of transmission line

Consider a lossless line of length l, characteristic impedance Z₀ and propagation constant jβ, with a source (V_s, Z_s) at the sending end and a load Z_L at the receiving end. Let z be the distance from the load toward the source.

 Z_s        I_in                      I_L
  o---/\/\--->--------- l ------------>---o
 V_s ~     V_in      Z0, β            V_L   Z_L
  o----------------------------------------o

Voltage and current on the line

V(z) = V⁺ e^{jβz} + V⁻ e^{−jβz}
I(z) = (V⁺ e^{jβz} − V⁻ e^{−jβz}) / Z₀

At the load (z = 0): V_L = V⁺ + V⁻, I_L = (V⁺ − V⁻)/Z₀, and Z_L = V_L/I_L. Solving:

V⁺ = (I_L/2)(Z_L + Z₀),   V⁻ = (I_L/2)(Z_L − Z₀)

Input impedance at z = l

V(l) = (I_L/2)[(Z_L + Z₀)e^{jβl} + (Z_L − Z₀)e^{−jβl}]
     = I_L [Z_L cos βl + jZ₀ sin βl]

I(l) = (I_L/2Z₀)[(Z_L + Z₀)e^{jβl} − (Z_L − Z₀)e^{−jβl}]
     = I_L [cos βl + j(Z_L/Z₀) sin βl]

Dividing, and then dividing the numerator and denominator by cos βl:

Z_in = V(l)/I(l) = Z₀ (Z_L + jZ₀ tan βl) / (Z₀ + jZ_L tan βl)

For a lossy line, replace jβ by γ and tan βl by tanh γl: Z_in = Z₀ (Z_L + Z₀ tanh γl)/(Z₀ + Z_L tanh γl).

In terms of the reflection coefficient: Z_in = Z₀(1 + Γ_in)/(1 − Γ_in), with Γ_in = Γ_L e^{−j2βl}.

Effect of source impedance Z_s

The source impedance and the input impedance form a voltage divider:

V_in = V_s Z_in / (Z_s + Z_in)
I_in = V_s / (Z_s + Z_in)

The power delivered to the line is P_in = ½ Re(V_in I_in*).

Special cases: Z_L = Z₀ gives Z_in = Z₀ for any l (matched); l = λ/4 gives Z_in = Z₀²/Z_L; l = λ/2 gives Z_in = Z_L; a short circuit gives Z_in = jZ₀ tan βl; an open circuit gives Z_in = −jZ₀ cot βl.