When we are sending an electrical signal from one point to another we need to worry about the wave nature of the signal, if the length of the cable between the two points is large compared to a fraction of the wavelength of the signal. The wavelength of the signal is proportional to 1/T where T is the frequency. Note that this frequency is not the same as the periodic frequency of the signal: If you are working with a 10-MHz digital signal with 1-ns risetimes, the frequencies required to reproduce them are several GHz. To get an idea of the order of magnitude, consider that the wavelength of a 100-MHz signal is about 3 m. Therefore, if you are dealing with a signal at 10 MHz, the wavelength is about 30 m, and if your circuit involves distances greater than about a meter, you need to be concerned about transmission line effects. At 100 MHz, the distance at which you need to start being concerned is just a few centimeters.

Transmission line theory involves some complex mathematics. I am not going to dwell on a lot of the mathematics in this paper, but rather state results. There are many texts available, which will go into plenty of depth on the mathematical foundations of the results presented here.

If we take a small piece of a cable, (small compared to the wavelength of the signals that we are dealing with), we can model it with the circuit shown below:

Considering the impedances L, C, RI and R2 to be impedances per unit length we can calculate the impedance seen when looking into an infinite length of this cable to be:

Where j is the imaginary number, and T is the frequency (v = 2p T.)

In most cable R1 is very close to 0, and R2 is very close to ý . This is known as lossless cable and the impedance equation will reduce to:

Where Z₀ is known as the characteristic impedance. This means that driving an infinite length cable looks the same to the driver as when it is driving a fixed resistance of value Z₀. Most types of cable have characteristic impedances that fall between about 50 W and 500 W . As an example, a common cable used in cable TV installations is RG59 which has an inductance of 370 nH/m and a capacitance of 67 pF/m. Using the equation above, yields Z₀ = 74.3 W . RG59 is commonly referred to as a 75-W cable.

If a transmission line is of finite length (and most of them are), and you place a lumped resistor of the value of the characteristic impedance at the end of the transmission line, then the line appears to be infinite in length to the driver. This is known as terminating the line. A terminated line is electrically the same as an infinite length line.

In free space an electromagnetic wave travels at the speed of light, c, or 3x10⁸ m/s. Inside a transmission line, the speed at which the waves travel is slower due to their interaction with the materials of the transmission line. The velocity of propagation is equal to:

The wavelength of the signal, l , can be calculated from:

therefore for RG59, the speed of propagation is about 2x10⁸ m/s or about 67% the speed of light and a signal with a frequency of 1 GHz, has a wavelength of about 20 cm.

When a signal traveling down a transmission line of characteristic impedance, Z₀, encounters a load of impedance Z_L, if Z_L Z₀ then there will be a reflection, which will combine with the signal going down the line to generate standing waves. There are various ways of quantifying this phenomenon, including reflection coefficient (r ), return loss (RL), and Voltage Standing Wave Ratio (VSWR).

If we send a signal with power P_IN into a transmission line, and P_REF is reflected back, then we can define return loss (RL) as P_IN/P_REF. In the ideal situation, where the transmission line is terminated in the characteristic impedance, P_REF = 0 and RL is infinite. Return loss can be expressed as a ratio or in dB.

A similar measurement is reflection coefficient, r , which is expressed in terms of the voltages. If V_IN is the voltage corresponding to P_IN, and V_REF is the voltage corresponding to P_REF then r =V_REF/V_IN.

VSWR is the Voltage Standing Wave Ratio, and is the ratio of the maximum value to the minimum value of the standing wave caused by the reflection. In the ideal sense, where there is no reflection, VSWR= 1. The maximum value of the standing wave is V_IN+V_REF and the minimum value of the standing wave will be V_IN-V_REF VSWR is therefore V_IN+V_REF /V_IN-V_REF .

Equations relating RL, p, Z₀, Z_L and VSWR to one another are:

RL = 1/r ²

VSWR =(1 + r )/(1 - r )

r = (Z_L-Z₀)/(Z_L+Z₀)

r = 1/ RL = (VSWR-1)/(VSWR+1)

In addition, RL and r are often expressed in dB rather than as linear numbers, this of course changes the equations that relate these parameters to one another.

When designing high-frequency circuits, where transmission-line effects come into play the best way to keep things working as you would expect is to make certain that all of the transmission lines are terminated in matching impedances. This usually involves designing a small network(often just one resistor) that can be used to terminate the line. For more complex matching circuits, and a method for determining them, see the application note from Hewlett Packard.