MICROVE DEVICES

3. Further Transmission Lines Characteristics

3.2. Standing Wave and Standing Wave Ratio

3.1 Standing Wave and Standing Wave Ratio

3.1.1 Standing Wave

The general solutions of the transmission-line equation consist of two waves traveling in opposite directions with unequal amplitude as discussed in subsection 2.1.2. Equation 2.1.22 can be written as

V(z) = V+e−αze−jβz +V−eαzejβz(3.1.1)

= V+e−αz[cos(βz)− jsin(βz)] +V−eαz[cos(βz) + jsin(βz)]

=V+e−αz +V−eαz cos(βz)− jV+e−αz −V−eαz sin(βz)

With no loss in generality it can be assumed that V+e−αzand V−eαzare real. Then the voltage-wave equation can be expressed as

Vs = V0e−jφ(3.1.2)

This is called the equation of the voltage standing wave, where

V0 =

V+e−αz +V−eαz 2cos2(βz)− jV+e−αz −V−eαz 2sin2(βz)o12(3.1.3)

which is called the standing-wave pattern of the voltage wave or the amplitude of the

standing wave, and

φ = arctan

V+e−αz +V−eαz

V+e−αz −V−eαztan(βz) 15

(3.1.4)

which is called the phase pattern of the standing wave.

The maximum and minimum values of equation 3.1.3 can be found as usual by differen tiating the equation with respect to βz and equating the result to zero. By doing so and substituting the proper values of βz in the equation, we find that;

(i) The maximum amplitude is

Vmax = V+e−αz +V−eαz = V+e−αz(1+|Γ|)

this occurs when βz = nπ, where n = 0,±1,±2,......

(ii) The minimum amplitude is

Vmin = V+e−αz −V−eαz = V+e−αz(1−|Γ|)

this occurs when βz = (2n−1)π/2, where n = 0,±1,±2,......

(iii) The distance between any two successive maxima or minima is one-half wavelength,

since

βz = nπ z =nπβ=nπ

2π/λ= nλ2n = 0,±1,±2,......

then z1 =λ2. It is evident that there are no zeros in the minimum.

Similarly,

Imax = I+e−αz +I−eαz = I+e−αz(1+|Γ|)

Imin = I+e−αz −I−eαz = I+e−αz(1−|Γ|)

The standing-wave patterns of two oppositely traveling waves with unequal amp litude in lossy or lossless line are shown in figures 3.1.1 and 3.1.2

(iv) When V+ 6= 0 and V− = 0, the standing-wave pattern becomes V0 = V+e−αzand when V+ = 0 and V− 6= 0, the standing-wave pattern becomes V0 = V−eαz

(v) When the positive wave and the negative wave have equal amplitudes (i.e.,V+e−αz = V−eαz) or the magnitude of the reflection coefficient is unity, the standing-wave pattern with a zero phase is given by

Vs = 2V+e−αzcos(βz) (3.1.5)

which is called a pure standing wave. Similarly for the current we have Is = −j2YoV+e−αzsin(βz) (3.1.6)

Figure 3.1.1: Standing-wave pattern in a lossy line.

Figure 3.1.2: Voltage standing-wave pattern in a lossless line.

Equations 3.1.5 and 3.1.6 show that the voltage and current standing wavesare 90o out of phase along the line. The points of zero current are called the current nodes. The voltage nodes and current nodes are interlaced a quarter wavelength apart.

The voltage and current may be expressed as real functions of time and space: vs(z,t) = Re Vs(z)ejωt = 2V+e−αzcos(βz) cos(ωt) (3.1.7)

is(z,t) = Re is(z)ejωt = 2Y0e−αzcos(βz) cos(ωt) (3.1.8)

The amplitudes of equations 3.1.7 and 3.1.8 vary sinusoidally with time; the voltage is a maximum at the instant when the current is zero and vice versa as shown in the figure 3.1.3

Figure 3.1.3: Pure standing waves of voltage and current.

3.1.2 Standing Wave Ratio

Standing waves result from the simultaneous presence of waves traveling in opposite dir ections on a transmission line. The ratio of the maximum of the standing-wave pattern to the minimum is defined as the standing-wave ratio (ρ).i.e.,

Standing Wave Ratio(ρ) = Maximum voltage or current

Minimum voltage or current=|Vmax|

|Vmin|=|Imax|

|Imin|

The standing-wave ratio results from the fact that the two traveling-wave components of 18

the wave add in phase at some points and subtract at other points. The standing-wave ratio of a pure traveling wave is unity and that of a pure standing wave is infinite.

NB:When the standing-wave ratio is unity, there is no reflected wave and the line is called a flat line. The standing-wave ratio cannot be defined on a lossy line because the standing wave pattern changes markedly from one position to another. On a lowloss line the ratio remains fairly constant, and it may be defined for some region. For a lossless line, the ratio stays the same throughout the line.

Since the reflected wave is defined as the product of an incident wave and its reflection coefficient, the standing-wave ratio ρ is related to the reflection coefficient Γ by

ρ =1+|Γ|

1−|Γ|(3.1.9)

and vise versa

|Γ| =ρ −1

ρ +1(3.1.10)

The plot in figure shows the relationship between reflection coefficient |Γ| and standing wave ratio ρ. Since |Γ| ≤ 1, the standing-wave ratio is a positive real number and never

Figure 3.1.4: Standing wave ratio against reflection coefficient

less than unity, ρ ≥ 1 and the magnitude of the reflection coefficient is never greater than unity. Example:

A transmission line has a characteristic impedance of 50+ j0.01Ω and is terminated in a load impedance of 73− j42.5mega. Compute (a) the reflection coefficient; (b) the

standing wave ratio.

Solution

(a) the reflection coefficient

Γ =Zl −Zo

Zl +Zo=(73− j42.5)Ω−(50+ j0.01)Ω

(73− j42.5)Ω+ (50+ j0.01)Ω= 0.377

(b) the propagation constant

ρ =1+|Γ|

1−|Γ|= 2.21

3.2 Line Impedance and Admittance

3.2.1 Line Impedance

The line impedance of a transmission line is the complex ratio of the voltage phasor at any point to the current phasor at that point. It is defined as

Z =V(z)

I(z)(3.2.1)

Figure 3.2.1 shows a diagram for a transmission line. The voltage or current along a Figure 3.2.1: Diagram of a transmission line

line is the sum of the respective incident wave and reflected wave in the line. i.e., V(z) = Vinc +Vre f = V+e−γz +V−eγz(3.2.2)

I(z) = Iinc +Ire f = Y0V+e−γz −V−eγz (3.2.3)

At the sending end z = O; thus equations 3.2.2 and 3.2.3 becomes

IsZs = V+ +V−

IsZ0 = V+ −V−

when these to equations are solved simultaneously, we get

V+ =Is2(Zs +Z0) (3.2.4)

V+ =Is2(Zs −Z0) (3.2.5)

Substituting the value of V+ and V− in equations 3.2.2 and 3.2.3 we get that, V(z) = Is2 (Zs +Z0)e−γz + (Zs −Z0)eγz (3.2.6)

I(z) = Is 2Z0

(Zs +Z0)e−γz −(Zs −Z0)eγz (3.2.7)

Then the line impedance at any point z from the sending end in terms of Zs and Zo is

expressed as

Z = Z0(Zs +Z0)e−γz + (Zs −Z0)eγz

(Zs +Z0)e−γz −(Zs −Z0)eγz(3.2.8)

At z = l the line impedance at the receiving end in terms of Zs and Zo is given by Zl = Z0(Zs +Z0)e−γl + (Zs −Z0)eγl

(Zs +Z0)e−γl −(Zs −Z0)eγl(3.2.9)

Equations 3.2.8 and 3.2.9 are simplified by replacing the exponential factors with either hyperbolic functions or circular functions. The hyperbolic functions are obtained from

e±γl = cosh(γz)±sinh(γz) (3.2.10)

Substitution of the hyperbolic functions in equation 3.2.8 yields the line impedance at any point from the sending end as,

Z0 cosh(γz)−Zs sinh(γz)= Z0Zs −Z0 tanh(γz)

Z = Z0Zs cosh(γz)−Z0 sinh(γz)

Z0 −Zstanh(γz)(3.2.11)

For a lossless line, γ = jβ; and by using the following relationships between hyperbolic 21

and circular functions

sinh(jβz) = jsin(βz) and cosh(jβz) = j cos(βz) (3.2.12)

the impedance of a lossless transmission line (Zo = Ro) can be expressed in terms of the circular functions:

Z = R0Zs cos(βz)− jR0 sin(βz)

R0 cos(βz)− jZs sin(βz)= R0Zs − jR0 tan(βz)

R0 − jZstan(βz)(3.2.13)

The normalized impedance of a transmission line is defined as

|z| =ZZ0=1+Γ

1−Γ(3.2.14)

The normalized impedance for a lossless line has the following significant features i. The maximum normalized impedance is

R0=|Vmax|

zmax =Zmax

R0Imin=1+|Γ| 1−|Γ|= ρ

Here zmax is a positive real value and it is equal to the standing-wave ratio ρ at the location of any maximum voltage on the line.

ii. The minimum normalized impedance is

zmin =Zmin

R0=|Vmin|

R0Imax=1−|Γ|

1+|Γ|=1ρ

Here zmin is a positive real number also but equals the reciprocal of the standing wave ratio at the location of any minimum voltage on the line.

iii. For every interval of a half-wavelength distance along the line, zmax or zmin is

repeated:

zmax(z) = zmax

and

zmin(z) = zmin

z±λ2

iv. Since Vmax and Vmin are separated by a quarter-wavelength, zmax is equal to the reciprocal of zmin for every λ/4 separation:

zmax

z±λ2 =1 zmin(Z)