Standing Waves on transmission lines and waveguides

To convert Nepers N, into decibels dB, use is made of the relationship, . The imaginary part of g ¢ is known as the image phase-change coefficient b ¢ , expressed in radians. Thus, for the general case,

g ¢ = a ¢ + jb ¢

For the special case of a symmetrical network, the source impedance is equal to the load impedance, and so g ¢ is then known as the propagation coefficient of the network. Note that the propagation coefficient is dimensionless. If the propagation coefficient is expressed in terms of length, then g is in units of /length, where g l = g ¢ . Equation 1-1 permits the receiving end voltage V_L, to be expressed in terms of the sending end voltage V_i, on a transmission line, by

(1-1)

and similarly for the receiving end current

(1-2)

Equation 1-1 shows that the magnitude of the line voltage decreases exponentially with the distance l from the sending end of the line. The phase angle of the line voltage is always a lagging angle and this angle increases in direct proportion to the length of the line from the sending end. Similar remarks apply to the magnitude of the line current, as given in equation 1-2.

Consider a uniform lossy transmission line excited by a general instantaneous sinusoidal voltage which is attenuated as it travels down the line away from the source, that is,

v(z,t) = Ve^{-a z} cos(w t- b z) and current i(z,t) = I e^{-a z} cos(w t- b z + j ). If we express these equations in phasor form

= 

and similarly,

= 

If we had chosen a sine wave instead of a cosine wave, we would take instead the imaginary part of the phasor or . For a more general form of the instantaneous sinusoidal excitation on the transmission line, we can employ the use of the phasors and .

A phasor solution may be written as

(1-3)

where the propagation coefficient g , is given by

(1-4)

The propagation coefficient is in general complex and in general is given by

(1-5)

where the propagation coefficient per unit length g , is given in units of per unit length, the attenuation coefficient per unit length a , in units of Nepers per unit length and the phase-change coefficient b , in radians per unit length.

The term represents the forward voltage wave on the line and the term represents the reflected wave which may exist if the line is improperly terminated and not infinite in length. If the line is infinite in length and thus the reflected wave zero, then is the same as of equation 1-1. In this case, where the line is not infinite and is improperly terminated there will be exist the reflected voltage wave component . In the steady state condition, at the sending end of the line (z=0), and thus, . That is, the excitation voltage at the sending end of the line must be equal to the sum of the forward and reflected wave. Similarly,

(1-6)

Where and the characteristic impedance Z_o is given for travelling waves, by . Note that this expression is not an approximation.

The reciprocal of Z_o is defined as the characteristic admittance (Z_o) of the line and therefore

siemens (1-7)

Since the voltages and currents in equations 1-3 and 1-6 are phasor quantities, they are usually complex quantities and are dependent on the conditions of the generator and load.
 
 

VELOCITY OF PROPAGATION

v_p = l f kilometres/second

In one wavelength, a phase change of 2p radians occurs. Hence the phase change for a length of line, l kilometers long, where there are (l/l ) wavelengths, will be 2p (l/l ) radians. This is equal to the phase change over a length of line l. The phase change coefficient b , of the line is the change in phase which would occur, in this case, in one kilometer. Thus

radians/kilometer (1-8)

or

kilometers (1-9)

and

kilometers/second (1-10)

Since, 

kilometers/second (1-11)

MISMATCHED TRANSMISSION LINES

 Figure 1 Forward currents and voltages on a lossy mismatched transmission line

Consider figure 1 below, which shows a lossy line whose output terminals are terminated in a load impedance Z_L which is not equal to the characteristic impedance of the line. For ease in description, the generator impedance is taken to equal the characteristic impedance of the line and thus is matched. Thus, the voltage into the line V_s is half of the generator e.m.f V_g , and the current into the line I_s is given by V_g/(2Z_o). As the voltage and current waves propagate along the line, they experience a change in amplitude and phase. At the end of the line, the forward travelling waves amplitudes have been reduced by the factor eg ^l.

(1-12)

There are two reflection coefficients, one for voltage r _v, and one for current r _I. At the load, which is a specific point of z, some fraction of the voltage or current which has arrived at the load is being sent back towards the source in the negative direction. This is given by r _vL the fraction of the voltage and r _iL the fraction of current. The voltage reflection coefficient, at the load, can be defined with the aid of , the amount of reflected voltage at the load, and , the amount of voltage arriving at the load, by

(1-13)

Equation 1-13 shows the reflection coefficient r _vL to be complex due to g . As the load is complex for a mismatched line, there will also be a phase shift between the forward and reflected voltages . This phase shift is given by the argument of as will be shown below.

The current reflection coefficient, at the load, can be similarly be defined from , the amount of reflected current at the load and the amount of current arriving at the load, by

(1-14)

The phasor solutions for voltage and current on the tra nsmission line, at any distance z from the source, to the differential equations, were given by equations 1-3 and 1-6, and are again given for convenience below.

(1-3)

(1-6)

These equations show that the reflected current is in antiphase with the reflected voltage. To reconcile this the reflection coefficient of the current is taken as the negative of the reflection coefficient of the voltage

(1-15)

If equations 1-3 and 1-6 are taken with the voltage reflection coefficients inserted and also, the distance is taken from the load end in the negative z direction by the insertion of d = l - z, then, with the aid of equation 1-12

(1-16)

(1-17)
 
 

Therefore, the phasor solutions for voltage and current on the transmission line, at any distance d from the load are given by equations 1-16 and 1-17.

From equations 1-16and 1-17, at any point on the line, at some distance d, from the load, the forward voltage is given by and the reflected voltage is given by , thus in general, the voltage reflection coefficient at any arbitrary point d on the line can be written as

(1-18)

which can be represented by

where j = arg() (1-19)

Equation 1-19 shows that the reflection coefficient r is complex, and that when r _L is known, the reflection coefficient r can be determined at any point on the line d, from the load.

Figure 2 shows the forward and the reflected currents and voltages at the extreme ends of the line.

From equations 1-16 and 1-17 together with equation 1-19, the phasor solution at any point on the line d, from the load is given by

(1-20)

(1-21)
 
 

 Figure 2 Reflected currents and voltages on a lossy mismatched transmission line, referenced from the load at some distance d = l - z
 
 

The load is at d=0, so from equations 1-20 and 1-21, the division of the phasor voltage by the phasor current at d=0 must equal the load impedance, thus

(1-22)

therefore,

(1-23)

where is defined as the normalized load impedance, which will be used extensively when dealing with Smith charts. Equation 1-23 is particularly useful if the mismatched impedance and characteristic impedance of the line is known, as the voltage reflection coefficient and current reflection coefficient at the load can be determined. Note that the reflection coefficients are usually complex, because the load impedance is complex. If the following admittances are defined, then the voltage reflection coefficient can be defined in terms of admittances. That is then

(1-24)

the general form for the reflection coefficient at any point on the line can be found from equations 1-20 and 1-21, so that

(1-25)
 
 

In Figure 3, below the first diagram shows a travelling wave moving down a transmission line with no reflections. The distance axis z, and the time axis t, are both shown to visually describe how the wave the wave propagates. The remaining diagrams show cases where reflected waves of different amplitudes exist and combine with the forward travelling wave to form standing waves, as given by .

Figure 3 shows that as the amplitude of the reflected wave is increased the maximum and minimum amplitude of the standing wave at a particular point on the line varies with time, that is, at a particular point on the line, the standing wave pulsates. This is particularly apparent in the last of the diagrams, where the reflected wave is the same amplitude as the forward wave. What is not apparent from the diagrams is the scale of the amplitude of the maximum and minimum amplitudes. Again. the last of the diagrams shown indicate that there is complete cancellation at one time, where no voltage would exist along the line, and at another time, complete wave addition occurs, where an amplitude would exist at some point which is twice the amplitude of the forward wave. For reflections which are not of the same amplitude as the forward wave, a maximum standing wave voltage less than twice the amplitude of the forward wave exists and a minimum standing wave voltage of greater than zero exists. At the point where the forward wave maximum (or minimum) adds to the reflected wave maximum (or minimum), that is where both waves are in phase, we find from

that 

and so, as r is complex, taking its amplitude only,

(1-26)

At the point where the forward wave maximum (or minimum) adds to the reflected wave minimum (or maximum), that is where both waves are out of phase,

(1-27)

The voltage standing wave ratio (VSWR) is defined as

(1-28)

This definition of VSWR shows that VSWR takes values between unity and infinity. If no reflections exist, then = 0, and the VSWR is unity. For full reflection of the travelling wave, = 1, and the VSWR is infinite. From equation 1-28, can be found as

(1-29)

For a lossy line  (1-30)

showing that the reflection coefficient is not constant but reduces with the distance from the load by an amount . Normally, when measuring VSWR, a lossless line is assumed, so that the reflection coefficient is of constant amplitude and equal to the reflection coefficient no matter what the value of d, the distance from the load along the line. As the phase-change coefficient of the reflection coefficient changes by 2b , compared to the travelling wave,

which changes only by b , from equation 1-8, the wavelength of the standing wave is half that of the forward travelling wave. This is more easily seen looking at the last diagram of figure 3, and following the peaks of the standing wave diagonally across the diagram, as time and distance increases, rather than trying to consider the wave at one instant in time.