Let's imagine a small, infinitesimally short segment of a transmission line of length Δz. This segment can be modeled as a series impedance and a parallel admittance, based on our primary parameters.

● Series Impedance of segment: Zs = RΔz + jωLΔz = (R + jωL)Δz
● Shunt Admittance of segment: Ysh = GΔz + jωCΔz = (G + jωC)Δz

Now, let's apply Kirchhoff's Laws to this small segment:

1. Kirchhoff's Voltage Law (KVL) around the loop: Consider the voltage at point z, V(z), and the voltage at point z + Δz, V(z + Δz). As current I(z) flows through the series impedance of the segment, there's a voltage drop. V(z + Δz) = V(z) - I(z)(RΔz + jωLΔz)

Rearranging: V(z + Δz) - V(z) = -I(z)(R + jωL)Δz Dividing by Δz and taking the limit as Δz → 0 (which turns the difference into a derivative):
dzdV(z) = - (R + jωL)I(z) (Equation 2.2.1 - Voltage Differential Equation) This equation tells us that the rate of change of voltage along the line is proportional to the current and the series impedance per unit length.

1. Kirchhoff's Current Law (KCL) at the node: Consider the current entering the segment at z, I(z), and the current leaving at z + Δz, I(z + Δz). Some current also "leaks" through the shunt admittance due to the voltage V(z). I(z + Δz) = I(z) - V(z)(GΔz + jωCΔz)

Rearranging: I(z + Δz) - I(z) = -V(z)(G + jωC)Δz Dividing by Δz and taking the limit as Δz → 0:
dzdI(z) = - (G + jωC)V(z) (Equation 2.2.2 - Current Differential Equation) This equation tells us that the rate of change of current along the line is proportional to the voltage and the shunt admittance per unit length.

Now, we have two coupled first-order differential equations. To solve them, we can differentiate one with respect to z and substitute the other:

Differentiate Equation 2.2.1 with respect to z:
dz2d2V(z) = - (R + jωL)dzdI(z)
Now, substitute Equation 2.2.2 into this:

dz2d2V(z) = - (R + jωL)[- (G + jωC)V(z)]
dz2d2V(z) = (R + jωL)(G + jωC)V(z)
Let's define the propagation constant squared as γ²: γ² = (R + jωL)(G + jωC)
So, the voltage equation becomes a standard second-order linear differential equation:
dz2d2V(z) = γ²V(z)

The general solution for this type of differential equation is a sum of two exponential terms: V(z) = V0+ e^{-γz} + V0− e^{+γz} (General Voltage Solution)
Similarly, if we differentiated Equation 2.2.2 and substituted Equation 2.2.1, we would get the same form for the current:
dz2d2I(z) = γ²I(z)
With the general solution: I(z) = I0+ e^{-γz} + I0− e^{+γz} (General Current Solution).

We've already introduced these in detail in Section 2.1, but let's re-emphasize their importance and provide precise formulas:

● Characteristic Impedance (Z0):
○ Definition: The impedance seen looking into an infinitely long line, or a line terminated with its characteristic impedance. It's the ratio of voltage to current for a pure forward-traveling wave.
○ Formula: Z0 = G + jωCR + jωL (Ohms, Ω)
○ For lossless lines (R = 0, G = 0): Z0 = CL (real and purely resistive)

● Propagation Constant (γ):
○ Definition: A complex number describing how a wave changes in both amplitude and phase per unit length as it propagates.
○ Formula: γ = (R + jωL)(G + jωC) (per meter, 1/m)
○ It is composed of two parts: γ = α + jβ.

● Attenuation Constant (α):
○ Definition: The real part of γ, representing the rate at which the wave's amplitude decays due to losses (resistance in conductors, leakage in dielectric).
○ Formula: α = Re((R + jωL)(G + jωC)) (Nepers/meter, Np/m)
○ For lossless lines: α = 0

● Phase Constant (β):
○ Definition: The imaginary part of γ, representing the phase shift per unit length as the wave propagates. It dictates the wavelength and phase velocity on the line.
○ Formula: β = Im((R + jωL)(G + jωC)) (Radians/meter, rad/m)
○ For lossless lines: β = ωLC.