# `Transmission Lines`

## Introduction to Transmission Lines

How do you connect a receiver or transmitter to an antenna? Simple - a transmission line. You are no doubt familiar with transmission lines (sometimes abbreviated as tx lines). If you plug any electric device into a wall outlet, the wires that connect the wall outlet to the device is a transmission line.

However, transmission lines behave very oddly at high frequencies. In traditional (low-frequency) circuit theory, wires connect devices, but have zero resistance. There is no phase delay across wires; and a short-circuited line always yields zero resistance.

For high-frequency transmission lines, things behave quite differently. For instance, short-circuits can actually have an infinite impedance; open-circuits can behave like short-circuited wires. The impedance of some load (ZL=XL+jYL) can be transformed at the terminals of the transmission line to an impedance much different than ZL. The goal of this tutorial is to understand transmission lines and the reasons for their odd effects.

Let's start by examining a diagram. A sinusoidal voltage source with associated impedance ZS is attached to a load ZA (which could be an antenna or some other device - in the circuit diagram we simply view it as an impedance called a load). The load and the source are connected via a transmission line of length L:

In traditional low-frequency circuit analysis, the transmission line would not matter. As a result, the current that flows in the circuit would simply be:

However, in the high frequency case, the length L of the transmission line can significantly affect the results. To determine the current that flows in the circuit, we would need to know what the input impedance is, Zin, viewed from the terminals of the transmission line:

The resultant current that flows will simply be:

Since antennas are high-frequency devices (in the sense that their size is on the order of a half wavlength or more), transmission line effects are often VERY important. That is, if the length L of the transmission line significantly alters Zin, then the current into the antenna from the source will be very small. Consequently, we will not be delivering power properly to the antenna. The same problems hold true in the receiving mode: a transmission line can skew impedance of the receiver sufficiently that almost no power is transferred from the antenna.

Hence, a thorough understanding of antenna theory requires an understanding of transmission lines. A great antenna can be hooked up to a great receiver, but if it is done with a length of transmission line at high frequencies, the system will not work properly.

Examples of common transmission lines include the coaxial cable, the microstrip line which commonly feeds patch/microstrip antennas, and the two wire line:

### When are transmission line effects significant?

We know that at low frequencies, transmission lines don't affect power transfer in practical applications we use every day. However, at high frequencies, even short lengths of transmission lines will affect the power transfer. Why is there this difference?

### It is not the length of the transmission line, or what frequency we operate at that determines if a transmission line will affect a circuit. What matters is how long the transmission line is, measured in wavelengths at the frequency of interest.

If a transmission line has a length greater than about 10% of a wavelength, then the line length will noticably affect the circuit's impedance. Let's look at some examples to make this clear:

Let's say you plug your vaccuum cleaner into a wall outlet. The chord (transmission line) that connects the power to the motor is 10 meters long. The power is supplied at 60 Hz. Should transmission line effects be taken into account?:

Answer: The wavelength at 60 Hz is 5000 km (5 million meters). Hence, the transmission line in this case is 10/5,000,000 = 0.000002 wavelengths (2*10^-6 wavlengths) long. As a result, the transmission line is very short relative to a wavelength, and therefore will not have much impact on the device.

Example #2. Suppose a wireless device is transmitting at 4 GHz. Suppose also that a receiver is connect to a microstrip antenna via a microstrip transmission line that is 2.5 centimeters (cm) long. Should transmission line effects be taken into account?

Answer: The wavelength at 4 GHz (4*10^9 Hz) is 7.5 cm. The transmission line is 2.5 cm long. Hence, the transmission line is 0.33 wavelengths long. Since this is a significant fraction of a wavelength (33%), the length of the line must be taken into account in analyzing the reciever/transmission line/antenna system.

Hopefully we now understand when a transmission line will affect the circuit. If you don't, read the last two examples again until you understand. What we don't understand yet, is how the transmission line messes things up. This will be covered in the next sections.

Next: Transmission Line Models and Z0

Top: Transmission Lines (Home)

Antenna Tutorial (Home)