MICROVE DEVICES

2. Transmission Lines

Topic 2

Transmission Lines

Conventional two-conductor transmission lines are commonly used for transmitting mi crowave energy. If a line is properly matched to its characteristic impedance at each ter minal, its efficiency can reach a maximum.

In ordinary circuit theory it is assumed that all impedance elements are lumped constants. This is not true for a long transmission line over a wide range of frequencies.

Frequencies of operation are so high that inductances of short lengths of conductors and capacitances between short conductors and their surroundings cannot be neglected. These inductances and capacitances are distributed along the length of a conductor, and their ef fects combine at each point of the conductor.

Since the wavelength is short in comparison to the physical length of the line, distributed parameters cannot be represented accurately by means of a lumped-parameter equivalent circuit.

Thus microwave transmission lines can be analyzed in terms of voltage, current, and im pedance only by the distributed-circuit theory. If the spacing between the lines is smaller than the wavelength of the transmitted signal, the transmission line must be analyzed as a waveguide.

2.1 Transmission Line Equations and Solution

2.1.1 Transmission Line Equation

A transmission line can be analyzed either by the solution of Maxwell’s field equations or by the methods of distributed-circuit theory. The solution of Maxwell’s equations involves three space variables in addition to the time variable. The distributed-circuit method, how ever, involves only one space variable in addition to the time variable.In this section the latter method is used to analyze a transmission line in terms of the voltage, current, imped ance, and power along the line.

Based on uniformly distributed-circuit theory, the schematic circuit of a conventional two conductor transmission line with constant parameters R, L, G, and C is shown in figure 2.1.1.

Figure 2.1.1: Elementary section of a transmission line

The parameters are expressed in their respective names per unit length, and the wave propagation is assumed in the positive z direction.

Using Kirchhoff’s voltage law, the summation of the voltage drops around the central loop is given by

∂t+v(z,t) + ∂ v(z,t)

v(z,t) = i(z,t)R∆z+L∆z∂ i(z,t)

∂ z∆z (2.1.1)

Rearranging this equation, dividing it by ∆z, and then omitting the argument (z, t), we

obtain

−∂ v∂ z= Ri+L∂ i∂t(2.1.2)

Using Kirchhoff’s current law, the summation of the currents at point B in figure 2.1.1 can 6

be expressed as

i(z,t) = v(z+∆z,t)G∆z+C∆z∂ v(z+∆z,t)

∂t+i(z+∆z,t) (2.1.3)

v(z,t) + ∂ v(z,t) ∂t∆z

G∆z+C∆z∂∂t v(z,t) + ∂ v(z,t)

∂t∆z

+i(z,t) + ∂ i(z,t)

∂ z∆z

By rearranging the above current equation, dividing it by ∆z, omitting (z, t), and assuming

∆z equal to zero, we have

−∂ i∂ z= Gv+C∂ v∂t(2.1.4)

Differentiating equation 2.1.2 with respect to z and 2.1.4 with respect to t and combining the results, the final transmission-line equation in voltage form is

−∂2v

∂ z2= RGv+ (RC +LG)∂ v∂t+LC∂2v

∂t2(2.1.5)

Also, by differentiating equation 2.1.2 with respect to t and 2.1.4 with respect to z and combining the results, the final transmission-line equation in current form is

∂ z2= RGi+ (RC +LG)∂ i∂t+LC∂2i

−∂2i

∂t2(2.1.6)

All these transmission-line equations are applicable to the general transient solution. The voltage and current on the line are the functions of both position z and time t.

The instantaneous line voltage and current can be expressed as

v(z,t) = Re V(z)ejωt(2.1.7)

i(z,t) = Re I(z)ejωt(2.1.8)

The factors V(z) and V(z) are complex quantities of the sinusoidal functions of position z on the line and are known as phasors. The phasors give the magnitudes and phases of the sinusoidal function at each position of z, and they are expressed as

V(z) = V+e−γz +V−eγz(2.1.9)

I(z) = I+e−γz +I−eγz(2.1.10)

where γ = α + jβ is the propagation Constant, V+ and I+ indicate amplitudes in the pos itive z direction, V− and I− complex amplitudes in the negative z direction, α is the atten

uation constant in nepers per unit length, and β is the phase constant in radians per unit length.

Substituting jω for ∂∂tin equations 2.1.2, 2.1.4, 2.1.5 and 2.1.6 and divide each by ejωt, the transmission-line equations in phasor form of the frequency domain becomes

dz= −ZI (2.1.11)

dz= −YV (2.1.12)

d2V

dz2= γV (2.1.13)

d2I

dz2= γI (2.1.14)

where the following substitution have been made

Z = R+ jωL (ohms per unit length) (2.1.15)

Y = G+ jωC (mhos per unit length) (2.1.16)

γ =√ZY = α +β j (Propagation constant) (2.1.17)

For a lossless line, R = G = 0, and the transmission-line equations are expressed as

dz= −jωLI (2.1.18)

dz= −jωCV (2.1.19)

d2V

dz2= −ω2LCV (2.1.20)

d2I

dz2= −ω2LCI (2.1.21)

NB: Equations 4.2.5 and 4.2.6 for a transmission line are similar to equations of the elec tric and magnetic waves, respectively. The only difference is that the transmission-line equations are one-dimensional.

2.1.2 Solutions of the Transmission Lines

One possible solution for equation 4.2.5 is

V(z) = V+e−γz +V−eγz = V+e−αze−jβz +V−eαzejβz(2.1.22) 8

The term involving e−jβzshows a wave traveling in the positive z direction, and the term with the factor ejβzis a wave going in the negative z direction. The quantity βz is called the electrical length of the line and is measured in radians.

Similarly one possible solution for equation 4.2.6 is

I(z) = YoV+e−γz −V−eγz = Yo V+e−αze−jβz −V−eαzejβz (2.1.23)

In equation 4.2.7, the characteristic impedance of the line is defined as Zo =1Yo≡rZY=sR+ jωL

G+ jωC= Ro ± jXo (2.1.24)

At microwave frequencies R <<< ωL and G <<< ωC. By using the binomial expan sion, the propagation constant can be expressed as

γ =√ZY =p(R+ jωL)(G+ jωC)

(jω)2LC

1+R jωL

1+G jωC

= jω√LC 1+12R jωL

1+12G jωC

= jω√LC 1+12 R

jωL+G

=12

L+G

rL C

jωC

+ jω√LC

From the above equations the attenuation and phase constants are, respectively, given by

α =12

L+G

rL C

(2.1.25)

and

β = jω√LC (2.1.26)

Similarly, the characteristic impedance is found to be Zo =1Yo≡rZY=sR+ jωL

G+ jωCu

C(2.1.27)

From equation 2.1.26, the phase velocity is

νp =ωβ=1

√LC(2.1.28)

The product of LC is independent of the size and separation of the conductors and depends only on the permeability µ, and permittivity of ε of the insulating medium.

If a lossless transmission line used for microwave frequencies has an air dielectric and contains no ferromagnetic materials, free-space parameters can be assumed. Thus the numerical value of equation 2.1.28 for air-insulated conductors is approximately equal to the velocity of light in vacuum. That is

νp =ωβ=1

√LC=1

√µoεo= c = 3.0×108 m/s (2.1.29)

When the dielectric of a lossy microwave transmission line is not air, the phase velocity is smaller than the velocity of light in vacuum and is given by

νε =1

√µε=c

√µrεr(2.1.30)

In general, the relative phase velocity factor can be defined as

Velocity Factor =actual phase velocity

velocity of light in a vacuum

νr =νεc=1

√µrεr(2.1.31)

A low-loss transmission line filled only with dielectric medium, such as a coaxial line with solid dielectric between conductors, has a velocity factor on the order of about 0.65.

Example:

A transmission line has the following parameters;R = 2 Ω/m, G = 0.5 mhos/m, f = 1 GHz, L = 8 nH/m and C = 0.23 pF. Calculate: (a) the characteristic impedance; (b) the propagation constant.

Solution

(a) the characteristic impedance

Zo =

R+ jωL

G+ jωC= 179.44+ j26.50 10

(b) the propagation constant

γ =p(R+ jωL)(G+ jωC) = 0.051+ j0.273

2.2 Reflection and Transmission Coefficients

2.2.1 Reflection Coefficient

Figure 2.2.1 shows a transmission line terminated in an impedance Zl. It is usually more convenient to start solving the transmission-line problem from the receiving rather than the sending end, since the voltage-to-current relationship at the load point is fixed by the load impedance. The incident voltage and current waves traveling along the transmission line

Figure 2.2.1: Transmission line terminated in a load impedance.

are given by

V(z) = V+e−γz +V−eγz(2.2.1)

I(z) = I+e−γz +I−eγz(2.2.2)

in which the current wave can be expressed in terms of the voltage by I(z) = V+

Zoe−γz −V+

Zoeγz(2.2.3)

If the line has a length of l, the voltage and current at the receiving end become

Vl = V+e−γl +V−eγl(2.2.4)

Zoe−γl −V+

Il =V+

Zoeγl(2.2.5) 11

The ratio of the voltage to the current at the receiving end is the load impedance. i.e. Zl =VlIl= ZoV+e−γl +V−eγl

V+e−γl −V+eγl(2.2.6)

The reflection coefficient Γ, is defined as

Reflection Coefficient(Γ) = reflected voltage or current

Incident voltage or current

Γ =VrVi=−Ir

Ii(2.2.7)

Solving equation 2.2.6 for the ratio of the reflected voltage at the receiving end, which is V−eγl, to the incident voltage at the receiving end, which is V+eγl, the result is the reflection coefficient at the receiving end:

Γ =V−e−γl

V+e−γl=Zl −Zo

Zl +Zo(2.2.8)

If the load impedance and/or the characteristic impedance are complex quantities, as is usually the case, the reflection coefficient is generally a complex quantity that can be expressed as

Γ = |Γ|eiθl(2.2.9)

where |Γ| is the magnitude and not greater than unity. i.e. |Γ| ≤ 1. θlis the phase angle between the incident and reflected voltages at the receiving end. It is usually called the phase angle of the reflection coefficient.

2.2.2 Transmission Coefficient

A transmission line terminated in its characteristic impedance Zo is called a properly ter minated line. Otherwise it is called an improperly terminated line. As described earlier, there is a reflection coefficient Γ at any point along an improperly terminated line. Ac cording to the principle of conservation of energy, the incident power minus the reflected power must be equal to the power transmitted to the load. This can be expressed as

1−Γ2l =Zo

ZlT2(2.2.10)

Where T is the transmission coefficient defined as

Transmission Coefficient(T) = Transmitted voltage or current

Incident voltage or current=Vtr

Vi=−Itr

Figure 2.2.2: Power transmission on a line.

Figure 2.2.2 shows the transmission of power along a transmission line where Pinc is the incident power, Pre f the reflected power, and Ptr the transmitted power. Let the traveling waves at the receiving end be

V+e−γl +V−eγl = Vtre−γl(2.2.11)

Zoe−γl −V+

V+

Zoeγl =Vtr

Zle−γl(2.2.12)

Multiplication of equation 2.2.12 by Zl and substitution of the result in equation 2.2.11 yield

Γl =V−e−γl

V+e−γl=Zl −Zo

Zl +Zo(2.2.13)

which on substitution of the resulting equation in 2.2.10 we get

T =Vtr

V+=2Zl

Zl +Zo(2.2.14)

The power carried by the two waves in the side of the incident and reflected waves is

Pcr = Pinc −Pre f =

V+e−αl 2

2Zo−

V−eαl 2

2Zo(2.2.15)

The power carried to the load by the transmitted waves is

Ptr =

Vtre−αl 2

2Zl(2.2.16)

Setting Pcr=Ptr and using equations 2.2.13 and 2.2.14, we get

T2 =Zl

Zo(1−Γ)2l(2.2.17)

This relation verifies the previous statement that the transmitted power is equal to the dif ference of the incident power and reflected power. Example:

A certain transmission line has a characteristic impedance of 75+ j0.01Ω and is ter minated in a load impedance of 70+ j50Ω. Compute (a) the reflection coefficient; (b) the transmission coefficient.

Solution

(a) the reflection coefficient

Zl +Zo=(70+ j50)Ω−(75+ j0.01)Ω

Γ =Zl −Zo

(b) the propagation constant T =2Zl

(70+ j50)Ω+ (75+ j0.01)Ω= 0.08+ j0.32

Zl +Zo=2(70+ j50)Ω

(70+ j50)Ω+ (75+ j0.01)Ω= 1.08+ j0.32 14

Topic 3

Further Transmission Lines Characteristics

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)