# Transmission lines and SWR.

When the phrase "transmission line" is encountered most people will automatically think of coaxial cable. However, two wire line can also be a transmission line. Ordinary lamp cord, speaker wire, the old style 300 ohm TV twin lead, open wire line, (sometimes called mouse ladder), or the twisted pair of wires used by land line telephones, all of these are transmission lines and each kind has its own characteristic impedance.

## Audio Cable and Transmission Lines.

There are those who believe that speaker cable and shielded audio cable need to have a characteristic impedance equal to that of the load. A length of shielded audio cable or a pair of wires connecting an amplifier to a speaker becomes a transmission line only when the cable becomes about 1/10 wavelength at the highest frequency to be sent over the line. The wave length of a frequency of 20,000 Hz is,

Wavelength in meters = 3 x 108 / f

Wavelength = 3 x 108 / (2 x 104) = 1.5 x 104
Or 15 km.

1/10 of this is 1.5 km which is 0.932 miles.

But there's more as the infomercials say. The velocity of waves along a transmission line is usually about 60% of the speed of light. This is a somewhat pessimistic value as most coaxial cable comes in at 0.66 and open wire line at about 0.97.

The length of the waves are shorter when traveling down a transmission line than they are in free space. This velocity factor as it is known depends on the material with which the wires making up the line are insulated.

Taking the velocity factor into account the critical length of line becomes 0.559 miles. So, your speaker cable impedance doesn't need to match the speaker impedance, and the impedance of the shielded cable between components doesn't need to match the input impedance of your amplifier unless these cables are a little over half a mile long.

## When to Match?

Impedance matching is required when the transmission line is more than 1/10 of a wavelength. Radio frequencies are higher and so have shorter waves than audio and this is where impedance matching is important. The frequency of the top end of the AM broadcast band is 1700 kHz so lets see what length of line constitutes a transmission line.

Wavelength = 3 x 108 / (1700 kHz) = 176.5 m = 579 ft.

Taking the velocity factor into account a 1/10 wavelength of line is 34.7 ft. If you are planning to build some kind of outdoor AM antenna you will need to match impedances. If you have a couple of feet of wire between a large indoor loop antenna and your radio you don't need to worry.

## What is Matching.

A matched system consists of a source of signal which has a resistive impedance which is equal to the characteristic impedance of the line and a load which is also resistive and equal to the characteristic impedance of the line.

Most data transmission systems have either the load or the source impedance matched to the line but not both. However, when it comes to RF transmission, especially if there is power of 1 watt or more, matching of both source and load are imperative.

Now, I'm going to contradict what I just said. There are conditions under which the source and load do not have to be matched to the line impedance. They DO have to be matched to each other. Before I can explain this situation I must explain what happens on a transmission line when impedances are not matched.

The input impedance of a transmission line is a function of the length of the line in wavelengths and the nature of the termination.

An open ended transmission line will show an infinite impedance at DC. As the frequency is increased until the line is 1/4 wave long the impedance will be very low. It would be zero if there were no losses in the line. At 1/2 wave the impedance is back near infinity. Not at infinity due to losses. After that as frequency is increased it keeps repeating, low at 3/4 wave, high at 1 wave length, Low again at 1-1/4, and so on and so on.

If the line is shorted the impedance is zero at DC, high at 1/4, low at 1/2, high at 3/4, and so on.

Now suppose the line is terminated with a resistor which is not equal to the characteristic impedance of the line. Suppose it is a 50 ohm line but the termination is 100 ohms. At DC the input impedance is 100 ohms. At 1/4 wavelength the input impedance is 25 ohms. This comes from the equation

Z1 Z2 = Zo2

where Z1 is the input impedance of the line, Z2 is the terminating impedance, and Zo is the characteristic impedance of the line. At 1/2 wave length the input impedance is 100 ohms. This repeats, 25 ohms at 3/4 wavelength, 100 ohms at 1 wavelength, and so on.

What happens in between is that the impedance is between the values at 1/4 and 1/2 wave but reactive. You need a smith chart to figure it out. Also if the termination is reactive you need a smith chart. Smith chart calculators are available at this location.

Now for the final example if the terminating impedance is equal to the characteristic impedance, 50 ohms in our example, the input impedance is 50 ohms regardless of the length of the line and the frequency neglecting losses. This is known as a matched line.

If you have an unknown piece of coaxial cable and some way of measuring impedance at RF such as the Millen impedance bridge or the Heathkit noise bridge you can determine the characteristic impedance as follows. Connect a resistor of a value that you are sure isn't the same as the line impedance. Start at a low frequency and work your way up until the input impedance becomes resistive and read the bridge. Apply the equation given above. I have measured the characteristic impedance of some Radio Shack audio cable as 25 ohms.

As you might imagine shorted transmission lines can be, and are, used in place of tuned circuits in VHF and UHF equipment. 1/4 wave length transmission lines can be used to transform impedances in a single frequency or narrow band system. 1/2 wave or multiples there of lines can be used to reflect the terminating impedance to the line input regardless of the impedance of the load or the characteristic impedance of the line. Within reason. This is known as a mismatched line. Such a line has standing waves on it.

## What is a Standing Wave.

Tie a rope to some stationary object. It's better if the rope is vertical so a tree limb is probably called for. Hold the free end of the rope in your hand and make a single motion in one direction. You will see a wave go up the rope and when it reaches the top it will return back down the rope. This is an example of a single pulse being reflected.

If you wave the rope in a sinusoidal motion you will send waves up the rope that will be reflected back to you. If you have good eye hand coordination you can find a frequency at which the middle of the rope is almost stationary while the rest is waving about.

If you send radio waves up a transmission line and the termination is matched to the characteristic impedance of the line all of the energy will be absorbed. If you could measure the voltage on the center conductor of a coaxial line at random points along its length you would measure a constant voltage all the way along.

If there is a mismatch some of the energy will be reflected back along the line. The return wave combines with the incident wave and in some places they reinforce and in others they partially cancel. If you could measure the center conductor voltage along such a line you would see peaks and valleys.

## Mismatched Systems and SWR.

When there is a mismatch between the load impedance and the characteristic impedance of the line the term SWR comes into use. It stands for Standing Wave Ratio. The SWR is the ratio of the maximum voltage to the minimum voltage. Some engineering texts use the phrase Voltage Standing Wave Ratio abbreviated VSWR. SWR can be defined in terms of maximum and minimum currents in the conductors of the line so we will use the more general, and shorter, term SWR.

A mismatched transmission line is not automatically a bad thing. Suppose after arriving home with that roll of coaxial cable you bought for a song at a ham fest you find that it has a characteristic impedance of 125 ohms. All you have to do to use it on your 40 meter dipole is to make it 1/2 wavelength or 1 wavelength, taking velocity factor into account and the impedance at the transmitter end will be the same as the impedance of the antenna feed point.

To arrive at the proper length use rope to estimate how long the coaxial cable to the antenna needs to be. Then decide if the coax needs to be 1/2, 1, or 1-1/2 wavelengths long. Don't forget to look up the velocity factor for that type of cable and take it into account. Now cut the cable several feet longer than the required length and install a connector on one end. Because you are going to be removing short lengths of cable you will have to use some kind of temporary jury rig connector on the other end. The error this will introduce at 7 MHz is insignificant. Now connect your 50 ohm dummy load to one end and your transmitter with an SWR bridge to the other. Read the SWR and write it down. Cut off a few inches, less than you made the cable too long, and again read the SWR. It had better be less. If it is more you have committed a major booboo. Keep trimming the cable until the SWR gets below 1.1 to 1. You may find that it starts to go back up with additional trimming. Most ham station SWR meters are not that accurate. If it does continue to fall you have a good meter. When the length is right feed the cable through the hole in the wall and connect it to your antenna's feed point.

## Why is the SWR Meter reading very close to 1 to 1?

There will be a definite SWR on the line. So why is the SWR meter reading a very low value. That's because it isn't really an SWR meter. To truly determine the SWR on a line you have to measure the voltage at many points along it for at least 1/2 wavelength. This is done at microwave frequencies with a device called a slotted line. Such a device isn't really practical for waves longer than this. The so called SWR meter is really a bridge that is fixed tuned to 50 ohms and zero reactance. When there is a 50 ohm resistive load on it it reads 1 to 1 SWR. This is true even if the line between the load and meter is not matched and has a fairly large SWR.

## Calculating SWR.

Although SWR at HF cannot be measured directly it can be calculated. If the characteristic impedance of the line and the load impedance are known. Lets look at the case where Zo is greater than ZL. At 1/4 wave back down the line from the load the impedance is given by,

Z = Zo2 / ZL.

Since Zo is greater than ZL the value Z will be greater than Zo. Therefore Z is the maximum impedance and ZL is the minimum impedance along the line. The impedance ratio, IR, is given by,

IR = Z / ZL = Zo2 / ZL / ZL.

IR = Zo2 / ZL2

Since voltage is proportional to the square root of impedance the voltage ratio is given by,

VSWR = Zo / ZL, if Zo > ZL

and

VSWR = ZL / Zo if Zo < ZL.

In our example above we never said what kind of antenna the ham was using but lets say it was a straight dipole which has an impedance of approximately 75 ohms.

SWR = Zo / ZL = 125 / 75 = 1.67 to 1.

## What's all the Fuss About?

If a so called SWR meter doesn't really measure the SWR on a line and it's alright to have some standing waves on the line what's all the fuss about? I am tempted to say it's all much ado about nothing but somebody has already used that phrase. A length of coaxial cable with a high SWR will introduce more loss than a cable with no standing waves, known as a flat line. At HF the losses are usually not large enough to be worried about. I think there is a way to calculate losses on a line with a high SWR but I don't remember how to do it.

Contrary to popular belief a high SWR does not by itself cause RF on the mike or key. In improperly constructed antenna or inadequate ground are the most common causes of this unpleasant problem.

The problem with a tuned transmission line is that you are restricted to operating in a narrow range of frequencies. The 40 meter example sighted would have to be tuned either for the phone or CW band. Most hams prefer to use 75 ohm cable for a straight dipole antenna or 52 ohm cable for an inverted V. There may be a slight mismatch but it is small enough that you don't need to lose any sleep over it. In summary then unless the line is terminated with a resistive load equal to the characteristic impedance of the line, the input impedance will vary all over the place as frequency is changed.

## A Popular Misconception.

Many hams have the mistaken idea that when there is a mismatch that power is being reflected back down the line from the antenna. It is believed that this so called reflected power causes RF on the mike or key, or is absorbed by the final amplifier increasing its dissipation. This misconception has been reinforced by many articles and advertisements which speak of reflected power.

The energy or power is not being reflected back to the transmitter, the fact is that the power is never taken from the final in the first place. A mismatched transmission line presents a resistive component and a reactive component. The resistive component takes power but the reactive component does not. Not only that but the reactive component is tuned out by the output matching circuit in the transmitter so the tube or transistor is presented with a purely resistive load. It is possible that there is so much reactance that the transmitter can't tune it all out but if a Johnson Matchbox is between the final and the transmission line the transmitter final can work into a 50 ohm nonreactive load.

## Conclusion.

This is by no means a complete and rigorous presentation. Because transmission lines have distributed capacitance and inductance they lend themselves well to the use of calculus. There are some rather thick books on the subject which make liberal use of those arcane mathematics. If you have the necessary knowledge, the time, and the stomach for it, get hold of an electrical engineering book and have at it. The above is intended for those who just want the bottom line without the pain.