# Mutual impedance

April 1, 2001

Matters of impedance are deeply significant in radio engineering. In audio, with lower frequencies, the actual component values are generally considerably larger than those used in RF work, but the effects of wrong component values are frequently more aurally apparent than similar failures in RF. In the latter, incorrect components may affect impedance and produce catastrophic problems, putting the station off the air.

The audio engineer has been accustomed to working with values of 150V to 600V for many years. The shift to bridging inputs has brought input impedances up to 10kV or more. Today, however, values of 75V and 110V are used with digital audio.

The RF engineer is still mainly concerned with 50V transmission lines, which makes life a little easier and more consistent. Correct line, load and operating impedances are of vital importance to directional antenna designers and operating engineers.

All individual towers have their own characteristic impedance, which is approximated using this formula:

Z0 =60(ln 2G/A-1)V

Where ln = natural logarithm
A = equivalent radius of antenna
G = height of antenna

In the case of a single non-directional antenna system, we are not normally concerned with characteristic impedance at this point, because the approximate base operating impedance can be obtained from published curves. The base operating impedance will be measured after all construction is completed, all appendages are added and the antenna is energized for the first time. The desired antenna current will be determined based on I2R and the transmitter operating at authorized power.

However, whenever an antenna is placed within the field of another antenna, the operating impedance of each antenna will be changed. The degree of change is controlled by such things as proximity, current in the antenna, antenna length and the method of connecting the base into the system. This means that, for the directional antenna designer, a new term has to be considered. This is known as mutual impedance.

A directional AM broadcast array consists of two or more radiators. These may be arranged in almost any formation as needed to produce the desired radiation pattern. The array will have one reference tower and as many additional towers as needed. The design of the system will call for each tower to have unique specifications of current, phase and base operating impedance to produce the desired pattern.

Usually, a change in one tower's operating parameters will result in a small (or large) change in other towers. How many towers are affected, and how much, depends on the location of the altered tower and the amount of change.

If incorrect antenna system operation is suspected, it is very important not to change the phasor settings of the reference tower. Any change made in the reference tower's parameters will affect every other tower and cause a lot of unnecessary work. Occasionally, in the case of a two-tower array, very small adjustments may be made to the reference tower in order to “tweak” a reading. But this is a special case, where the resulting parameter changes can be easily controlled, and it should not be done as normal practice.

During the design of the directional array, it becomes necessary to determine the mutual impedance arising in the array. A directional antenna designed with a very high mutual impedance will tend to be unstable and oversensitive to adjustments. Therefore, it is important to check this parameter early in the design stage. Short tower spacing of less than about 65 degrees is usually an indication that mutual impedance may be excessive, which can lead to feeder and phasor design problems.

The relationship between the towers is rather like that in a transformer, where the coupling between the windings controls the degree of mutual inductance. The voltage developed in a second tower is related to the phase angle, and it depends on their spacing. This requires that the coupling factor be described in terms of angle and magnitude — in other words, impedance.

It's been a long time since I heard of any engineer doing this, but it is possible to measure the mutual impedance of a pair of towers. Knowledge of a method might come in handy one day in the field.

## Measuring mutual impedance

Assume two towers: North (N) and South (S). Float tower S and measure the base impedance of tower N. Thus: ZN = RN +jXN.

Then, float tower N and measure the base impedance of tower S. Thus: ZS = RS + jXS.

Ground the base of tower S with a reactance equal to — jXN. This will self resonate the south tower. Measure the base impedance of the north tower. Thus: Zg = Rg + jXg.

It is usual to refer to the mutual impedance as ZNS.

ZNS = =BS (ZN-Zg)

This becomes a complex number, and the root can be obtained by halving the angle and finding the square root of the magnitude. This gives one number for magnitude, which can be either positive or negative. It is generally possible to determine which sign is correct by inspection and comparison with the tower data, or a chart such as Figure 1.

E-mail John at batcom@bright.net.

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