Transmission lines

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Definitions for transmission lines.
Microwaves for Satcoms.
Power flow on transmission lines.
Stub tuner matching.
Simple stub match example with figures
The SMITH chart - a description.
Standing wave patterns.
Microwave measurements.
Scattering parameters and the S-matrix.
Problems One with answers.
Problems Two with answers.
Waveguides and Cavity Resonators.
Problems Three with answers.
Problems Four with answers.
Problems Five on antennas, with answers.
Exam paper solutions 94-95 (DJJ's questions)
Exam paper solutions 95-96 (DJJ's questions)
Exam paper solutions 96-97 (DJJ's questions)
Exam paper solutions 97-98 (DJJ's questions)
Exam paper solutions 98-99 (DJJ's questions)
Exam paper solutions 99-00 (DJJ's questions)
Transmission line junctions.
Antennas. And onward links.
Four port S-matrix relations and directional couplers.
An example of pulses on transmission line.

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The limitations of circuit theory.

At low frequencies, circuit theory is generally adequate to explain the behaviour of collections of electronic components interconnected by wires.

As frequency increases, the circuit approximation becomes progressively less good, for a number of reasons. The energy stored in reactive components is held in the space around the components, and different components can have "fields" which overlap. Wires are also reactive components which store energy. The division of the circuit into separate reactive "components" interconnected by non-reactive "wires" is only an approximation; it is useful in that often it relates quite closely to how the circuit is constructed topographically, but is less good in describing how it behaves electromagnetically. A more important limitation is imposed by the relatively slow speed of light.

Modern media processor chips need the capacity to read and write DRAM at a refresh rate of 60 Hz, and at a total memory size of 64 M-bytes. The latest generation chips to do this have clock speeds around 1GHz; this is only possible by paying careful attention to the minimisation of transit time delays across a chip of area 100 square mm. It is found that the traditional interconnect technology of aluminium on insulator is too slow; preferred technology is to use gold wires suspended in air.

Action at a distance is illusory. Just as it takes time to travel from Hazel Farm to the University Campus, so it takes time for an electric disturbance to travel around or across a circuit. The further apart the component elements of the circuit are, the longer it takes for an event at one place to affect the conditions at another. Thus at a typical motor-bus speed it takes 15 min to get from Hazel Farm to campus, but it would take 90 min to get from Kingston to campus.

In an electronic circuit the absolute maximum speed with which events can travel is limited to the speed of light in free space, or vacuum, which is 30 cm per nanosecond.

It is traditional to think of the speed of light as being enormously large; but actually on the time-scale of modern electronics it is really quite slow. In human terms we notice events happening on the order of seconds as being separated in time, but reasonably close together. Modern electronics can separate events which occur within nanoseconds of each other. Imagine a kind of "human scale relativity" in which the maximum speed anything could travel was limited to 30 cm per second....You would rapidly get exceedingly frustrated. This is the kind of limitation with which we are asking modern electronics to cope.

A very good and readable first introduction to the limitation imposed by having a maximum velocity of propagation of 30 cm per nanosecond is to be found in the book "Special Relativity" by the late F N H Robinson, published by World Scientific. 1996.

Microwaves - working definition

In practical circuits, on copper coated circuit board for example, the speed is closer to 20 cm per nanosecond.

Thus if you can imagine a computer with a clocked electronic bus around the various parts, perhaps processor, memory, I/O ports etc, which runs at a clock frequency of 120MHz in a modern PC, one complete clock cycle occupies a distance of 167 cm along the bus. For half this distance (83.5 cm) the clock state is logic 1, and for the next 83.5 cm the clock state is logic 0. Surprisingly, the voltage isn't the same everywhere along hypothetical resistance-less bus or wire.

Wave motion

On a transmission line, waves travelling at the wave velocity have voltage or current which depends on distance along the line. Imagine a long washing line, shaken up and down at one end, and tied to a tree at the other. One can see the waves travelling. The waves are displacement transversely (up and down) from the length of the washing line, and can be observed. Of course, we can't see electrical waves on a transmission line but they are every bit as real.

[A very good place to see the wave motion at slow velocity on a mechanical transmission line is in the Discovery centre in Guildford Town (Ward Street, off North Street, by the Surrey Advertiser offices.)]

Thus we see that the voltage at one end of a wire is not instantaneously conveyed to the other end of a wire. Circuit theory ignores this phenomenon, and assumes that in a resistance-less wire, the voltage is the same everywhere along it at any given instant. On a transmission line carrying alternating current signals, the current and voltage vary sinusoidally along the line as well as in time at a fixed point on the line. The repetition time is called the period, and the repetition distance is called the wavelength. The velocity of waves on the line is given by the expression velocity = (wavelength)/(period). The waves travel a distance of one wavelength during a time of one period.

As a rule of thumb, if it takes time of over 1/10 of a cycle for an ac signal (travelling at the velocity of electromagnetic waves) to cross a circuit, then circuit theory is suspect at this frequency, and it will progressively break down at higher frequencies.

Exercise for the reader.

A building is wired with 10MHz Ethernet cable. Assuming a velocity of propagation 20 cm per nanosecond, calculate the number of wavelengths of the 10MHz signal along a 30 metre length of cable.

Answer ........ One cycle at 10MHz is 100 nsecs occupying a distance of 2000 cm or 20 metres, so there are 1.5 whole cycles or wavelengths along the cable. One could not run arbitrarily spaced synchronous computers on a cable like this at a clock speed of 10 MHz, without including some synchronising apparatus in hardware or software that would impose a speed penalty on the system. The preferred technology for modern high speed communications at a distance between digital machines is called Asynchronous Transfer Mode (ATM is the acronym). This technology is designed to circumvent the delay and synchronisation problems.

An example.

My new Pentium PC has dimensions 22 by 53 by 44 cm The maximum diagonal distance inside the box is 72 cm. At 30 cm per nanosecond this represents a time delay of about 2.4 nanoseconds so at a frequency for which 2.4 nanoseconds is 1/10 cycle we begin to be worried. That is 42 MHz and the clock speed is 450 MHz, 11 times faster. Thus this piece of equipment is "microwave" as well as being "digital circuit". The definition of "microwave" we are using is that the apparatus is comparable in size with a wavelength of radiation at the frequency of operation. Of course, the processor clock speed is confined to the processor chip only, so it is really the motherboard master clock at 33-50MHz with its fast rising and falling edges that produces most of the radiation inside the box.

Reflections on a transmission line.

Clearly, on a long straight line, waves can travel in both directions. In the Discovery exhibit (where there is a mechanical transmission line suspended vertically and hung from the ceiling) this is seen by setting off a pulse from the bottom of the line. After a time equal to twice the transit time, the pulse reflects from the ceiling and returns to your hand. This takes a few seconds. The pulse has been reflected at the top because the line is anchored there at zero displacement for all time. In an electronic transmission line, this is equivalent to holding the voltage at zero for all time by using a short circuit across the line. If you look at the direction of displacement in the pulse, it reverses on reflection. Thus, a wave of displacement to the right is returned as a wave of displacement to the left. In an electronic transmission line, a square pulse of 1 volt amplitude is returned as a square pulse of -1 volt amplitude.

The total disturbance at your hand is the sum of the disturbance now, and the disturbances two, four, six, etc, transit times earlier.

The fact that a transmission line has "memory" of what happened earlier was put to good use in computer memory technology in former years. Data could be stored as pulses, travelling slowly as acoustic waves along a transmission line, and collected at the end and re-launched after amplification at the input, to provide indefinite storage time. Of course, the access time was of the order of the transit time for a single pulse. The capacity also was limited.

Energy storage and transmission.

Transmission lines store energy, and convey it from one place to another. There is an (imperfect) analogy with terrestrial road transport systems. Imagine the section of the A3 from Guildford to Ripley. At any one time there will be many cars, say 80, in this section. If you wait at Ripley, all these cars will pass you on their way North east. The road "stores cars". It is a "lossless road" because all the cars entering at Guildford eventually come out at Ripley.

Suppose now you close the road North of Ripley, take all the cars off the northbound carriage-way and turn them back onto the southbound carriage-way. The closed road represents a non-existent transmission line; if it were a coaxial cable you would have cut the end off. There is nowhere for the cars to go other than to come back down the other carriage-way. [In a transmission line where "superposition" occurs, both waves exist on the same piece of wire, and can "pass through" each other. It is not necessary to have a separate wire for the return path.] All the energy transmitted up the lossless road or line is returned to the source.

This is a fundamental property of lossless transmission lines; they absorb energy or power from a generator, convey it elsewhere, but don't dissipate it. What goes in must come out. If it can't come out anywhere else it must come back to the source ("generator" end).


Suppose now you close the road as well to the south of Guildford, and for the sake of argument you only have one car on the section, travelling at the speed limit 70MPH. It goes North to Ripley, turns round and comes South to Guildford, turns round and goes back to Ripley again, and so on. It has nowhere else to go, and you don't allow it to stop. If you sit somewhere along the road on the central reservation it will pass you going North at intervals of "twice a transit time". You have made a "transmission line resonator."

Now consider the UHF television tuner. It runs at 800MHz, say, and it has a rod resonator which is long enough that the round trip takes 1 cycle at 800 MHz (1.25 nanoseconds), travelling at 30 cm per nanosecond. A little thought shows that we need to cut the rod 30*1.25/2 = 18.75 cm long. Of course, there are other details to a practical version of this circuit, but it is clear that useful tuned circuits can be made from rod transmission line.

The velocity of electromagnetic waves on transmission line is equal to 1/(squareroot(inductance*capacitance)) where the inductance and capacitance are taken for unit distance only (Henries per metre and Farads per metre). Using these SI units for inductance and capacitance, the velocity is also expressed in SI units; metres/sec.

To slow down the waves on the transmission line, all you need to do is to increase either or both of the inductance/metre or the capacitance/metre. The capacitance/metre is increased most easily by encasing the conductors in a dielectric having permittivity greater than unity. The inductance/metre can also be increased by enclosing the conductors in a lossless non-conducting magnetic material (maybe ferrite) but this is more difficult. Another way to slow the waves down is to coil up one of the conductors. This is used to make slow wave structures in helix travelling wave tubes (TWTs).

The advantage of having slower waves on transmission line is that the resonant structures can be physically smaller. The disadvantage is that if the structure is smaller, the energy stored per unit volume is less, so the power handling capability is less.

Characteristic Impedance

When a generator is suddenly connected to a length of transmission line, there will be a time delay until the signal travels to the other end of the line. During this time, the generator has no knowledge of how long the line might be; to all intents and purposes there is no difference (during this time) between a short line and a very long line. The generator has to supply energy to the line, to establish the electromagnetic fields around the conductors. Energy is stored in the capacitance between the conductors, proportional to the square of the voltage between the wires, and in the magnetic field around the conductors which represents series inductance, proportional to the square of the current on the conductors. Even though the line may be open circuit at the other end, the generator does not (yet) know this and supplies current; the product of current and voltage is the rate at which energy is supplied to the line; this is got by multiplying the stored electric plus magnetic energy per unit length of line, by the velocity at which the signals travel.

The ratio of voltage (between the wires) to current (along one wire and back along the other) has dimensions of impedance or resistance. At a single frequency, on a lossless line, the current is in phase with the voltage and the impedance is real. It is called the "Characteristic Impedance". [The usual algebraic symbol for the Characteristic Impedance is Zo.] It does not depend on what is connected to the ends of the line, but only on the line geometry and material construction.

The Characteristic Impedance, although real and looking like a resistance, is actually a "lossless, non-dissipative" impedance. Nothing gets hot as a result of supplying energy to this resistance. All that happens is that energy is transferred from the generator and stored temporarily in the transmission line. At some later time, possibly a great many transit times later, it can be extracted and returned to the generator, or used to make a real resistive dissipative load get hot. There is an analogy here with writing a cheque to the building society. This is a drain on your bank balance, but non-dissipative; the money is stored reversibly and can be got back at a later time; possibly to be used for the irreversible purchase of a "meal out".

Impedance Matching

Suppose we think about a very long length of transmission line, and somewhere along it we make an imaginary join between the left half and the right half. The right half presents a real lossless impedance (equal to the characteristic impedance) to waves approaching the junction from the left half. But there is no actual discontinuity at the imaginary join, so there is nothing to give rise to a reflection here. If we replace the right half transmission line with a resistor having resistance equal to the real lossless characteristic impedance of the line, there is no way the waves arriving at the junction can tell the difference between the resistor and the long right-half transmission line it has replaced. Thus there can be no reflection. We have "matched" the impedance of the line with the same value resistance. All the power delivered by the generator is dissipated in the resistance; there is no reflected wave amplitude or power, and there is no backward travelling wave. As far as the generator is concerned, there is no way it can know how long the length of line is, or even that there is a line there at all; it cannot tell the difference between a resistive load Zo and a very long matched line having characteristic impedance Zo.

Over a wide range of frequencies, in lossless line, the impedance Zo is substantially independent of frequency. Thus if a line is matched at one frequency, it will also be matched at another frequency, under these conditions. There are other conditions on impedance matching under which transmission line is matched only at a single frequency, or over a small range of frequencies.

Directions of travel

There are only two directions of travel on a transmission line, namely from generator to load, and from load to generator. These are termed "forward" and "backward" waves respectively. There is no physical difference you can measure between the ends of a line, so Zo must be the same if either end is driven. But the current in the backward wave is in the negative direction compared to the forward wave, for a generator connected to the "load" end pushes its current into the transmission line's hot conductor in the same way as the generator connected to the "generator" end does. However, the generators are facing in opposite directions and therefore so are the currents they supply to the transmission line also in opposite directions. The voltages, of course, are in the same sense in the forward and backward waves, being between the "hot" and "return" conductors. If we take the convention for current flow to be from "generator" end to "load" end, the backward wave has mathematically a characteristic impedance -Zo. The sign reversal has physical meaning; the currents are directed oppositely as we have seen, although the voltages have the same sense. Considering the product of current and voltage, power is delivered by the generator to the forward wave, but absorbed by the generator from the backward wave. A forward wave absorbs power from the generator end but delivers it to the load end; a real impedance that sources power is a negative impedance. Looking in to the load end of the cable, connected at the other end of the cable to a generator having series internal resistance Zo, we see a matched impedance -Zo because power is being "forced down our throat".

More...Reflections on transmission line.

As we have seen, the transmission line conveys energy from one place to another. If we short circuit the line, there is no energy dissipation in the short; the voltage times current across the short is zero; the energy is all reflected as it has to go somewhere; and a return wave is set up having the same size or amplitude as the incident wave. At the short circuit the voltage has to be zero for all time; so the sum of forward wave voltage and backward wave voltage must be zero. This can only happen for non-zero forward wave amplitude if the reflected wave has opposite polarity to the forward wave. There is no constraint on the current however; so the current in the short is not zero, being the sum of the currents in forward and backward wave.

If we open-circuit the line (cut the end off), the current is zero, but there is no constraint on the voltage. The sum of forward wave current and backward wave current is zero for all time; the backward wave voltage is equal to the forward wave voltage amplitude.

Mathematically, we say if the forward wave voltage amplitude is written V+, the backward wave amplitude is written V-, and similarly for the currents I+ and I-, then because of the -Zo property of backward wave impedance we have V+ = Zo I+ and V- = -Zo I-. The total voltage at the load is (V+ + V-) and the total current at the load is (I+ + I-) because the currents are measured in the same direction along the line, namely in the direction of travel of the forward wave. You should show that if V+ = V- then the total current (I+ + I-) is zero; and that if I+ = I- then the total voltage (V+ + V-) is zero.

For other conditions of load than short circuit or open circuit, the load impedance ZL sets the ratio of total voltage to total current at the load attachment point. Thus, ZL = (V+ + V-)/(I+ + I-). Taken with the equations above V+ = Zo I+ and V- = -Zo I- it is left as an exercise for the reader to show that ZL/Zo = (V+ + V-)/(V+ - V-) and that hence the ratio of backward wave voltage amplitude V- to forward wave voltage amplitude V+ is given by (V-/V+) = (ZL - Zo)/(ZL + Zo). This ratio is called the "complex reflection coefficient". It is also left as an exercise for the reader to show that the complex modulus of the reflection coefficient is unity if ZL is either zero or infinity, or pure imaginary. (The load in the latter case is a pure reactance and dissipates no energy).

In general the wave amplitudes V+ V- I+ I- are complex amplitudes in respect of waves which are sinusoidal or single frequency waves. The response of the transmission line to a complex signal is got by making a Fourier superposition of different frequencies in the usual manner. Fourier superpositions can be made in the spatial domain as well as the time domain; the spatial frequency is the reciprocal of the wavelength in the same way that the temporal frequency is the reciprocal of the period.


For many practical transmission lines it is a good approximation to take Zo to be real, and independent of frequency. On lossy lines Zo can be frequency-dependent and complex. The wave velocity can also be frequency-dependent and complex. A complex wave velocity leads to wave attenuation; a frequency-dependent wave velocity is called "dispersion" and lines having this property are called "dispersive lines". The word "dispersion" comes from the idea that the different frequency components of a complex signal which is propagating along a line, travel at different speeds, and spread out in distance along the line or "disperse".

Transformation of impedance.

Suppose we take an instantaneous picture of a wave travelling from generator to load. The voltage and current will vary along the transmission line. If the signal is complex the variation in space may be similar to the shape of the time dependent wave when observed on a one-shot oscilloscope trace. If the signal is sinusoidal, only a single frequency is present. That means that only a single wavelength is observed. The load end of the picture shows the voltage which set out from the generator at an earlier time; the generator end of the picture shows the voltage setting out now. We are observing a "time history" of the waveform of the generator, spread out along the line.

For waves travelling from generator to load, the load end shows earlier history and the generator end comparatively later. The situation is reversed for waves travelling from load to generator, possibly produced by a reflection at the load.

In what follows, the term "forward wave" is taken to include the sum of all the disturbances travelling in the forward direction, no matter how many times they may have been reflected from the ends of the line. Similarly, "reverse wave" includes the sum of all disturbances travelling in the backward direction.

Considering the forward waves, the phase angle of the sinusoidal waves becomes less as we move towards the load. This is because the phase of a sinusoidal oscillation is continually advancing in time as time progresses [exp(j omega t)] with phase angle (omega t) and so at earlier times, ie towards the load, the phase is less.

Considering the backward waves, the situation is reversed, and towards the load the phase is more.

Now we add the forward and backward wave voltages, and similarly the forward and backward wave currents, to get the total voltage and current on the line. It is the ratio of this total voltage to this total current which is the impedance we would measure if we cut the line and put it on an impedance bridge.

Because V+ = Zo I+ but V- = -Zo I-, a little mathematical experimentation shows that the measured impedance must depend on position along the line unless V- = 0 everywhere. Refer to any standard text on transmission line theory for the precise mathematical details.

Open circuit or short circuit? At an open circuit, the total current is zero and I+ + I- = 0 so I+ = -I- and V+ = V-. If we travel a quarter of a wavelength towards the generator, the phase angle of V+ has advanced by 90 degrees, whereas that of V- has retarded by 90 degrees, and so the total voltage is now zero (they are the same size and there is 180 degrees phase difference between them). The currents, which were 180 degrees out of phase, are now in phase. The total current is large and the total voltage zero, so looking into the "generator" end of a 1/4 wavelength of transmission line, with an open circuit at the "load" end, we see a perfect short circuit.

Another way of looking at this surprising result is, that at the load end there is a real open circuit: As we move towards the generator, along the line, we add the shunt capacitance between the line conductors, and the series inductance of the conductor loop. When we have moved 1/4 wavelength towards the generator these components form a series tuned circuit at the generator frequency across the generator terminals, which looks like a perfect short circuit. There is danger in this point of view, as the inductance and capacitance of the line are "distributed" along the line rather than being concentrated at a single point. It is necessary to solve the transmission line equations to get a proper understanding of the phenomenon.

Nevertheless, we see the danger of having unconnected pieces of circuit board track or wire floating around in a microwave circuit. If they become resonant they act as shorts and can cause catastrophic misbehaviour of the intended circuit. The rule is, if in doubt, terminate all lengths of unused transmission line in the line's characteristic impedance (resistance).

Some practicalities

Some of the factors which determine whether (or not) a practical transmission line behaves as intended include, but are not limited to, the following:-

Typically, twin feeder is very much more subject to variability in the wet than is coaxial cable. However, a common cause of coaxial cable failure is the wicking of water up the semi-air-spaced dielectric separating inner conductor from sheath.

All outdoor cables, and some indoor cables, suffer from the effects of light. The insulators harden and crack and can let in water.

Cables which are overstressed, electrically, can develop carbonised conducting paths between the conductors of the cable. Failure is then progressive.

Transmission line with exposed conductor surfaces can experience oxidation and an increase in losses due to skin effect heating.

Different cable technologies have their own range of practical characteristic impedance values.

Copyright © D.Jefferies 1996, 1997, 1999, 2000, 2001, 2002.

D.Jefferies email
11th May 2003.