A transmission line is, in the field of electrical engineering, a device over which electrical signals are transmitted. The term transmission line is typically used to describe circuits that operate in the radio frequency (RF) regime, where the wavelength is on the order of, or less than, the physical size of the transmission line. At these frequencies, the voltages and currents present at the input and output of a transmission line are no longer simple functions (i.e. V=IR), their magnitude and phase now vary over the length of the line.

```i(z, t)                 i(z + Δz, t)
-->                         -->
o-------R-----L-----------------o
|     |
+                 |     |       +
V(z, t)           G     C      V(z + Δz, t)
-                 |     |       -
|     |
o-------------------------------o

|----------- Δz ----------------|
```

Consider the above figure, which represents a small transmission line segment of length Δz. This transmission line is represented as a parallel two-wire line, but real transmission lines may be of many various shapes. We model this small length of the transmission line as a lumped-element circuit. The lumped element parameters are:

• R: the series resistance of the line.
• L: the series inductance of the line. This is due to the magnetic field created around the two conductors. The magnetic field varies depending on the configuration of the line.
• C: the shunt capacitance between the two conductors. This capacitance also varies depending on the type of line.
• G: the shunt admittance between the two conductors. This also varies depending on the line. For conductors separated by air this may be quite low. For a coax cable with a foam dielectric filling, it may be higher.

A transmission line can be thought of as a number of these small lumped elements connected together in series.

From the diagram above, applying the Kirchoff voltage law gives us

```v(z, t) - R Δz i(z, t) - L Δz δi(z,t)/δt - v(z + Δz, t) = 0
```

and applying the Kirchoff current law gives us

```i(z, t) - G Δz v(z + Δz, t) - C Δz δv(z + Δz, t)/δt - i(z + Δz, t) = 0
```

Dividing by Δz and taking the limit as Δz approaches zero, we derive the relationships for a differential transmission line segment

```δv(z, t)/δz = -R i(z, t) - L δi(z, t)/δt
δi(z, t)/δz = -G v(z, t) - C δv(z, t)/δt
```

If we consider these time domain equations in the sinusoidal steady state (frequency or phasor domain), they can be written as

```dV(z) / dz = -(R + jwL) I(z)
dI(z) / dz = -(G + jwC) V(z)
```

where w is the radian frequency. Simultaneous solution of these equations yield the homogeneous wave equations

```d2V(z) / dz2 - γ2 V(z) = 0
d2I(z) / dz2 - γ2 I(z) = 0
```

where γ = α + j β = sqrt((R + jwL)(G + jwC))

These wave equations have solutions

```V(z) = Vo+ e-γz - Vo- e+γz
I(z) = Io+ e-γz - Io- e+γz
```

The above two equations represent waves of voltage and current traveling in the +z and -z directions along the line. This means that the transmission line will support waves traveling in both directions simultaneously.

Note that the exponential term γ has both real and complex values. The R and G terms act to create loss along the transmission line. A wave will therefore have an exponential amplitude envelope along a lossy transmission line.

In the case of a lossless transmission line, i.e. R = G = 0, γ reduces to

```γ = jw sqrt(LC) = jβ
```

where β is called the wavenumber or propagation constant on the line. The wavelength on the line is 2π/β.

It follows from the above discussion that if a sinusoidal voltage is applied to one end of a transmission line, a sinusoidal voltage and current wave will propagate down the line. The voltage at the other end of the line is a function of time, distance from the input, and the constituent parameters of the line.

Now, what physically happens when a voltage is applied at the end of the transmission line? This is the part that most EE professors fail to drive home. This voltage excites an electromagnetic wave that propagates down the line between the two conductors in very specific spatial field configuration called a mode. Depending on the frequency of the input signal, multiple modes can be excited along the line. The line becomes, effectively, an electromagnetic waveguide. Power transmission along the line occurs totally through the EM wave formed between the conductors; it is not transmitted through the conductors themselves.

Sources:

D. M. Pozar, Microwave Engineering, Addison Wesley, 1990.