Microstrip bộ lọc cho các ứng dụng lò vi sóng RF (P2)

On the other hand, it is useful to be able to describe the op-eration of a microwave network such as a filter in terms of voltages, currents, andimpedances in order to make optimum use o

CHAPTER 2

Network Analysis

Filter networks are essential building elements in many areas of RF/microwave gineering Such networks are used to select/reject or separate/combine signals atdifferent frequencies in a host of RF/microwave systems and equipment Althoughthe physical realization of filters at RF/microwave frequencies may vary, the circuitnetwork topology is common to all

en-At microwave frequencies, voltmeters and ammeters for the direct measurement

of voltages and currents do not exist For this reason, voltage and current, as a sure of the level of electrical excitation of a network, do not play a primary role atmicrowave frequencies On the other hand, it is useful to be able to describe the op-eration of a microwave network such as a filter in terms of voltages, currents, andimpedances in order to make optimum use of low-frequency network concepts

mea-It is the purpose of this chapter to describe various network concepts and provideequations that are useful for the analysis of filter networks

Most RF/microwave filters and filter components can be represented by a two-port

network, as shown in Figure 2.1, where V1, V2and I1, I2are the voltage and current

variables at the ports 1 and 2, respectively, Z01and Z02are the terminal impedances,

and Esis the source or generator voltage Note that the voltage and current variablesare complex amplitudes when we consider sinusoidal quantities For example, a si-nusoidal voltage at port 1 is given by

v1(t) = |V1|cos(␻t + ␾) (2.1)

We can then make the following transformations:

v1(t) = |V1|cos(␻t + ␾) = Re(|V1|e j( ␻t+␾) ) = Re(V1e j ␻t) (2.2)

7

Microstrip Filters for RF/Microwave Applications Jia-Sheng Hong, M J Lancaster

Copyright © 2001 John Wiley & Sons, Inc ISBNs: 0-471-38877-7 (Hardback); 0-471-22161-9 (Electronic)

where Re denotes “the real part of ” the expression that follows it Therefore, one

can identify the complex amplitude V1defined by

Because it is difficult to measure the voltage and current at microwave frequencies,

the wave variables a1, b1and a2, b2are introduced, with a indicating the incident waves and b the reflected waves The relationships between the wave variables and

the voltage and current variables are defined as

V n=
Z0n (a n + b n)

n = 1 and 2 (2.4a)

In = (a n – b n)or

The above definitions guarantee that the power at port n is

Pn= 1Re(V n ·I n*) = 1(a nan * – b nbn*) (2.5)

where the asterisk denotes a conjugate quantity It can be recognized that a nan*/2 is

the incident wave power and b n b n */2 is the reflected wave power at port n

Z 0n

Two-port network

S11= ᎏb

a

1 1

ᎏa2=0

S12= ᎏb

a

1 2

ᎏa2=0 S22= ᎏb

a

2 2

ᎏa1=0

where a n= 0 implies a perfect impedance match (no reflection from terminal

im-pedance) at port n These definitions may be written as

where the matrix containing the S parameters is referred to as the scattering matrix

or S matrix, which may simply be denoted by [S]

The parameters S11 and S22are also called the reflection coefficients, whereas

S12 and S21 the transmission coefficients These are the parameters directly

mea-surable at microwave frequencies The S parameters are in general complex, and it

is convenient to express them in terms of amplitudes and phases, i.e., S mn =

|S mn |e j ␾mn for m, n = 1, 2 Often their amplitudes are given in decibels (dB), which

turn loss at port n Instead of using the return loss, the voltage standing wave ratio VSWR may be used The definition of VSWR is

VSWR = (2.10)

Whenever a signal is transmitted through a frequency-selective network such as afilter, some delay is introduced into the output signal in relation to the input signal.There are other two parameters that play role in characterizing filter performancerelated to this delay The first one is the phase delay, defined by

ᎏ

␻

1 + |S nn|ᎏ

where ␾21is in radians and ␻is in radians per second Port 1 is the input port andport 2 is the output port The phase delay is actually the time delay for a steadysinusoidal signal and is not necessarily the true signal delay because a steady si-nusoidal signal does not carry information; sometimes, it is also referred to as thecarrier delay [5] The more important parameter is the group delay, defined by

This represents the true signal (baseband signal) delay, and is also referred to as theenvelope delay

In network analysis or synthesis, it may be desirable to express the reflection

pa-rameter S11in terms of the terminal impedance Z01and the so-called input

imped-ance Z in1 = V1/I1, which is the impedance looking into port 1 of the network Such

an expression can be deduced by evaluating S11in (2.6) in terms of the voltage andcurrent variables using the relationships defined in (2.4b) This gives

Equa-to its terminal impedances

The S parameters have several properties that are useful for network analysis For

a reciprocal network S12= S21 If the network is symmetrical, an additional property,

S11= S22, holds Hence, the symmetrical network is also reciprocal For a losslesspassive network the transmitting power and the reflected power must equal to the to-tal incident power The mathematical statements of this power conservation condi-tion are

2.3 SHORT-CIRCUIT ADMITTANCE PARAMETERS

The short-circuit admittance or Y parameters of a two-port network are defined as

in which V n = 0 implies a perfect short-circuit at port n The definitions of the Y

pa-rameters may also be written as

where the matrix containing the Y parameters is called the short-circuit admittance

or simply Y matrix, and may be denoted by [Y] For reciprocal networks Y12= Y21 In

addition to this, if networks are symmetrical, Y11= Y22 For a lossless network, the Y

parameters are all purely imaginary

The open-circuit impedance or Z parameters of a two-port network are defined as

ten as

The matrix, which contains the Z parameters, is known as the open-circuit ance or Z matrix and is denoted by [Z] For reciprocal networks, Z12= Z21 If net-

imped-works are symmetrical, Z12= Z21and Z11= Z22 For a lossless network, the Z

para-meters are all purely imaginary

Inspecting (2.18) and (2.20), we immediately obtain an important relation

B = ᎏ–

V I

1 2

I I

1 2

rameters have the following properties:

AD – BC = 1 For a reciprocal network (2.24)

A = D For a symmetrical network (2.25)

If the network is lossless, then A and D will be purely real and B and C will be

pure-ly imaginary

If the network in Figure 2.1 is turned around, then the transfer matrix defined in(2.23) becomes

where the parameters with t subscripts are for the network after being turned

around, and the parameters without subscripts are for the network before being

turned around (with its original orientation) In both cases, V1and I1are at the left

terminal and V2and I2are at the right terminal

The ABCD parameters are very useful for analysis of a complex two-port

net-work that may be divided into two or more cascaded subnetnet-works Figure 2.2 gives

the ABCD parameters of some useful two-port networks

Since V2= –I2Z02, the input impedance of the two-port network in Figure 2.1 is

giv-en by

B A

D C

A C

V1

I1

Zin1= = (2.27)

Substituting the ABCD parameters for the transmission line network given in Figure

2.2 into (2.27) leads to a very useful equation

2.6 TRANSMISSION LINE NETWORKS 13

FIGURE 2.2 Some useful two-port networks and their ABCD parameters

where Z c, ␥, and l are the characteristic impedance, the complex propagation

con-stant, and the length of the transmission line, respectively For a lossless line, ␥= j␤

and (2.28) becomes

Besides the two-port transmission line network, two types of one-port transmissionnetworks are of equal significance in the design of microwave filters These areformed by imposing an open circuit or a short circuit at one terminal of a two-porttransmission line network The input impedances of these one-port networks arereadily found from (2.27) or (2.28):

Z in1|Z02=0= = Z ctanh ␥l (2.31)Assuming a lossless transmission, these expressions become

Z in1|Z02=0 = jZ ctan ␤l (2.33)

We will further discuss the transmission line networks in the next chapter when

we introduce Richards’ transformation

Often in the analysis of a filter network, it is convenient to treat one or more filtercomponents or elements as individual subnetworks, and then connect them to deter-mine the network parameters of the filter The three basic types of connection thatare usually encountered are:

con-Analogously, the networks of Figure 2.3(b) are connected in series at both their

input and output terminals; consequently

of cascaded two-port components For simplicity, consider a network formed by thecascade connection of two subnetworks, as shown in Figure 2.3(c) The followingterminal voltage and current relationships at the terminals of the composite networkwould be obvious:

FIGURE 2.3 Basic types of network connection: (a) parallel, (b) series, and (c) cascade

It should be noted that the outputs of the first subnetwork N⬘ are the inputs of the following second subnetwork N⬙, namely

 =  

If the networks N⬘ and N⬙ are described by the ABCD parameters, these terminal

voltage and current relationships all together lead to

Thus, the transfer matrix of the composite network is equal to the matrix product ofthe transfer matrices of the cascaded subnetworks This argument is valid for anynumber of two-port networks in cascade connection

Sometimes, it may be desirable to directly cascade two two-port networks using

the S parameters Let S⬘ mn denote the S parameters of the network N⬘, S⬙ mndenote the

S parameters of the network N ⬙, and S mn denote the S parameters of the composite network for m, n = 1, 2 If at the interface of the connection in Figure 2.3(c),

It is important to note that the relationships in (2.37) imply that the same terminal

impedance is assumed at port 2 of the network N⬘ and port 1 of the network N ⬙ when S⬘ mn and S⬙ mnare evaluated individually

From the above discussions it can be seen that for network analysis we may use ferent types of network parameters Therefore, it is often required to convert one

dif-type of parameter to another The conversion between Z and Y is the simplest one, as

given by (2.21) In principle, the relationships between any two types of parameterscan be deduced from the relationships of terminal variables in (2.4)

B D

A C

V2–I2

For our example, let us define the following matrix notations:

[a] = 1([
Y0]·[Z] + [
Z0])·[I]

[b] = 1([
Y0]·[Z] – [
Z0])·[I]

Replacing [b] by [S]·[a] and combining the above two equations, we can arrive at

the required relationships

way For convenience, these are summarized in Table 2.1 for equal terminations Z01

= Z02= Z0and Y0= 1/Z0

If a network is symmetrical, it is convenient for network analysis to bisect the metrical network into two identical halves with respect to its symmetrical interface

sym-When an even excitation is applied to the network, as indicated in Figure 2.4(a), the

symmetrical interface is open-circuited, and the two network halves become the twoidentical one-port, even-mode networks, with the other port open-circuited In a

similar fashion, under an odd excitation, as shown in Figure 2.4(b), the symmetrical

2.9 SYMMETRICAL NETWORK ANALYSIS 19

TABLE 2.1 (a) S parameters in terms of ABCD, Y, and Z parameters

–Y21

ᎏᎏ

Y11Y22– Y12Y21

1 ᎏ

Z21

–(Y11Y22– Y12Y21) ᎏᎏ

interface is short-circuited and the two network halves become the two identicalone-port, odd-mode networks, with the other port short-circuited Since any excita-tion to a symmetrical two-port network can be obtained by a linear combination ofthe even and odd excitations, the network analysis will be simplified by first analyz-ing the one-port, even- and odd-mode networks separately, and then determining thetwo-port network parameters from the even- and odd-mode network parameters

For example, the one-port, even- and odd-mode S parameters are

S 11e= ᎏb

a e e

ᎏ

(2.42)

S 11o=

where the subscripts e and o refer to the even mode and odd mode, respectively For

the symmetrical network, the following relationships of wave variables hold

a1= a e + a o a2= a e – a o

(2.43)

b1= b e + b o b2= b e – b o Letting a2= 0, we have from (2.42) and (2.43) that

a1= 2a e = 2a o

b1= S 11e a e + S 11o a o

b2= S 11e a e – S 11o a o Substituting these results into the definitions of two-port S parameters gives

The last two equations are obvious because of the symmetry

Let Z ine and Z inorepresent the input impedances of the one-port, even- and mode networks According to (2.14), the refection coefficients in (2.42) can be giv-

odd-en by

By substituting them into (2.44), we can arrive at some very useful formulas:

(2.46)

where Y ine = 1/Z ine , Y ino = 1/Z ino and Y01= 1/Z01 For normalized

impedances/ad-mittances such that z = Z/Z01and y = Y/Y01, the formulas in (2.46) are simplified

–

ino

1+ 1)

Networks that have more than two ports may be referred to as the multiport

net-works The definitions of S, Z, and Y parameters for a multiport network are similar

to those for a two-port network described previously As far as the S parameters are concerned, in general an M-port network can be described by

b2

ᎏ

a1

1ᎏ2

b1

ᎏ

a1

2.10 MULTIPORT NETWORKS 21

 =  ·  (2.48a)which may be expressed as

where [S] is the S-matrix of orderM × M whose elements are defined by

Sij= a k= 0(k⫽j and k=1,2, M) for i, j = 1, 2, M (2.48c)

For a reciprocal network, S ij = S ji and [S] is a symmetrical matrix such that

where the superscript t denotes the transpose of matrix For a lossless passive

net-work,

where the superscript * denotes the conjugate of matrix, and [U] is a unity matrix

It is worthwhile mentioning that the relationships given in (2.21), (2.40), and(2.41) can be extended for converting network parameters of multiport networks The connection of two multiport networks may be performed using the following

method Assume that an M1-port network N⬘ and an M2-port network N⬙, which aredescribed by

[b ⬘] = [S⬘]·[a⬘] and [b ⬙] = [S⬙]·[a⬙] (2.51a)

respectively, will connect each other at c pairs of ports Rearrange (2.51a) such that

(2.51b)

where [b⬘] c and [a⬘] c contain the wave variables at the c connecting ports of the network N⬘ , [b⬘ ] p and [a⬘ ] p contain the wave variables at the p unconnected ports

of the network N⬘ In a similar fashion [b⬘⬘] c and [a⬘⬘] ccontain the wave variables

at the c connecting ports of the network N⬘⬘, [b⬘⬘] q and [a⬘⬘] qcontain the wave

vari-ables at the q unconnected ports of the network N⬘⬘ ; and all the S submatrices tain the corresponding S parameters Obviously, p + c = M and q + c = M It is

con-[a⬙]q [a⬙] c

[S⬙]qc [S⬙] cc

[S⬙]qq [S⬙] cq

[b⬙]q [b⬙] c

[a⬘]p [a⬘] c

[S⬘]pc [S⬘] cc

important to note that the conditions for all the connections are [b⬘] c = [a⬘⬘] cand

[b⬘⬘] c = [a⬘] c, or

where [0] and [U] denote the zero matrix and unity matrix respectively Combine

the two systems of equations in (2.51b) into one giving

In order to make a parallel or series connection, two auxiliary three-port

net-works in Figure 2.5 may be used The one shown in Figure 2.5(a) is an ideal parallel junction for the parallel connection, and its S matrix is given on the right; Figure

[b⬘] p [b⬙] q

[a⬘] p [a⬙] q

[a⬘] c

[a⬙] c

[a⬘] c [a⬙] c

2.10 MULTIPORT NETWORKS 23