Davis W.A.Radio frequency circuit design.2001

70 MULTIPORT CIRCUIT PARAMETERS AND TRANSMISSION LINES

(real) and lossy (imaginary) parts:

The revised propagation constant is found by substituting this into Eq. (4.89). The result can be simplified by taking the first two terms of the Taylor series expansion since ε00r − ε0r:

The power loss is proportional to exp 2˛z .

Coaxial Conductor Loss. The power loss per unit length, P$, is obtained by taking the derivative of the power at a given point along a transmission line:

For a low-loss conductor where the dielectric losses are negligible, Eq. (4.87) becomes, with the help of Ohm’s law,

where . is the metal conductivity. This would be the same as Eq. (4.88) if

With this substitution the wave impedance becomes the metal surface impedance:

At the surface there will be a longitudinal electric field of ZmJs directed in the zO direction. Thus, in a lossy line, the fields will no longer be strictly TEM. This longitudinal electric field produces energy flow into the conductor proportional

The power, P, transmitted down the line is found by the Poynting theorem:

The attenuation constant associated with conductor loss is found from Eq. (4.95):

The total loss is found from Eq. (4.67).

4.7.4Microstrip Transmission Line

Microstrip has been a popular form of transmission line for RF and microwave frequencies for some time. The microstrip line shown in Fig. 4.18 consists of a conductor strip of width w on a dielectric of thickness h above a ground plane. Part of the electric field between the strip and the ground plane is in the dielectric and part in the air. The field is more concentrated in the dielectric than in the air. Consequently the effective dielectric constant, εeff, is somewhere between εr and 1, but closer to εr than 1. A variety of methods have been used to find εeff. However, rather than provide a proof, a simple empirically based procedure for synthesizing a microstrip line will be given. Microstrip line is not strictly a TEM type of transmission line and does have some frequency

72 MULTIPORT CIRCUIT PARAMETERS AND TRANSMISSION LINES

w

t

h

ε r

FIGURE 4.18 The microstrip transmission line.

dispersion. Unless microstrip is being used in a multi-octave frequency range application, the TEM approximation should be adequate. The synthesis problem occurs when the desired characteristic impedance, Z0, and the dielectric constant of the substrate, εr, are known and the geometrical quantity w/h is to be found. The synthesis equations, given by [4], are simple, and they give an approximate solution to the microstrip problem:

The analysis equations given below by [5] are more accurate than the synthesis equations. The value given by Eqs. (4.102) and (4.103) provides an initial value for w/h that can be used in an iterative procedure to successfully solve the synthesis problem. This process depends on knowing εr and the conductor thickness, t. The solution results in the effective dielectric constant, εeff, needed to determine electrical line lengths and Z0. The procedure for the analysis procedure is shown below:

From the given value of t and trial solutions of w/h, Eqs. (4.104) to (4.107) give unique values for ua and ur. The characteristic impedance and effective dielectric constant are obtained using Eqs. (4.108) through (4.112):

Since w/h increases when Z0 decreases, and vice versa, one very simple and effective method for finding the new approximation for w/h is done by using the following ratio:

This procedure has been codified in the program MICSTP. The effect of dielectric and conductor loss has been found [6,7,8]:

The value for the loss caused by the dielectric is

current. The generator and the load are characterized by a reflection coefficient,G and i, respectively. The wave bG from the generator undergoes multiple reflections until finally the reflected wave from the load, b1, is obtained:

This last expression is the sum of a geometric series. Since i D b1/a1, an expression for the wave entering into the load can be found:

The power actually delivered to the load is then

From the definition of i and Eq. (4.121), the delivered power can be found:

and when matched so that i D ŁG:

The latter is the available power from the source. Similar expressions could be found for an n-port circuit.

Now the wave values will be related to the terminal voltages and currents. With reference to Fig. 4.19, Ohm’s law gives

A forward-going voltage wave, VC, can be related to the forward-going current by

This is based on the reason that if Z D ZŁG, V D VC, because V D 0. Similarly

76 MULTIPORT CIRCUIT PARAMETERS AND TRANSMISSION LINES

The forward-going voltage and current can be expressed by means of Ohm’s law as follows:

These voltages and currents represent rms values, so the incident power is

jVCj2 Ð <fZGg

D 4.130 jZGj2

The incident power, Pinc, is proportional to jaj2, and the reflected power, Pref, to jbj2. Taking the square root of a number to get a complex quantity is, strictly speaking, not possible mathematically unless a choice is made regarding the phase angle of the complex quantity. This choice is related to choosing ZŁG for a and ZG for b:

From Eqs. (4.131) through (4.134) the forwardand reverse-going voltages and currents can be found in terms of the waves a and b. The total voltage and total current are then found:

These are now in the form of two equations where a and b can be solved in terms of V and I. Multiplying Eq. (4.136) by ZG and adding to Eq. (4.135) gives the result

2a<fZGg

V C ZGI D <fZGg

which can be solved for a. In similar fashion b is found with the following results:

Ordinarily the generator impedance is equal to the characteristic impedance of a transmission line to which it is connected. The common way then to write Eqs. (4.137) and (4.138) is in terms of Z0, which is assumed to be lossless:

1

a D p V C Z0I 4.139 2 Z0

1

b D p V Z0I 4.140 2 Z0

The ratio of Eqs. (4.140) and (4.139) is

b D V/I Z0 D

aV/I C Z0

where is the reflection coefficient of the wave. The transmission coefficient is defined as the voltage across the load V due to the incident voltage VC:

For the two-port circuit shown in Fig. 4.20, there are two sets of ingoing and outgoing waves. These four quantities are related together by the scattering matrix:

78 MULTIPORT CIRCUIT PARAMETERS AND TRANSMISSION LINES

The individual S parameters are found by setting one of the independent variables to zero:

S11 D

b1

a1 a2D0

S12 D

b1

a2 a1D0

S21 D

b2

a1 a2D0

S22 D

b2

a2 a1D0

Thus S11 is the reflection coefficient at port-1 when port-2 is terminated with a matched load. The value S12 is the reverse transmission coefficient when port-1 is terminated with a matched load. Similarly S21 is the forward transmission coefficient, and S22 is the port-2 reflection coefficient when the other port is matched.

The formulas for converting between the scattering parameters and the volt– ampere relations are discussed in Section 4.1 and given in Appendix D. In each of these formulas there is a Z0 because a reflection or transmission coefficient is always relative to another impedance, which in this case is Z0. This is further corroborated by Eqs. (4.139) and (4.140) where the wave values are related to a voltage, current, and Z0.

4.9THE INDEFINITE ADMITTANCE MATRIX

Typically a certain node in a circuit is designated as being the ground node. Similarly, in an n-port network, at least one of the terminals is considered to be the ground node. In an n-port circuit in which none of the terminals is considered the reference node, it can be described by the indefinite admittance matrix. This matrix can be used, for example, when changing the y parameters of a common emitter transistor to the y parameters of a common base transistor. The indefinite admittance matrix has the property that the sum of the rows D 0 and the sum