Lossless Transmission Lines (Part-IV)

Ṽ_g

(i)

Z_g

(ii)

A

(iii)

A’

(iv)

B

(v)

B’

(vi)

Z(d)

(vii)

Z_0

(viii)

C

(ix)

C’

(x)

Z_L

(xi)

d = l

(xii)

d

(xiii)

0

(xiv)

Z(d) = Z0((1 + Γ(e^-j2βd))/(1 - Γ(e^-j2βd)))

Wave Impedance Expression: Wave impedance in terms of reflection coefficient

Z(d) = Z0((1 + Γ_d)/(1 - Γ_d))

Wave Impedance Using Phase-Shifted Reflection Coefficient: Alternate form of wave impedance

Γ_d = Γ(e^-j2βd)

Phase-Shifted Voltage Reflection Coefficient: Reflection coefficient at distance d

Γ_d = |Γ|e^(j(θ_r - 2βd))

Expanded Form of Γ_d

Z0 = (V0^+)/(I0^+) = -((V0^-)/(I0^-))

Characteristic Impedance Z0: Ratio relating voltage and current of each traveling wave individually

Impedance seen when “looking” toward the load from a point on the transmission line.

Wave Impedance Interpretation

Equivalent-Circuit Principle for Transmission Lines

Any section of transmission line to the right of a point can be replaced by an equivalent impedance Z(d)

Z_in = Z(d = l)

Input Impedance: Wave impedance measured at the generator end of the line

Z_in = Z0((1 + Γ_l)/(1 - Γ_l))

Input Impedance Formula: Input impedance in terms of the reflection coefficient at the input

Γ_l = Γ(e^-j2βl)

Input-End Reflection Coefficient: Reflection coefficient evaluated at the source end

Γ_l = |Γ|e^(j(θ_r - 2βl))

Expanded Input-End Reflection Coefficient: Input-end reflection coefficient in magnitude-angle form

e^jβl = cos(βl) + j(sin(βl))

Euler Relation (Positive Exponential)

e^-jβl = cos(βl) - j(sin(βl))

Euler Relation (Negative Exponential)

z_L = (Z_L)/(Z_0)

Normalized Load Impedance: Load impedance normalized to characteristic impedance

Z_in = Z0(((z_L)cos(βl) + jsin(βl))/((z_L)cos(βl) + jsin(βl)))

Input Impedance in Terms of Normalized Load Impedance: Transmission-line input impedance equation

Z_in = Z0(((z_L) + jtan(βl))/(1 + jtan(βl)))

Tangent Form of Input Impedance: Alternate input impedance equation