...The Coaxial Line
- impedance matching
Transmission lines carry RF power from one place to another. At radio frequencies if the line is long compared to the wavelength, it will radiate power, making it inefficient, so that less power is received at the other end. The longer the wire, the more power is lost as the line acts like a long wire antenna radiating along its length. If the line is really lossy, more power may be lost than is lost in the resistance of conductors and dielectric insulating materials. While it is not possible to eliminate power lost by resistance completely, that lost through radiation can and should be avoided...
This is achieved by using two conductors arranged and operated so the electromagnetic field from one is balanced everywhere by an equal and opposite field from the other. There are two basic types of transmission line: lines which have conductors in parallel, and the coaxial line which achieves balance by having two conductors, one inside the other.
In the coaxial line, the current flowing on the inner conductor is balanced by an equal current flowing in the opposite direction on the inside surface of the outer conductor. Because of skin effect, the current on the inner surface of the outer conductor does not penetrate far enough to appear on the outside surface.
With the currents flowing on the conductors inside, the total electromagnetic field outside the coaxial line is always zero. This is because the outer conductor acts as a shield at radio frequencies.
Theoretically speaking, if a pulse of current travels along two parallel wires at the speed of light, it will travel 30 metres in 0.1 microseconds, 60 metres in 0.2 microseconds, 90 metres in 0.3 microseconds, and so on.
With a pulse of alternating current at the frequency of 10 MHz (10 million cycles per second), each cycle will take 0.1 microseconds, which means there will be one complete cycle every 30 metres. This is the distance of one wavelength. At it travels the distance of one wavelength, the applied voltage goes through a cycle of values. Assuming there are no losses, each cycle will be the same. If you insert an ammeter in either conductor, you will get the same reading at any point along the line. This is because an ammeter averages the current over the whole cycle. (The difference in phases that exists cannot be shown by an ammeter.)
However, in reality, current can only travel at the speed of light if it is in a vacuum. When the insulation between the conductors is air, the speed comes quite close to the speed of light. With other dielectrics, the speed is less, with the result that the current will not travel as far in one cycle as it would in free space. How much less it travels will depend upon the type of dielectric.
Because a line has both capacitance and inductance - in effect, is composed of a whole series of small inductors and capacitors - each inductance limits the rate at which each immediately following capacitor can be charged, creating resistance. As a result the current flowing along a line depends directly on the voltage. This apparent resistance is known as characteristic impedance.
The value of the characteristic impedance of a line is influenced by the diameter of the conductor and the amount of space between the conductors. A line with large conductors spaced close together will have a relatively low characteristic impedance. On the other hand, a line with thin conductors spaced wide apart will have a high characteristic impedance. In reality, a coaxial line can have a characteristic impedance ranging from 30 to 100 W, whereas parallel conductor lines will vary from around 200-800 W.
When a line is terminated at the output end in a load that is a pure resistance of a value equal to the characteristic impedance of the line, the line is matched. The load looks like more transmission line of the same characteristic impedance and all the current is completely absorbed. The current in the line is equal to the applied voltage divided by the characteristic impedance.
Where the load at the end of the line differs, the line is mismatched and the larger the variation, the greater the mismatch and the smaller the amount of current absorbed. In a mismatched line, only part of the current is absorbed. This is known as incident or forward power. Power that is not absorbed then travels back down the line as reflected power. The ratio of voltage to current of the reflected power is the same, because it also, is determined by the characteristic impedance of the line.
In the case of AC power, both the incident and reflected voltage are simultaneously present all along the line. The actual voltage at any point along the line comprises the sum of the incident and reflected components, as does the current. The greater the mismatch, the greater the amount of reflected power. In the most extreme cases - a short circuit and an open circuit - all of the power is reflected.
Where there is mismatch, standing waves are created as the two waves of equal frequency and intensity travel in opposite directions along the line. Where the load is greater than the characteristic impedance, the voltage is highest and the current lowest at the load. Where the load is less than the characteristic impedance, the opposite occurs. At 180° (1/2 wave) the voltage and current have the same values as they do at the load. At 90° (1/4 wave) and 270°(3/4 wave) these values are reversed. These values are duplicated at every uneven multiple of 1/4 wave and any multiple of 1/2 wave from the load.
In this diagram the load is less than the Characteristic Impedance Zo. Voltage is represented by ‘E’ and current by ‘I’. The E1 vector positions relate to the incident voltage and E2 to the reflective voltage. Similarly I1 relates to the incident current and I2 to the reflective current. Because it takes time for the power to be transferred along the line, the reflective voltage E2 and reflective current I2 lag the incident voltage and current at E1 and I1.
VSWR & Mismatch
The ratio of the maximum to minimum voltage (or current) can be measured to obtain the VSWR. If the load contains no reactance, the SWR is equal to the ratio between the load resistance and the characteristic impedance of the line. Where the load resistance is higher than the characteristic impedance, the load resistance value is divided by that of the characteristic impedance to obtain the SWR. Where the load resistance is lower, the equation is inverted.
Forward and Reflected Power
The greater the difference between the characteristic impedance and load resistance values, the greater the mismatch and the larger the SWR. With open and short circuits the SWR is infinite as the voltage and current are zero at the minimum points, total reflection occurs at the load and both the incident and reflected power have equal amplitudes.
In the case of the perfectly matched line, the voltage and current is constant and does not change in value, having a ratio of 1:1. Graphically, this is represented by a straight line.
where the load is less than the characteristic impedance
(I = Current, E = Voltage)
In a transmission line, voltage and current are in phase at the load, at every point that is a multiple of 90° from the load, and also at every point that is a multiple of 90° from the load. The value of the resistance measured in a terminated line depends on the length of the line. Where the length is 90° or an uneven multiple of 90° (high voltage/low current), the resistance is greater than at the load. Where the length is 180° or a multiple of 180°, the voltage and current are the same as at the load, showing a resistance equal to the actual load resistance.
The current and voltage are exactly in phase at points that are multiples of 90° from the load (R), showing pure resistance. Elsewhere the current is either ahead or behind the voltage. Where the current is behind the voltage, the input impedance of the line has inductive reactance, where it is ahead, it has capacitive reactance.
Voltage (E) and Current (I) where the load is less than the characteristic impedance (I)
Where the load resistance is LESS than the characteristic impedance, (R < Zo), the reactance is inductive in every uneven 90° of the line from the load to the power source and capacitive in every even 90° section towards the power source. Where the load resistance is HIGHER, (R > Zo) the reverse occurs.
Voltage (E) and Current (I) where the load is greater than the characteristic impedance
The input impedance is mainly resistive at all line lengths when the SWR is low, but can have a relatively high reactive component when it is high. This can be shown by a series of resistances and reactances, usually denoted as the load ±jX, the j indicating that the two values cannot be added directly together and X representing the value:”+ jX” denotes inductive reactance, “- jX” denotes capacitive reactance.
In the case of a transmission line coupled to an antenna designed to work at a single spot frequency, the resonant frequency, the load can be fairly close to being a pure resistance. However, with an antenna that operates across a range of frequencies, this will only be true at the resonant frequency. At all other frequencies, there will be a certain amount of reactance as well as resistance, which will result in a higher SWR. The reactance in the load shifts the phase of the current in relation to the voltage and affects both the incident and reflected current, and also the resultant voltage. When a load has inductive reactance, the point of maximum voltage and minimum current is shifted towards the load. When it is capacitive, the opposite occurs.
In addition, a line will also have some inherent attenuation or loss due to the resistance of the conductors, some absorption from dielectric conductors and some loss through line radiation.
While these losses will have an affect on the characteristic impedance, usually this is not to any great degree. However, if the line very lossy, whether it be due to length, a high loss per unit length, a high SWR, or combinations of these factors, the input impedance may be affected significantly.
A line will lose a percentage of its input power over a given length, depending on the operating frequency used. This is usually expressed in terms of decibels per 30.5 metres or 100 feet. Every 3dB down means that half of the available power is lost. As frequency is increased, more power is lost through conductors and dielectrics. Where long lines are unavoidable, therefore, low loss cables are essential to ensure efficient radiation.
Power losses in a line are lowest where the line is terminated in a resistance equal to its characteristic impedance. As the VSWR increases, so do the losses. Increased current creates greater losses in conductors, while increased voltage leads to greater losses in dielectrics. These losses may or may not significantly affect signal quality.
At frequencies up to 30 MHz, if the VSWR at the load is 2:1 or less, the additional loss caused by the standing waves compared to a perfectly matched line will only be around 0.5 dB, even where long lines are concerned. Such a loss will have little appreciable affect on signal strength and, as far as losses go, in practice amount to almost a perfect match. At VHF and UHF frequencies, however, as cable types normally have quite high line losses, even a slight mismatch can have a serious impact.
Where the line is perfectly matched but the line is very lossy, less power is able to reach the load reducing the level of reflected power. The reflected power measured at the input will then be further attenuated due to the additional loss as it travels back along the line. For example, in a line with 6dB loss, only 25% of input power will reach the load (half power is lost with every 3dB loss: 50% + 25%). If the VSWR at the load is 4:1, only 9% of the input power will be reflected. With further line losses of 6dB as it returns, only around 2% of the input power will reach the input terminals. If the VSWR is measured all along the line, VSWR measurements will show a progressive decrease from the load to the input terminals.
Obviously this can be very misleading. So, to know the VSWR of the antenna, it is important to take readings at the load.
An antenna as described above would not give good performance but the chances are that the low VSWR reading at the input terminals would not result in the transmitter winding back power to compensate, as it would with a low loss line, as a perfectly matched low loss line with high VSWR can result in large increases in power. However, this may only be small in comparison with the power delivered at the load. A VSWR of 10:1 on a perfectly matched line with 0.3dB loss will only cause an additional 1dB loss which would only result in a minor decrease in signal strength.
In phasing or matching of antenna arrays, it is sometimes useful to terminate a line with a nonresistive load, for example when reversing the beam heading of an array by remote switching of reactive terminations on sections of line. This is because nonresistive terminations do not consume power at all, but reflect all of the energy arriving at the end of the line. Short and open Circuits are nonresistive terminations that can be useful in stub matching and in evaluating the quality of a section of transmission line. The impedance of the short circuit termination is 0 + j0 whereas the impedance of the open circuit is infinite.
When a line is an exact multiple of a 1/4 wavelength long, the input impedance is purely resistive when the termination is a pure resistance. Conventional inductors and capacitors can be replaced by short or open circuited lines as the input impedance is mostly pure reactance where line losses are low. Moreover, sections of line can be cut in or out of a line without changing operating conditions where line losses are negligible, if they are an exact multiple of a 1/2 wave in length, as the existing input and output are simple repeated.