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Chapter 2Characterization of Microwave Networks
2.1 Introduction2.1 Scattering Parameters
2.1.1 Definition2.1.2 Advantages of Scattering Matrix Representation2.1.3 Some Characteristics of S-matrices
2.2 Scattering Matrices for Some 2-port Networks2.2.1 S-matrices of Symmetrical 2-port Networks2.2.2 General Method for Determining S-matrices2.2.3 Analysis of arbitrarily connected 2-ports2.2.4 Analysis of Arbitrarily Connected Networks using Multiport
Connection Method 2.3 Shift in Port Reference Planes2.4 Signal-flow Graph
2.3.1 Rules for Constructing Signal-flow Graphs2.3.2 Simplification of Signal-flow Graphs2.3.3 Applications of Signal-flow Graphs
2.5 Cascaded Two-ports (ABCD Parameters)2.6 Immittance Matrices2.7 Measurement of S-parameters using Vector Network Analyzer
2.7.1 Calibration of Network Analyzer2.7.2 De-embedding
Appendix-2AAppendix-2BAppendix-2C
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Chapter 2Characterization of Microwave Networks
The microwave circuits have the following distinguishing features primarily due to the their size being comparable to the wavelength of operation.(a) Radiation from the circuit: We know from radiation theory that power radiated by a short wire varies as (l / λ )2. Therefore, longer the size of the circuit element, larger will be the power radiated. The radiated power not only gives rise to loss of signal it may give rise to undesired coupling between neighboring circuit elements affecting the circuit performance.
(b) Coupling: Any two open transmission lines running in close proximity to each other gives rise to coupling of fields between them. This results in transfer of power from one line to the other. The coupling is not significant at low frequencies because the coupling length is a small fraction of wavelength. However, the coupling becomes significant at microwave frequencies and this phenomenon is used to design directional couplers, power splitters and combiners.
(c) Parasitic: Parasitic is defined as something undesirable. The microwave circuits include junctions and bends of transmission lines, open ends, shorts and so many other types of discontinuities. These discontinuity effects can be characterized in terms of additional reactances. Since these reactances are unintentional at the initial circuit design stage, they are called parasitics and affect the accuracy of design. In the accurate or integrated circuit design, the parasitics are included in the design process.
(d) Skin effect and losses: The skin depth is inversely proportional to the square root of frequency of operation. As a result, the penetration of fields in the good conductors is limited to a few skin depths only. The conductors therefore have higher loss at microwave frequencies than at lower frequencies.
(e) Reference planes: The circuits at microwave frequencies are enclosed between reference planes at the ports of the network. This is due to the fact that voltages and currents have spatial variations, and therefore the network parameters [Z], [Y], or [S] depend on the reference planes. Any shift from the reference planes results in change in network characteristics.
At lower frequencies, networks are characterized invariably in terms of voltages and currents at various terminals. This leads to the formulation of immitance (impedance or admittance) parameters or ABCD parameters, more formally known as A-parameters. However, at microwave frequencies it is difficult to measure immitance matrices of circuits. Therefore, microwave circuits are described theoretically by wave variables a and b at various ports, rather than by voltages and currents. Relationships among variables a and b are expressed by means of a scattering matrix, whose elements are known as S-parameters. It does not mean that immitance or ABCD-parameters are not used at all at microwave frequencies. We shall see, however, that ABCD-parameter representation is very useful for analyzing two-port circuits.
2.1 Generalized Scattering Parameters2.1.1 Definition
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The microwave networks are best analyzed in terms of wave propagation. It is therefore appropriate to characterize microwave networks using wave amplitudes at the ports. Consider an N-port network or component shown in Fig. 2.1. Let Z01 , Z02 ,..
Z0 Nbe the characteristic impedances of the transmission lines connected to these ports. In place of the voltages and currents at the terminals of low frequency networks, we employ the incident and the reflected wave amplitudes at the ports to determine the characteristics of microwave networks. The incident and reflected wave amplitudes at the n-th port is denoted by an and bn, respectively; which may be defined in terms of the voltages as
an=V n
+¿
√Z0n
¿, bn=V n
−¿
√Z0n
¿ (2.1)
where V n+¿¿and V n
−¿ ¿ are the voltage wave amplitudes in the transmission line (or waveguide) connected to the nth port. The voltages are complex numbers with magnitude and phase. The generalized S-matrix refers to the S-matrix of a network with different port impedances, that is, in general Z01≠ Z02≠ …Z0 N ≠ Z0. The S-matrix of a network with Z01=Z02=… Z0 N=Z0 is readily obtained from the generalized S-matrix.
To justify the definition (2.1) for wave variables we compute power flow into and out of ports. For this purpose, port voltage V nand port current I n at the nth port (see Fig. 2.1) may be written as: (Chapter1, (1.10), (1.11) and (1.17) with z = 0 at the port plane),
V n=V n+¿¿+ V n
−¿=√Z0n ( an+b n)¿ (2.2)and
I n=¿¿- V n−¿ ¿/ Z0 n=(an−bn ) /√Z0 n (2.3)
The negative sign in (2.3) is due to the direction assigned to the backward current as a convention. The net power flowing into the nth port may be expressed as
W n=12
ℜ (V n I n¿ )=1
2 ( an an¿−bn bn
¿ ) (2.4)
This shows that W nis equal to the power incident at the nth port via the incident wave 12 (an an
¿ ) less the power sent back 12 (bnbn
¿ ) via the reflected wave traveling away from
the port. Relation (2.4) shows that the definition (2.1) for a and b is consistent with the calculation of power flow in the network.
Mathematically, scattering matrix [S] expresses the relationship between a’s and b’s at various ports of a network through the matrix equation
b = [S]a (2.5)where b and a are n-element column vectors (n×1matrices) and [S] is an n×n square matrix. For a two-port network, (2.5) may be written as
b1=S11a1+S12 a2 (2.6a)b2=S21a1+S22a2 (2.6b)
It may be noted that the coefficients along the main diagonal of the scattering matrix are voltage reflection coefficients. For an n-port network,
Snn=bn
an|ai=0 (i ≠n )
=V n
−¿
V n+¿¿
¿ (2.7)
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Equation (2.7) indicates that the voltage reflection coefficientSnn may be determined by finding bn for a unit value of an when the incident wave amplitudes at all other ports are set to zero. Off-diagonal terms of the scattering matrix represent the voltage transmission coefficients of the network. For example,
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Snm=bn
am|ai=0 (i ≠ m)=
V n−¿/√Z0 n
V m+¿/√Z0m ¿
¿ (2.8)
Thus Snm represents the voltage transmission coefficient from port m to port n, and may be determined by finding bnfor am=1, with incident wave amplitudes at all other ports equal to zero. Note that the order of subscripts in Snmis opposite to the direction of power transmission. For a network with ports impedances same, Snm=V n
−¿/V m+¿¿ ¿.
2.1.2 Advantages of Scattering Matrix RepresentationApart from the physical interpretation of scarring matrix parameters in terms of voltage reflection and transmission coefficients as discussed above, the scattering matrix representation is convenient also because of the following considerations.
Existence of S-matrix: A scattering matrix exists for every linear, passive and time-invariant network. This is not true of admittance or impedance matrices. We will come across an example of this while discussing the representation of a transmission line section later.
Convenience of measurement: One of the main advantages associated with the use of S-parameters is the ease with which they can be measured at microwave frequencies. At lower frequencies, measurements of Z- or Y-parameters are carried out by using open-circuit or short-circuit terminations at some of the ports and measuring the voltage and/or current at the port(s) associated with the particular matrix element being measured. At microwave frequencies, an ideal open-circuit or short-circuit is difficult to realize because of the parasitic reactance associated with such conditions. Also, for some active circuits such as transistor amplifiers, open-circuit and short circuit terminations may disturb the stability of the amplifier. The S-parameter measurements on the contrary, are carried out by terminating the ports with matched loads (usually the normalizing impedance Z0 n) so that the input wave amplitudes at these ports are zero. The matched loads are easier to realize and do not in general affect the circuit stability.
Shift of reference plane: Another important advantage of the S-parameter approach emerges from the fact that the S-parameters are defined in terms of traveling waves; and unlike currents and voltages, the wave magnitude does not vary along a uniform lossless line. This feature allows the S-parameters of a network or device to be measured relatively far away from the physical location of its ports because at times the physical ports are not accessible. This is equivalent to shifting the reference planes from the port locations and can be compensated for by modifying the phase of the measured data. On the other hand, if ABCD or Z- or Y-parameters are used, they would vary with distance away from the port terminals not only in phase but in magnitude also.
2.1.2 Characteristics of S-matricesSome of the important characteristics of S-matrices are listed below:
(i) For a reciprocal network, the S-matrix is symmetric, that is, [ S ]=[ S ] t (2.9)
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where the superscript t indicates the transpose of a matrix. This relationship can be proved by starting from the fact that for reciprocal networks the Z-matrix is symmetrical. Then use the inter-relationship between Z- and S-parameters.
(ii) For lossless passive networks, the power conservation property leads to two interesting characteristics of S-matrices*. Power conservation implies that the sum of the net power flowing into or out of the network equals zero, that is,
∑n
(an an¿−bn bn
¿ )=0 (2.10)(a) One of the results that can be derived from the above relation is
∑n=1
N
|S¿|2=∑
n=1
N
S¿S¿¿=1 (for all i = 1,2,3…N) (2.11a)
that is, the inner product of any column of the scattering matrix with its own conjugate equals unity i.e. the sum of reflected and transmitted power coefficients equals unity for unit power fed to the port.
(b) Another property of S-parameters arising from power conservation is
∑n=1
N
Sns Snr¿ =0 (for all s, r = 1,2,3…N, s≠r) (2.11b)
This orthogonality constraint implies that the inner product of any column of a scattering matrix with the complex conjugate of any other column is zero.
Derivations of (2.11a) and (2.11b) are given in the Appendix-2A. These two relations are useful in the verification of the correctness of S-matrices for lossless, passive networks, and are also helpful in deriving the values of certain elements of S-matrix.
2.2 Scattering Matrices for Some 2-port NetworksTwo-port networks are relatively easy to design (if the network consists of sections of transmission lines/ waveguides connected in cascade) and constitute a major building block of microwave networks. Analysis for the S-matrix of 2-ports is discussed next.
2.2.1 S-matrices of Symmetrical 2-port NetworksA 2-port network is symmetrical when the interchange of ports 1 and 2 does not alter its characteristics in any manner. A simple example of such a network is a section of uniform transmission line shown in Fig. 2.2(a). The symmetry results in the following relationship between S-parameters:
S11=S22 , S12=S21 (2.12)that is, only the two out of total four parameters need to be determined. Let us consider two different modes of excitation of a symmetrical 2-port network shown in Fig. 2.2(b) and Fig. 2.2(c). The excitation in Fig. 2.2 (b) is called in-phase excitation because the ports are excited by sources with identical magnitude and phase. Because of the symmetry of the network and excitations, the voltage distribution across the network has even symmetry about the plane ss. Consequently, the gradient of voltage at the plane of symmetry is zero and no current crosses ( I∝dV /dz ) this plane. Thus the plane ss may be replaced by an open circuit plane without disturbing the voltage and current distribution anywhere in the network. The open circuit condition is equivalent to magnetic wall boundary condition at plane ss in the field theory.
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* For example, see R.E. Collin, Foundations for Microwave Engineering, McGraw Hill, 1966, pp. 174-175.
Introducing an open-circuit plane (or a magnetic wall) at ss allows us to determine b1e by considering the corresponding one-port network problem shown in Fig. 2.3.
For the even-voltage distribution or in-phase excitation, half-section of a transmission line of length ℓ, S11e (reflection coefficient at port 1) or b1 for a1=1 , may be
determined from the input impedanceZ¿ of transmission line of length ℓ/2 and using (1.25) to calculate Γ. We get,
Zine=− j Z0 cot ( β l /2 )and
S11e=Γ e=− j Z0 cot ( β l /2 )−Z0
− j Z0 cot ( β l /2 )+Z0=
− jcot ( β l /2 )−1− jcot ( β l /2 )+1
(2.13)
The second mode of excitation shown in Fig. 2.2(c) is out-of-phase excitation since the voltage sources are out of phase. This excitation produces a voltage distribution that is anti-symmetric with respect to the plane ss. Therefore, the voltage along ss is zero which can be replaced by a short circuit plane. The short circuit condition is equivalent to electric wall boundary condition at plane ss in field theory. The half sections of the networks corresponding to this excitation scheme are shown in Fig. 2.4. The reflection coefficient S11 ofor the anti-symmetric distribution is found as follows:
Zino= j Z0 tan ( β l /2 )and
S11 o=Γo=j Z0 tan ( β l /2 )−Z0
j Z0 tan ( β l /2 )+Z0= jtan (β l /2 )−1
jtan ( β l /2 )+1(2.14)
The characteristics of even-mode (in-phase) and odd-mode (out-of-phase) half sections may be combined to yield the characteristics of the 2-port symmetrical networks. If the excitations shown in Fig 2.2(b) and Fig. 2.2(c) are superimposed, we obtain the input and output wave amplitude values shown in Fig. 2.5. For this excitation, we can write S11 for the 2-port network as
S11=b1
a1|a2=0
=b1e+b1 o
2 |a2=0
=S11 e+S11 o
2 (2.15)
When we substitute the values of S11e∧S11 o from (2.13) and (2.24), respectively and
simplify (see Appendix-2B), we obtain S11=0.
From the values of a1and b2 shown in Fig. 2.5, the value of S21 may be obtained as:
S21=b2
a1|a2=0
=b2e+b2o
2 |a2=0
=b1e−b1o
2 |a2=0
=S11e−S11o
2 (2.16a)
Again substituting the values ofS11eandS11 o, and simplifying (as given in Appendix-2B),
S21=e− jβ l (2.16b)
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Thus, the S-matrix of the transmission line section shown in Fig. 2.2 may be written as
[S] = [ 0 e− jβ l
e− jβ l 0 ] (2.17)
Various terms in (2.17) may be interpreted as follows. Zeros along the diagonal indicate that the reflection coefficients at ports 1 and 2 are zero. This is obvious from the configuration since the normalizing impedances at ports 1 and 2 areZ0. The transmission coefficients (from port 1 to port 2 and vice versa) are given by e− jβl, which implies a phase delay of β l radians as the signal propagates through length l of the line.
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9
10
11
The above example of a transmission line section illustrates a useful method for determining the S-parameters of symmetrical 2-port networks. Some of the networks whose S-parameters may be determined in a similar manner are shown in Fig. 2.6.
The concept of symmetry may also be used for simplifying the analysis of symmetrical 4-port (in general 2n-port) networks. For a symmetrical 4-port network, one has to analyze only two 2-port even and odd-mode half-sections. Examples of this kind of network would be discussed later when we describe the design of branchline and coupled line directional couplers in Chapter 8.
Evaluation of S-matrices of symmetrical networks can also be described in terms of the solution of eigenvalue equation given below
[S]U n=snU n or ( [ S ]−sn[I ])U n=0 (2.18)where Un is the nth eigenvector and snis the corresponding eigenvalue. Comparing (2.18) with (2.5), it can be seen that the eigenvector Unrepresents a possible excitation of the network at the terminal planes with the excitation voltages proportional to the elements of the eigenvector. The eigenvalue snrepresents a reflection coefficient measured at any terminal plane. The elements of the S-matrix are obtained as linear combination of eigenvalues. Details of this approach are described in the book by Altman. [J.A. Altman, Microwave Circuits, Van Nostrand, New York, 1964].
2.2.2 General Method for Determining S-matricesAnother approach for calculating the scattering matrices is based on finding the relationships among wave variables by expressing them in terms of port voltages and port currents. We will illustrate this method by considering an ideal junction of two transmission lines of different impedances as shown in Fig. 2.7. In general, a junction discontinuity gives rise to disturbance in rf voltage and current across the junction and the effect of this disturbance may be expressed as a series reactance. Neglecting this junction effect, we can write using the Kirchoff’s laws:
V 1=V 2, I 1=−I 2 (2.19)From (2.2) and (2.3) we obtain
2an=V n
√Z0 n
+ I n √Z0 n (2.20a)
and
2bn=V n
√Z0 n
−I n √Z0 n (2.20b)
or V n=√Z0 n ( an+bn ) I n=( an−bn )/√Z0 n (2.21)
Thus, for the circuit in Fig. 2.7
2a1=V 1
√Z1
+ I1 √Z1 (2.22)
Using (2.19) in (2.22) gives
2a1=V 2
√Z1
−I 2 √Z1 (2.23)
Also
2b2=V 2
√Z2
−I 2 √Z2 (2.24)
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By definition
S21=b2
a1|a2=0=
V 2
√Z2
−I 2√Z2
V 2
√Z1
−I 2√Z1|a2=0
(2.25)
The condition a2=0 can be ensured by terminating port 2 in matched load ZL=Z0 . This gives V 2=−I 2 Z2. Substituting this in (2.25) yields
S21=
−Z2
√Z2
−√Z2
−Z2
√Z1
−√Z1
=2√Z1 Z2
Z1+Z2(2.26)
Similarly,
S21=b1
a1|a2=0=
V 1
√Z1
−I 1√Z1
V 1
√Z1
+ I1 √Z1|V 2=− I 2 Z2
S11=
−I 2 Z2
√Z1
+ I 2√ Z1
−I 2 Z2
√ Z1
−I 2 √Z1
=Z2−Z1
Z2+Z1
(2.27)Thus the S-matrix of the transmission line junction of Fig. 2.7 may be written as
[S] = [ Z2−Z1
Z2+Z1
2√Z1 Z2
Z1+Z2
2√Z1 Z2
Z1+Z2
Z1−Z2
Z2+Z1] (2.28)
In (2.28), S22 is obtained from S11 by interchanging Z1 and Z2.
The method used above can be extended for the circuits shown in Fig. 2.8. As we will discuss later in Chapter 4, the configuration in Fig. 2.8(a) is an equivalent circuit for a step discontinuity in strip lines. It may be noted that these networks are not symmetrical, and hence the in-phase and out-of-phase excitation approach cannot be used here. For these networks, relationships similar to (2.19) are obtained from Kirchoff’s laws. For the series impedance network of Fig. 2.8(a) we have
I 1=−I 2, and V 1=V 2+Z I 1 (2.29)whereas for the shunt admittance network of Fig. 2.8(b) we have
V 1=V 2, and I 1=Y V 1−I2 (2.30)These equations, along with (2.20) and (2.21) yield S-matrices for these networks.
2.2.3 Analysis of Arbitrarily Connected 2-ports [CAD of Microwave Circuits*]2-port components can be combined in series or parallel to yield a 2-port network. Four different configurations, which are possible for such connections, are shown in Fig. 2.9. These circuits can be analyzed by using Y-, Z-, H-, or G-matrices* also. The
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Y-matrix characterization for the parallel-parallel connection of Fig. 2.9(a) can be obtained simply by adding Y-matrices of the two components. For the series-series connection of Fig. 2.9(b), the individual Z-matrices are added to obtain the overall Z-matrix. Similarly, for the series-parallel connection of Fig. 2.9(c) and parallel-series connection of Fig. 2.9(d). Even though very little effort is spent in adding these matrices, these representations have the drawback that Y-, Z-, H- or G-matrices are computed from the S-matrices and vice-versa.
Fig. 2.10: Combinations of 2-port components(a) A parallel-parallel connection of two 2-port components(b) A series-series connection of two 2-port components(c) A series-parallel connection of two 2-port components(d) A parallel-series connection of two 2-port components
2-port components can be combined arbitrarily to yield a multiport network. Some examples of these connections are shown in Fig. 2.10. These circuits can be analyzed by characterizing such a connection as a dummy multiport component (indicated by dotted lines in the Figure), and then using any of the multiport connection methods*; one such method is described in the next section. To do so, the S-matrix of the multiport connection/junction is required.
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Fig. 2.10: Some examples of arbitrarily connected 2-port components to yield multiport networks.
2.2.4 Analysis of Arbitrarily Connected Networks using Multiport Connection MethodIn this method, the S-matrix of the overall network is determined from the S-matrices of individual components. This method is applicable when the network contains arbitrarily interconnected multiport components without independent generators. When one or more independent generators are present, these can be treated as existing outside the remaining network N as shown in Fig. 2.11, and the present method yields the S-matrix for the network N.
Fig. 2.11: An arbitrarily connected network with p-external ports and c-internal ports
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21 Y1 X 2
Consider a network N of multiport components shown in Fig. 2.11. The ports in the network are classified as internal ports or c-ports which are connected, and p-ports which are external to it. If there are m components in the network, the governing relations for all the components can be written together as
[b] = [S] [a] (2.31)Since the components are not yet connected, the associated S-parameter entries are filled by 0s.The rows and columns in (2.31) can be re-ordered so that the wave variables are separated into two groups; the first corresponding to the p-external ports, and the second to c-internally connected ports. Equation (2.31) can now be written as
[bp
bc ]=[ Spp S pc
Scp Scc ][a p
ac ] (2.32)
where b p and a p are the wave variables at the p external ports and bc and ac are the wave variables at the c internal ports. The interconnection constraints for the c-internal ports can be written as
[bc ]=[ Γ ] [ac ] (2.33)where Γ is the connection matrix and is defined next.
*Wienberg, L., “Scattering matrix and transfer scattering matrix,” in Amplifiers, R. F. Shea, Ed., McGraw Hill, New York, 1966.
Interconnection matrix: Let us consider cascade of two 2-port components as in Fig. 2.12. This connection results in a pair of connected ports as shown there. Across the junction, the outgoing wave variable at one port must equal the incoming wave variable at the other port provided the wave variables at the two connected ports are similarly normalized. Applying this argument to the pair of connected ports of Fig. 2.12, we have
Fig. 2.12: A cascade connection of two 2-port components showing the dummy 2-port model for interconnection constraints.
b2x=a1
y and b1y=a2
x (2.34)or
[b2x
b1y ]=[0 1
1 0] [a2x
a1y ] (2.35)
The matrix on the right hand side of (2.35) may be defined as the connection matrix Γ, i.e.bc=Γ ac (2.36)
The resultant network is supposed to have a different S-matrix than that of either component. The interconnection constraints are expected to play an important role in
2
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this respect; and we may model the interconnection constraints as a dummy 2-port as shown Fig. 2.12. The S-matrix description for this dummy 2-port may be written as
[a2x
a1y ]=[ S I ][b2
x
b1y ] or ac=S I bc (2.37)
The LHS in (2.37) is the outgoing wave vector on dummy 2-port and the RHS is the incident wave vector. Comparing (2.37) and (2.36) we may write that S-matrix for the interconnection is the inverse of connection matrix, i.e.
S I=Γ−1=[0 11 0]
−1
(2.38)
If the normalizing impedances for the S-matrices across the interconnection are not same, say Z1 and Z2 for components X and Y, respectively, the interconnection matrix in (2.36) is given by the inverse of S-matrix of (2.28).
The concept of interconnection matrix Γ can be extended to more than one junction, e.g. Fig. 2.11. For this, the relations similar to (2.36) are written together for all the interconnected ports in the network. The interconnection matrix Γ now describes the topology of the network. In each row of Γ, all elements are zero except an entry 1 in the column indicating the interconnection. If the element (j, k) of Γ is 1, it implies that the port j is connected to the port k. The size of Γ is equal to the number of connected ports. The interconnection matrix is therefore highly sparse, and the zero, nonzero pattern depends on the topology and is independent of the component characterizations.
The interconnection matrix for the network N of Fig. 2.11 may be written as[b¿¿c ]=[ Γ ] [a¿¿c ]¿¿ (2.39)
From (2.39) and lower half of (2.32) we getΓ ac =Scp a p+Scc ac
orac =( Γ−Scc )−1Scp a p (2.40)
Substituting this in upper half of (2.32), we getb p={S pp+S pc ( Γ−Scc )−1 Scp}a p (2.41)
or, the network scattering matrix Sp is given bySp=Spp+Spc ( Γ−Scc )−1 Scp (2.42)
After having obtained the S-matrix of a multiport network, the wave variables at the internal ports for an arbitrary excitation at the p-external ports a pmay be obtained from (2.39) and (2.40).
An exampleIn the example of a simple circuit shown in Fig. 2.13, the circuit contains a 2-port component A characterized by SA; a 3-port component B characterized by SB; and a 1-port component C characterized by SC . The ports are numbered as shown in the figure; local and global port numbers are used for components and assembled circuit, respectively. Characterizations for the three components are put together as in (2.43). The vectors a and b employ global port numbers whereas the component S-matrices employ local port numbers.
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[b1
b4
b2
b6
b5
b3
]=[S11
A 00 S22
BS12
A 00 0
0 0S23
B S21B
S21A 0
0 0S22
A 00 sC
0 00 0
0 S32B
0 S12B
0 00 0
S33B S31
B
S13B S11
B][a1
a4
a2
a6
a5
a3
] (2.43)
The interconnection matrix Γ defines the topology of the assembled network and can be written as
[b2
b6
b5
b3]=[0 0
0 00 11 0
0 11 0
0 00 0 ][a2
a6
a5
a3] (2.44)
Using (2.43), (2.44) in (2.42), the S-matrix of the composite network is given by
Sp=[S11A 0
0 S22B ]+[S12
A 00 0
0 0S23
B S21B ][−S22
A 00 −sC
0 11 0
0 11 0
−S33B −S31
B
−S13B −S11
B ]−1
[S21A 0
0 00 S32
B
0 S12B ]
(2.45)
Fig. 2.13: The network considered in the example.
2.3 Applications of S-ParametersThe S-matrix of a network may be employed to determine (i) the wave variables at
various ports when the network is subjected to an excitation at one or many external ports. One of the applications of S-matrix is to determine the (ii) new S-matrix when one or more of the reference planes of the network are shifted. This may be required for measurements or analysis of network. Another common application is (iii) to determine the S-matrix of loaded network e.g. for input impedance calculation. These types of applications are discussed next.
2.3.1 Shift in Reference PlanesS-parameters relate magnitude and phase of the travelling waves incident on and reflected/transmitted from microwave networks. Therefore, reference planes for phase determination of S-parameters must be specified for each port of the network. Any
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change in any of the reference planes gives rise to a change in the whole of the S-matrix of the network. However, this change in S-matrix can be determined analytically and the S-matrix for the new port planes can be obtained from the S-matrix for the old port planes.
Consider a 2-port microwave network of Fig. 2.14 where the original reference port planes are located at z1= 0 and z2= 0, for the 2-port. Then [b] = [S][a], where [S] is the scattering matrix of the network for this set of reference planes.
Fig. 2.14: Shift of port planes of a network
Consider, now a new set of port planes at z1=−l1 and z2=l2 on the transmission lines connected to the ports. Let the new S-matrix for the same network be defined as [b' ]= [S ' ][a ']. Then, in terms of the incident and reflected port voltages we have
¿ (2.41a)
¿ (2.41b)where the unprimed quantities refer to the original port planes at z1= 0 and z2= 0, and the primed quantities are referred to the new planes at z1=−l1 and z2=l2 .
From the theory of traveling waves on lossless lines, the incident wave amplitudes are related as
V n+¿=V n
'+¿ e− j θn¿ ¿
or¿ (2.42)
where θn=β ln. Similarly, the reflected wave amplitudes are related as
V 'n−¿=V n
−¿ e− jθn¿ ¿ ⟹ V n
−¿=V n'−¿ e
jθn ¿¿
or¿ (2.43)
Substituting for ¿ and ¿ in (2.41a) gives
[e j θ1 00 e j θ2]¿
or¿ (2.44)
Comparing this expression with (2.41b) gives
[ S' ]=[e− j θ1 00 e− jθ2] [S] [e− j θ1 0
0 e− jθ2] ¿ [ S11 e− j 2θ1 S12e− j (θ¿¿1+θ2 )¿S21e− j (θ¿¿1+θ2 )¿S22e− j 2 θ1 ]
(2.45)
1 2
l1 l2
z1 = 0 z2 = 0
19
2.3.2 S-Matrix of a Loaded NetworkOne of the most common uses of S-parameters is to determine the S-parameters of reduced network when the network is loaded at one or many ports, e.g. input impedance of a transmission line section when loaded at the other port. The effect of port loadings on the S-matrix can be included by the constraint between a and b vectors imposed by the load/loads. The reduced S-matrix may be obtained by including the effect of load as a = Γ Lb.
2.4 Signal-Flow GraphThe signal-flow graph is a convenient technique to graphically represent and analyze the transmission and reflection of waves in a microwave component. It permits the derivation of expressions such as power gains, voltage gains of complex microwave amplifiers, reflection coefficient, transmission coefficient, etc.
2.4.1 Rules for Constructing Signal-flow Graphs1. A flow graph has two nodes for each port, one for the incident wave and the
other for the reflected wave. The incident waves are independent wave amplitude nodes and the reflected waves are dependent amplitude nodes.
2. The S-parameters and reflection coefficients are represented by branches.3. Branches enter dependent amplitude nodes and emanate from independent
amplitude nodes.4. A dependent node is equal to the sum of the products of the branches
terminating at the node with the independent nodes they emanate from.5. Flow graph of cascaded networks is the cascade of flow graphs of individual
networks comprising it.
Example 1: b1=S11 a1+S12 a2
Here b1 is the dependent node, a1 and a2are independent nodes, and S11 and S12 are represented by the appropriate branches. The signal flow graph is shown in Fig. 2.9.
Example 2: b2=S21a1+S22a2
Proceeding in the same way as for example 1 we obtain the signal flow graph is shown in Fig. 2.10.
Example 3: Two-port networkThe signal flow graphs of Fig. 2.9 and Fig. 2.10 can be combined to yield the signal flow graph of a two-port network. This is shown in Fig. 2.11.
Example 4: Voltage-source with impedance ZS
The voltage-source with an interconnecting line is shown in Fig. 2.12(a). From here, the voltage at the terminal AA can be written as
V g=V S+ I g ZS (2.31)In terms of traveling waves on the connecting line,
V g=V g++V g
−(2.32)
and
20
I g=V g
+
Z0−
V g−
Z0 (2.33)Then (2.31) becomes
V g++V g
−=V S+( V g+
Z0−
V g−
Z0)ZS
(2.34)
Substitute for V gand I g in (2.31) and solving for V g−
, we obtainV g
−
√Z0=
V S √Z0
ZS+Z0+
V g+
√Z0
ZS−Z0
ZS+Z0 (2.35)If we denote
V g−
√Z0=bg
,
V g+
√Z0=ag
,bs=
V S√Z0
ZS+Z0 , and Γ s=
ZS−Z0
ZS+Z0 (2.36)equation (2.35) becomes
bg=bs+Γ s ag (2.37)Equation (2.37) can also be obtained by using multiple reflections procedure*. This
equation suggests that the incident wave bg is not equal to bs unless the source is
non-reflecting, which would give ag=0 or Γ s=0 (that is, a matched source with ZS=Z0 ). The signal flow graph for bg=bs+Γ s ag is shown in Fig. 2.12(b).
Example 3: Load impedance terminating a line
For the load impedance ZL terminating a transmission line of characteristic
impedance Z0 , as shown in Fig. 2.13(a), we can write
*G.D. Vendlin, Design of Amplifiers and Oscillators by the S-parameter Method, John Wiley, New York, 1982.
V L=ZL IL (2.38)In terms of traveling waves on the feeding line, we can write
V L=V L++V L
−(2.39)
and
I L=V L
+
Z0−
V L−
Z0 (2.40)
Substituting for V Land IL gives
V L++V L
−=ZL( V L+
Z0−
V L−
Z0)
(2.41)or
V L−
√Z0=
ZL−Z0
ZL+Z0
V L+
√ Z0 (2.42)Writing
21
aL=V L
+
√ Z0 ,bL=
V L−
√Z0 , andΓ L=
ZL−Z0
ZL+Z0 (2.43)we obtain
bL=aL Γ L (2.44)The signal flow graph for (2.44) is shown in Fig. 2.13(b).
We can combine the signal flow graph for the two-port network, with the signal flow graphs of signal generator and the load to yield the signal flow graph of a typical
microwave amplifier. This is shown in Fig. 2.14. Here bg , ag , bL and aL are replaced
by a1 , b1 , a2and b2 , respectively. S11 , etc. are the S-parameters of the transistor amplifier network.
2.4.2 Simplification of Signal-flow GraphsThe flow graph representation can be used to determine the ratio or transfer function of a dependent to an independent variable. For this we can use Mason’s non-touching loop rule*. An alternative procedure, that is more easy to remember, is the flow graph simplification technique proposed by Kuhn**. The simplification can be achieved using four rules that are stated as follows:
Rule 1 (Node elimination): The transfer function of two similarly directed branches in series is the product of the two as shown in Fig. 2.15. This eliminates the common node between the branches.
* S.J. Mason, “Feedback theory-further properties of signal flow graphs”, Proc. IRE, vol. 44, July 1956, pp. 920-926.** N. Kuhn, “Simplified signal flow graphs analysis”, Microwave J., vol. 6, Nov. 1963, pp. 59-66.Rule 2 (Branch elimination): The transfer function of two similarly directed branches in parallel is the sum of the two as shown in Fig. 2.16. This step eliminates a branch.
Rule 3 (Loop elimination): When a node possesses a self-loop with transfer function S, the self-loop can be eliminated by dividing the transfer function of all the input branches by (1-S) as shown in Fig. 2.17
Rule 4 (Node duplication): The transfer function remains unchanged if a node with an input branch and N output branches is split into two nodes. The input branch goes to each of the new nodes. There will be K<N output branches leaving one node and the remaining (N-K) branches leaving the second node. The dual graph with one output branch and several input branches also holds. This is shown in Fig. 2.18.
The rules, given above, can be proved very easily as done by Adam*. Simplifying flow graphs in this manner is particularly helpful in finding the transfer function of a circuit, e.g. gain of a transistor amplifier. Some of the applications of the flow graph are described next.
2.4.3 Applications of Signal-flow Graphs
22
Example 1: Input reflection coefficientFigure 2.19 shows the flow graph for a load connected at the output of a two-port network. The input reflection coefficient for this network is defined as
Γ in=b1
a1 (2.45)To determine this ratio we can simplify the flow graph as shown in Fig. 2.19. From the final flow graph we obtain using Rule 1
b1=S11 a1+S12 S21 Γ L
1−S22 Γ La1
(2.46)or
Γ in Δb1
a1=S11+
S12 S21 Γ L
1−S22 Γ L (2.47)
Example 2: Output reflection coefficientConsider a two-port network with a load connected at the input as shown in Fig. 2.20. The corresponding signal flow graph is also shown there, which can be simplified by
splitting node a1 to obtain
Γ out Δb2
a2=S22+
S12 S21 Γ s
1−S11 Γ s (2.48)
* S.F. Adam, microwave Theory and Applications, Prentice Hall, New Jersey, 1969.
Example 3: Available power from a generatorThe power available from a generator or a source is defined as the power delivered by the generator to a conjugately matched load. The signal flow graph of a generator
connected to a conjugately matched load, i.e. Γ L=Γ s¿
, is shown in Fig. 2.21. The power available from the generator (or the power delivered to the load through the terminals AA) is given by
PAVS=|bg|2−|ag|
2(2.49)
Also, because bg=bs+Γ s ag and ag=Γ s
¿ bg (Rule 1) (2.50)
bg=bs
1−|Γ s|2
(2.51)
ag=bs Γs
¿
1−|Γ s|2
(2.52)
Substitution in PAVS gives
23
PAVS=|bs|
2 (1−|Γ s|2 )
(1−|Γ s|2 )2
or
PAVS=|bs|
2
1−|Γ s|2
(2.53)
Substituting for bs and Γ s from (2.36) in (2.55) gives
PAVS=V s
2Z0
|Zs+Z0|2(1−|
Z s−Z0
Zs+Z0|2)
or
PAVS=V s
2Z0
|Zs+Z0|2−|Zs−Z0|
2=V s
2
4 Re(Zs ) (2.54)
where V s represents the rms voltage of the signal generator.
Example 4: Power delivered to a load (Γ L )The power delivered by the generator to the load is given by the difference between the incident and reflected powers, that is,PL=|aL|
2−|bL|2
orPL=|aL|
2 (1−|Γ L|2) (2.55)
From the flow graph of Fig. 2.22, aL=bg and
bg=bs
1−Γ s Γ L ⇒aL=
bs
1−Γ s Γ L (2.56)Therefore,
PL=|bs|
2
|1−Γ s Γ L|2 (1−|Γ L|
2)(2.57)
For conjugately matched load Γ L=Γ s¿
, the above relation reduces to (2.53) that is power delivered to the load is equal to the power available from the source.
Example 5: Transfer function of a two-port network with signal generator at the input and a load at the outport portSignal flow graph for this problem is shown in Fig. 2.23. It will be simplified to
determine the transfer function T defined as T=b2/bs . Various steps in the analysis are shown in Fig. 2.23. One obtains
b2
bs=
S21
(1−S11 Γs ) (1−S22 Γ L )−S12S21 Γ s Γ L (2.58)
24
2.5 Analysis of Cascaded Two-ports (ABCD Parameters)In several microwave circuits, two-port components are connected in cascade, that is, port 2 of one component is connected to the port 1 of next component (and so on), as shown in Fig. 2.24. If we consider a cascade combination of only two components A and B, S-matrix of the combination can be written in terms of the S-matrices of the components A and B. Let us write
[ SA ]=[ S11A S12
A
S21A S22
A ], and
[ SB ]=[S11B S12
B
S21B S22
B ](2.59)
We haveb1
A=S11A a1
A+S12A a2
A(2.60a)
b2A=S21
A a1A+S22
A a2A
(2.60b)and
b1B=S11
B a1B+S12
B a2B
(2.61a)b2
B=S21B a1
B+S22B a2
B(2.61b)
At the common portb2
A=a1B
, and a2A=b1
B(2.61)
The S-matrix of the combination circuit can be obtained by eliminating these common
wave amplitudes, that is, b2A , a1
B,a2
Aand b1
B from (2.60) and (2.61). We obtain
[b1A
b2B ]=[S11
A +S12
A S11B S21
A
1−S22A S11
B
S12A S12
B
1−S22A S11
B
S21A S21
B
1−S22A S11
BS22
B +S21
B S22A S12
B
1−S22A S11
B ][a1A
a2B ]
(2.62)This relation can be used repeatedly to obtain S-matrix of a cascade of more than two components.
The analysis of cascaded two-ports can be simplified, compared to (2.62), by using another representation for two-port components. This is known as ABCD or A-matrix and is defined as follows in terms of voltages and currents at the ports of the component.
[V 1
I1 ]=[A BC D ][ V 2
−I 2] (2.63)
The directions of I 1and I 2are as shown in Figs. 2.7 and 2.8, and explains the negative
sign with I 2 in (2.63). When two components x and y are cascaded as in Fig. 2.25 the interconnection relationships may be written as
V 1y=V 2
x, and I 1
y=−I 2x
(2.64)From (2.63) and (2.64) we can write
[V 1x
I1x ]=[ Ax Bx
Cx D x ][ V 2x
−I 2x ]=[A x Bx
Cx Dx ] [V 1y
I 1y ]
=[ A x Bx
Cx D x ][A y B y
C y D y ][ V 2y
−Iy ](2.65)
25
Thus ABCD-matrix of the combination xy is given by the product of [ ABCD x ] and [ ABCD y ]or
[A xy Bxy
Cxy Dxy ]=[ Ax Bx
C x D x ][A y B y
C y D y ](2.66)
The relationship (2.68) is very convenient from the computational point of view since it involves only multiplication of the matrices of individual components. This property makes ABCD-matrix representation very attractive for cascade connection of large number of two-port components.
The ABCD-matrix for a transmission line section may be derived from (1.19) and (1.20), which relate the voltage and current at one end of the line to those at the other end. We have*
[V 1
I1 ]=[A BC D ][ V 2
−I 2]=[cosh γℓ Z0 sinh γℓsinh γℓ /Z0 cosh γℓ ][ V 2
−I 2 ] (2.67)For a lossless line α=0 , γ= jβ and since cosh ( jβℓ )=cos ( βℓ) , and sinh( jβℓ )= jsin ( βℓ) , we have
[ ABCD ]=[cos βℓ jZ0 sin βℓjsin βℓ /Z0 cos βℓ ]
(2.68)-------------------------
* It may be noted that the direction ofI 2 used for defining [ABCD] is opposite to that used in (1.19) and (1.20).
ABCD-matrices for other networks may be obtained by deriving relationships between voltages and currents at the two ports. Alternatively, if the S-matrix for a given network is known, ABCD-matrix of the network may be derived there-from by using the following equivalences:
A=(1+S11−S22−ΔS )√Z01 /Z02 / (2 S21 ) (2.69a)B=(1+S11+S22+ ΔS )√Z01 Z02 /(2 S21 ) (2.69b)C=(1−S11−S22+ ΔS) / (2 S21√Z01 Z02 ) (2.69c)D=(1−S11+S22−ΔS )√Z02 /Z01 / (2 S21) (2.69d)
where ΔS=(S11 S22−S12 S21) and Z01 and Z02 are the impedances of the feed lines connected to the network. Alternatively, if the ABCD-matrix is known, S-matrix may be derived using the following formulas*:
S11=( AZ02+B−CZ01 Z02−DZ01 )/ ΔA (2.70a)S12=2 ( AD−BC ) √Z01Z02 / ΔA (2.70b)S21=2√Z01 Z02/ ΔA (2.70c)S22=(−AZ02+B−CZ01 Z02+DZ01) / ΔA (2.70d)
where
26
ΔA=( AZ02+B+CZ01Z02+DZ01 )
Physical interpretation of ABCD or A-parametersThe ABCD-parameters may be interpreted physically by noting that
A=
V 1
V 2|I2=0
,B=
V 1
−I 2|V 2=0
,C=
I1
V 2|I 2=0
, andD=
I 1
−I 2|V 2=0
(2.71)We note that A is a dimensionless parameter, being the ratio of the input voltage at port 1 to the voltage appearing at port 2 when port 2 is open circuited. D is likewise a dimensionless parameter, being the ratio of current flowing in port 1 to the current flowing out of port 2 when it is short circuited. Parameter B has the dimension of impedance, and is the ratio of voltage at port 1 to the current flowing out of port 2 when it is short circuited. Finally, C is an admittance given by the ratio of the current flowing into port 1 to the voltage appearing at port 2 when it is open circuited.
Some important properties of ABCD-matrices may be summarized as follows:i) For reciprocal networks
AD – BC = 1 (2.72)Unlike [S], [Z] and [Y] matrices, ABCD-matrix need not be symmetric for reciprocal networks.
ii) For symmetrical networks (which remain unaltered when the two ports are interchanged,A = D (2.73)
---------------------------* K.C. Gupta et. al., CAD of Microwave Circuits, Artech House, Mass., 1981, p.34.
2.6 Immittance MatricesImpedance and admittance matrices are also useful at microwave frequencies. Immittance description of transmission line components and discontinuities may be used to derive their lumped equivalent circuits. Also, impedance matrix description is convenient for two-dimensional (planar) components because of the availability of impedance Green’s functions for such components.
The impedance matrix Z for an n-port component is defined as
[V 1
V 2
⋮V n
]=[Z11 Z12 ⋯ Z1n
Z21 Z22 ⋯ Z2n
⋮ ⋮ ⋮ ⋮Zn 1 Zn2 ⋯ Znn
][I 1
I 2
⋮I n
](2.74)
The directions of currents are as shown in Fig. 2.7. For a two=port network,V 1=Z11 I 1+Z12 I 2 (2.75a)V 2=Z21 I1+Z22 I 2 (2.75b)
The entries in Z-matrix are the open circuit impedances, since
Z11=V 1
I1|I 2=0
,Z12=
V 1
I2|I1=0
,Z21=
V 2
I 1|I2=0
, andZ22=
V 2
I 2|I1=0
(2.76)For a reciprocal network,
27
Z12=Z21 ⇒[ Z ]=[ Z ]t(2.77)
and for a symmetrical two-port (where ports 1 and 2 are interchangeable)Z11=Z22 (2.78)
The following relationships holds between Z-matrix and ABCD-matrix for a two-port network (See Appendix-2C)
[ Z ]= 1C [A AD−BC
1 D ] (2.79)For a transmission line section, the ABCD-matrix is given in (2.68). Using (2.79), the Z-matrix may be written as
[ Z ]=− jZ0 [cot βℓ cos ec βℓcosec βℓ cot βℓ ] (2.80)
When ℓ=λ /4 , βℓ=π /2 ,
[ Z ]=[ 0 − jZ0
− jZ0 0 ] λ/4 (2.81)
But when ℓ=λ /2 , βℓ=π ,cot βℓ and (1 /sin βℓ ) tend to infinity, with the result that Z-matrix cannot be written for a λ /2 line length. This illustrates the point that Z-matrix might not exist for every circuit. On the other hand, S-matrix exists for every linear, passive, time invariant network.
One of the advantages of Z-matrix representation is that once the Z-matrix is known, a lumped equivalent circuit for the network can be obtained. For a symmetrical, reciprocal two-port the equivalent lumped network is shown in Fig. 2.26(a); when the two-port is not symmetrical, the equivalent circuit gets modified to that shown in Fig. 2.26(b). These equivalent circuits may be verified by the equivalent Z-parameters using (2.75) and (2.76).
Based on this approach we can write an equivalent lumped T-network for a given transmission line of length ℓ , with Z-parameters given in (2.79). We get
Z1=Z2=Z11=Z12=− jZ 0 cot ( βℓ )+ jZ0 /sin ( βℓ )
=− jZ0 (cot ( βℓ )−1 /sin ( βℓ ) )= jZ0
1−cos ( βℓ )sin ( βℓ )
= jZ 0 tan( βℓ2 )
(2.82)and
Z3=Z12=− jZ 0 /sin ( βℓ ) (2.83)
When βℓ<π , the series elements are purely inductive with equivalent L1 given by
L1=Z0
ωtan ( βℓ
2 ) (2.84a)and shunt capacitance
C3=sin( βℓ )/(ωZ0 ) (2.84b)The equivalent circuit is shown in Fig. 2.27. When the line length is so small that βℓ << 1 , (2.84) can be approximated as
28
L1=Z0 βℓ2ω ,
C3=βℓ
ωZ 0 (2.85)
A further approximation can be made for low values of Z0 . In this case, the equivalent circuit can be approximated by a shunt capacitor only since the series inductors are negligibly small.
Admittance Matrix RepresentationY-matrix is inverse of the Z-matrix, and is defined as
[ I1
I2
⋮I n
]=[Z11 Z12 ⋯ Z1 n
Z21 Z22 ⋯ Z2 n
⋮ ⋮ ⋮ ⋮Zn1 Zn 2 ⋯ Znn
]−1
[V 1
V 2
⋮V n
]=[Y 11 Y 12 ⋯ Y 1n
Y 21 Y 22 ⋯ Y 2n
⋮ ⋮ ⋮ ⋮Y n1 Y n2 ⋯ Y nn
] [V 1
V 2
⋮V n
](2.86)
For a two-port network,I 1=Y 11V 1+Y 12V 2 (2.87a)I 2=Y 21V 1+Y 22 V 2 (2.87b)
Y-matrix and ABCD-matrix are related by (see Appendix-2D)
[Y ]= 1B [ D −AD+BC
−1 A ] (2.88)The Y-matrix for a transmission line section may be derived from (2.88). We obtain
[Y ]=− jY 0 [cot βℓ −cosec βℓ−cos ec βℓ cot βℓ ] (2.89)
When ℓ=λ /4 , βℓ=π /2 ,
[Y ]=[ 0 jY 0
jY 0 0 ]λ/4 (2.90)Like the Z-matrix, the Y-matrix also becomes indeterminate for a half-wave transmission line section.
Once the Y-matrix of a network is known, a π section equivalent network may be
drawn as shown in Fig. 2.28. The admittances Y 1 , Y 2 and Y 3 are given byY 1=Y 11+Y 12 , Y 2=Y 22+Y 12 , Y 3=−Y 12 (2.91)
For a transmission line section, the lumped element equivalent π -network consists ofY 1=Y 2=− jY 0 cot (βℓ )+ jY 0 /sin ( βℓ )
=− jY 0 (cot ( βℓ )−1 /sin ( βℓ ) )= jY 0
1−cos ( βℓ )sin ( βℓ )
= jY 0 tan( βℓ2 )(2.92)
andY 3=−Y 12=− jY 0 /sin ( βℓ ) (2.93)
The lumped elements corresponding to a line length with βℓ << 1 are shunt capacitances and a series inductor with the values given by
C1=Y 0 βℓ2ω ,
L3=βℓ
ωY 0 (2.94)
29
The equivalent circuit is shown in Fig. 2.29. It may be observed from (2.85) and
(2.94) that these equations are dual of each other with L ↔C and Y 0↔ Z0 since the π - and T-networks are also dual of each other.
When the characteristic impedance of the transmission line section is very high, the equivalent circuit can be approximated by a series inductor only.
The lumped element equivalent networks obtained from the Y-matrix or Z-matrix representations are strictly true over a very narrow band of frequencies. Otherwise, the values of lumped elements vary with frequency.
2.6.1 Conversion between S-, Z- and Y-matricesFor some circuit configurations it is easier to derive impedance or admittance matrices. These matrices can be converted into S-matrix by using the relationship (2.20) and (2.21) between the wave amplitudes and the voltages and currents. The general method of determining the S-matrices is based on this approach.
We can write, for the normalized wave amplitudesa= 1
2 (V + I ) (2.95)and
b= 12 (V −I ) (2.96)
S-matrix from Z- or Y-matrix:For this conversion replace b by [ S ]a and V by [ Z ]I in the above equations to obtain
[ S ]a= 12 ([Z ]−[ I ]) I (2.97)
Using (2.95) in (2.97) gives[ S ] ([ Z ]+[ I ]) I =([ Z ]−[ I ] )I (2.98)
implying [ S ] ([ Z ]+[ I ])=([ Z ]−[ I ]) (2.99)
Post multiply (2.99) by ([ Z ]+[ I ])−1 to yield
[ S ]=([ Z ]−[ I ] ) ([ Z ]+[ I ])−1(2.100)
By virtue of [Y ]=[ Z ]−1, we obtain from above
[ S ]=([ I ]−[Y ])([ I ]+[Y ])−1(2.101)
Equations (2.100) and (2.101) provide conversion from Z- and Y-matrices to S-matrix.
Z- or Y-matrix from S-matrix:From (2.95) and (2.96) we can write
V=a+b (2.102a)and
I=a−b (2.102b)Again replace b by [ S ]a in (2.102) to obtain
V= ([ I ]+[ S ]) a (2.103)and
30
I=([ I ]−[ S ]) a (2.104)Now replace V=[ Z ]I in (2.103) and use (2.104) giving
[ Z ]([ I ]−[ S ]) a=([ I ]+[ S ] )a (2.105)implying
[Z ]([ I ]−[ S ])=([ I ]+[ S ] ) (2.106)
Post multiply (2.106) by ([ I ]−[ S ])−1 to give
[ Z ]=([ I ]+[ S ])([ I ]−[ S ])−1(2.107)
or[Y ]−1=([ I ]+[ S ] )([ I ]−[ S ])−1
or[Y ]=([ I ]−[ S ] )([ I ]+[ S ])−1
(2.108)Equations (2.107) and (2.108) can be used to convert S-matrix to Z- and Y-matrices, respectively.
31
Appendix-2A: Derivation of the S-matrix Relations (2.11) for Lossless NetworksThe power conservation condition (2.10) may be expressed as
∑n=1
N
bn bn¿=∑
n=1
N
an an¿
(2A.1)From (2.2)
bn=∑i=1
N
Sni a i(2A.2)
The power conservation condition (2A.1) may now be written as
∑n=1
N
|∑i=1
N
Sni ai|2=∑
n=1
N
an an¿
(2A.3)
Since an ’s are independent, we may choose all an ’s except a i to be zero. This leads to
∑n=1
N
|Sni ai|2=ai ai
¿
or
∑n=1
N
|Sni|2=∑
n=1
N
Sni Sni¿ =1
(2A.4)Since the index i is arbitrary, the above equation is valid for all values of i.
The other set of constraints (2.11b) may be derived from (2A.3). By selecting all an ’s
except as and ar to be zero (r≠s ), the relation (2A.3) may be written as
∑n=1
N
|Snsas+Snr ar|2=∑
n=1
N
(Sns as+Snr ar ) (Sns as+Snr ar)¿=|as|2+|ar|
2
(2A.5)Expanding the left side, and using (2A.4) to equate two of the terms to zero gives
∑n=1
N
(Sns Snr¿ as ar
¿+Sns¿ Snr as
¿ ar )=0(2A.6)
In view of the independent nature of as and ar , we may choose in the first instance, as=ar . This leads to
|as|2∑
n=1
N
(Sns Snr¿ +Sns
¿ Snr )=0(2A.7)
If, instead we choose as= jar with ar real, we obtain
j|ar|2∑
n=1
N
(Sns Snr¿ −Sns
¿ Snr )=0(2A.8)
Since neither as norar is zero, both (2A.7) and (2A.8) are satisfied only if
∑n=1
N
Sns Snr¿ =0
s≠r (2A.9)which is the orthogonality condition stated in (2.11).
32
Appendix- 2B: Derivation of (2.17)Let βℓ≡x , then from (2.13) and (2.14)
S11 e+S11o=− jcot ( x /2 )−1− jcot ( x /2 )+1
+j tan ( x /2 )−1j tan ( x /2 )+1
¿1− jcot ( x /2 )− j tan ( x /2 )−1+1+ j cot ( x /2 )+ j tan (x /2 )−12+ j ( tan (x /2 )+cot ( x /2 ) )
=0
Let x/2 = y, then
S11 e−S11 o=− j cot y−1− j cot y+1
− j tan y−1j tan y+1
¿1− jcot y− j tan y−1−(1+ j cot y+ j tan y−1 )2+ j ( tan y−cot y )
=−2 j (cot y+ tan y )2+ j (tan y−cot y ) (2B.1)
Since,tan ( x /2 )=(1−cos x ) /sin x and cot ( x /2)=(1+cos x ) /sin xtan( x /2 )−cot (x /2 )=−2 cos x /sin xtan( x /2 )+cot ( x /2)=−2 /sin x
Therefore, (2B.1) can be expressed as
S11 e−S11 o=−2 j (2 /sin x )
2+ j (−2cos x /sin x )= −2 j
sin x− j cos x= 2
e jx =2 e− jx
orS11e−S11o
2=e− jβℓ
33
Appendix-2C: Derivation of (2.79) and (2.88)
From the definition of ABCD-matrixV 1=AV 2−BI 2 (2C.1)I 1=CV 2−DI2 (2C.2)
Equation (2C.2) can be written asCV 2=I 1+DI 2
orV 2=( I 1+DI 2 )/C (2C.3)
Substituting for V 2 in (2C.1) one obtains
V 1=AC
I1+( ADC
−B)I 2(2C.4)
V 2=1C
I 1+DC
I 2(2C.5)
Equations (2C.4) and (2C.5) yield the impedance matrix as
[ Z ]= 1C [A AD−BC
1 D ] (2C.6)Also,
I 2=−V 1/B+( A /B )V 2 (2C.7)
Substituting for I 2 in (2C.2) one obtains
I 1=CV 2−D(−1B
V 1+AB
V 2)=DB
V 1+(C− ADB )V 2
(2C.8)Equations (2C.7) and (2C.8) yield
[Y ]= 1B [ D BC−AD
−1 A ] (2C.9)
34
Appendix-2D: ABCD, and S and T-Matrices for some commonly used two-ports
35
36
Important Results
1. For a multi-port network, the wave amplitudes an and bnare defined asan=V n
+ /√ Z0n , an=V n+ /√ Z0n
where Z0n represent normalizing impedances at various ports.
2. Scattering matrix [S] for an n-port network is defined by b=[S ]a .3. For the lossless, passive networks the following relationships exist
∑n=1
N
Sni Sni¿ =1
, for all i = 1,2,3,…N.
∑n=1
N
Sns Snr¿ =0
, for all s, r = 1,2,…N; s≠r .
4. For two-port symmetrical networks S11=S22 , and S-parameters may be
found from S11 e and S11 o for the even and odd-mode half-sections, respectively.S11=S22=
12 (S11e+S11 o) , S12=S21=
12 (S11e−S11o )
5. In general, S-parameters may be found from the voltages and currents at the various ports by using
an=12 ( V n
√Z0 n
+ I n√Z0 n), and
bn=12 ( V n
√Z0 n
−I n√Z0 n)6. ABCD-matrix representation is convenient for cascaded two-ports and is
defined as
[V 1
I1 ]=[ A BC D ][ V 2
−I 2]
37
1211 ZZ 1222 ZZ
2112 ZZ 1211 YY
2212 YY
2112 YY
7. Some general properties of ABCD-matrix are:
i) For cascaded two-port networks [ A ]=[ A ]1 [ A ]2 . . .[ A ]N
ii) For reciprocal networks ΔA=Ad−BC=1
8. Z- and Y-parameters may be used for writing T- and π -lumped equivalent circuits for a network, as shown below
38
Problems for Chapter 2
2.1 A lossless transmission line of characteristic impedance Z and electrical lengthβℓ is terminated at both ends by transmission lines of characteristic
impedance Z0 . Derive the expressions for S-matrix for this two-port.
2.2 Consider a lumped impedance Z in series with a transmission line of
characteristic impedance Z0 . The line extends on both the sides as shown in Fig. 2.6(b). Evaluate two-port S-matrix of this configuration.
2.3 Consider a shunt connected admittance Y as shown in Fig. 2.6(c). Calculate two-port scattering matrix for this configuration.
2.4 Find the scattering matrix for the symmetrical T-network shown in Fig. 2.6(f).
2.5 Find the scattering matrix for the symmetrical π -network shown in Fig. 2.6(g).
2.6 Calculate the S-matrix for a series impedance Z connected between two lines
of characteristic impedances Z1 and Z2 , respectively as shown in Fig. 2.8(a).
2.7 Find S-matrix for a shunt admittance Y connected as shown in Fig. 2.8(b).
2.8 Use the S-matrix to A-matrix transformation to find A-matrices for series impedance connected as in Prob. 2.2, and also when connected as in Prob. 2.6. Compare and comment on the two results obtained.
2.9 Calculate the A-matrix for an admittance Y connected in shunt between two
lines of characteristic admittance Y 0 as shown in Fig. 2.6(c).
2.10 A transmission cavity can be represented by a section of transmission line
(characteristic admittanceY 0 , length βℓ ) shunted by susceptance jB on either end as shown in Fig. 2.6(d). Find admittance parameters and derive an equivalent circuit therefrom.
2.11 A lossless transmission line of length ℓ1 (characteristic impedance Z0 and phase constant β ) is connected at the input terminals of a two-port network with S-
parameters S11 , S12 , S21 , S22 . Find the S-matrix for the composite network. If another
line of length ℓ2 is connected at port 2, what will be the resulting S-matrix?
2.12 Show that a symmetrical 3-port lossless, reciprocal transmission line junction cannot be matched at all the three ports. The properties of S-matrix for a lossless network may be used.
39
2.13 Consider a 3-port network consisting of a 3-db power divider matched at the
input port (say port 1). The reference planes at ports 2 and 3 are chosen such that S21
and S31 are real. Use the properties of S-matrix for lossless reciprocal networks to
show that |S22|=|S33|=|S23|=1/2
2.14
impedance ZL at port 2. Calculate the input reflection coefficient at port 1.
2.15
measurement of S-parameters by varying ZL ? How many known values of ZL will be required? Can all S-parameters be determined completely by this method?
2.16
and Z3 as shown in Fig. 2.30. Derive the S-matrix (ignoring parasitic reactance).
2.17 Consider a junction of three series-connected transmission lines of characteristic
impedancesZ1 , Z2 and Z3 , respectively, as shown in Fig. 2.31. Derive the S-matrix (parasitic reactance may be ignored).
2.18are connected in series at a junction J as shown in Fig. 2.32.. Derive the S-matrix of the junction assuming that the parasitic reactance at the junction may be ignored.
2.19 Four transmission lines of different characteristic impedancesZ1 , Z2 ,Z3 and Z4 are connected in series at a junction J as shown in Fig. 2.33.. Find the S-matrix of the junction when the parasitic reactances are ignored.
2.20
port 3 is terminated withZL . Find the S-matrix of the resulting two-port network when
(a) ZL=Z0 , where Z0 is the reference characteristic impedance at the ports.
(b) ZL=0 , and
(c) ZL=∞
2.21
(b) Show that for a shunt connected admittance of Fig. 2.6(c), S21=1+S11 .
40
2.22of a connector when two identical connectors are available. Here, ℓ is the uniform
transmission line of characteristic impedance Z0 . The signal flow graph of the circuit is also shown. Verify using the signal flow graph that the reflection and transmission coefficients can be expressed as
ρ=S11+
S12S22 S21e−2 jβℓ
1−S222 e−2 jβℓ
, and
T=
S12 S21 e− jβℓ
1−S222 e−2 jβℓ
2.23 Determine the S-matrix of an ideal T-junction consisting of transmission lines
of characteristic impedanceZ0 .
Ans.
[ S ]=[−1/3 2 /3 2/32/3 −1 /3 2/32/3 2 /3 −1/3 ]
2.24 Consider a symmetrical Tee-junction in the form of a 1:1 power splitter. The
through-arm has characteristic impedanceZ0=100 ohm and the stub-arm hasZ0=50 ohm. Determine the S-matrix of the junction. The effect of discontinuity may be ignored.
Ans.
[ S ]=[−1/2 1/2 1/√21/2 −1/2 1/√2
1/√2 1/√2 0 ]2.25 (a) Consider an inductor L= 1/6 nH embedded in series in a transmission line
withZ0 = 50 ohm as shown in Fig. 4(a). Determine the magnitude of reflection coefficient and transmission phase (delay or advance) produced by it at ω=30×109
rad/sec.(b) A capacitor with C = 1/15 pF is mounted in shunt across a transmission
line withZ0 = 50 ohm as shown in Fig. 4(b). Determine the magnitude of reflection coefficient and transmission phase (delay or advance) produced by it at ω=30×109
rad/sec.(c) Now make a combination of L and C as shown in Fig. 4(c) and insert it in the same transmission line and compute the magnitude of reflection coefficient and transmission phase (delay or advance) produced by the combination at ω=30×109
rad/sec.(d) Comment on the behavior of the circuit in (c) compared to its constituents behavior in (a) and (b).
41
Fig. 4(a) Fig. 4(b) Fig. 4(c)
2.26 This problem emphasizes that an unduly long interconnect line between the components may seriously affect the power transfer between them especially if the characteristic impedance of the interconnect is different from the impedances at its ends. A transmission line section of length ℓ is connected to the source at one end and the load at the other end as shown below
Show that the voltage V L across the load is given by the expression
V L=V 0 Z in
RS+Zin(cos( βℓ )− jsin ( βℓ)
Z0
Z in)
where
Zin=Z0
RL+ jZ0 tan( βℓ )Z0+ jR L tan( βℓ
Compute and plot |V L| for , , RS=20Ω ℓ=2cm , β=2 π / λ and V S=2volt . The frequency may be varied from 1MHz to 10 GHz.
See the solution at the end.
42
Table 2.1: Conversions between two-port parameters normalized to Z0=1¿
with ΔK=K11 K22−K 12 K21
S Z Y A
S [b1
b2 ]=[S11 S12
S21 S22 ][a1
a2 ]S11=
(Z11−1 ) (Z22+1 )−Z12 Z21
(Z11+1 ) (Z22+1 )−Z12 Z21
S12=2 Z12
(Z11+1 ) (Z22+1 )−Z12 Z21
S21=2 Z21
(Z11+1 ) (Z22+1 )−Z12 Z21
S22=(Z11+1 ) (Z22−1 )−Z12 Z21
(Z11+1 ) (Z22+1 )−Z12 Z21
S11=(1−Y 11) ( 1+Y 22)+Y 12Y 21
(1+Y 11) (1+Y 22 )−Y 12Y 21
S12=−2Y 12
(1+Y 11) (1+Y 22)−Y 12Y 21
S21=−2Y 21
(1+Y 11) (1+Y 22)−Y 12Y 21
S22=(1+Y 11) (1−Y 22 )+Y 12Y 21
(1+Y 11) (1+Y 22)−Y 12Y 21
2( ADBC )A+B+C+D
S21=2
A+B+C+D
S22=−A+B−C+D
A+B+C+D
Z
Z11=(1+S11 ) (1−S22 )+S12 S21
(1−S11 ) (1−S22 )−S12 S21
Z12=2 S12
(1−S11) (1−S22)−S12 S21
Z21=2 S21
(1−S11) (1−S22 )−S12 S21
Z22=(1−S11 )(1+S22 )+S12 S21
(1−S11) (1−S22 )−S12 S21
[V 1
V 2 ]=[Z11 Z12
Z21 Z22 ] [I 1
I 2 ]
Z11=Y 22
ΔY
Z12=−Y 12
ΔY
Z21=−Y 21
ΔY
Z22=Y 11
ΔY
Z11=AC
Z12=ΔAC
Z21=1C
Z22=DC
Y
Y 11=(1−S11) (1+S22)+S12 S21
(1+S11) (1+S22 )−S12 S21
Y 12=−2 S12
(1+S11) (1+S22)−S12 S21
Y 21=−2 S21
(1+S11) (1+S22)−S12 S21
Y 22=(1+S11) (1−S22 )+S12 S21
(1+S11) (1+S22)−S12 S21
Y 11=Z22
ΔZ
Y 12=−Z12
ΔZ
Y 21=−Z21
ΔZ
Y 22=Z11
ΔZ
[ I1
I 2 ]=[Y 11 Y 12
Y 21 Y 22 ][V 1
V 2 ]
Y 11=DB
Y 12=−ΔA
B
Y 21=−1B
Y 22=AB
Table 2.1 (continued)
A
A=(1+S11) (1−S22)+S12S21
2S21
B=(1+S11) (1+S22)−S12 S21
2 S21
C=(1+S11) (1−S22 )−S12 S21
2 S21
D=(1−S11) (1+S22 )+S12 S21
2 S21
A=Z11
Z21
B= ΔZZ21
C= 1Z21
D=Z22
Z21
A=−Y 22
Y 21
B=−1Y 21
C=−ΔYY 21
D=−Y 11
Y 21
[V 1
I1 ]=[A BC D ][ V 2
−I 2]
*All Z, Y, and A parameters are normalized with respect to Z0=Y 0=1 . This implies that normalizing impedances at two ports (for S-matrix representation) are equal. The normalized ABCD parameters given in this Table are defined as
[V 1
I1 ]=[ A Z0 BC / Z0 D ][ V 2
−I 2 ]
DCBADCBAS
11
43
Need for distributed parameter approachA lumped element circuit consists of inductor, capacitor, etc connected through wire or transmission lines called interconnect. The bases of low frequency circuit design are the Kirchhoff’s voltage and current laws, and ignore the spatial variation of voltage. The voltage variation on the interconnect between lumped elements may be described as
V ( z ,t )=V 0 cos (ωt−βz ) (2.1)
where V 0 is the maximum voltage, ω is the radian frequency and β is the phase constant of the signal. At low frequencies, where the length of interconnect is negligible compared to the wavelength, the associated voltage variation given by (2.1) is negligible and the circuit behaves according to the design. However, as the frequency of operation is increased the length of the interconnect begins to tell on the circuit performance. The deteriorating effect of the interconnect on the circuit performance can be reduced by eliminating them as far as possible. To understand the effect of interconnect with frequency, let us study the following experiment.
The experiment consists of a sinusoidal voltage source V 0 with internal resistance RS
connected to a lumped load RL by a length of wire ℓ as shown in Fig. 2.1. Let the wire or interconnect length ℓ be 2cm and is assumed lossless. Let this circuit be realized on a printed circuit board (PCB) with the interconnection wire in the form of a narrow strip of metallization. The strip and the ground metallization of PCB may be modeled as a
transmission line of characteristic impedance Z0 . Alternatively, if the hook-up wires are used to realize the circuit, the combination of interconnection and ground wires may again be
modeled as parallel wire transmission line of characteristic impedance Z0 . The circuit may now be analyzed according to the transmission line theory of Sec. 1.2.
Fig. 2.1: A transmission line fed by a source at one end and terminated in a load at the other end.
ForRS=RL=Z0 , it is found from the analysis of the circuit of Fig. 2.1 that the voltage
across the load, V L=
V 0
2e− jβℓ
, that is, interconnect affects the phase only and not amplitude.
For any other combination of Z0 , RL , RS , β and ℓ the voltage across the load will vary with frequency. The circuit may be analyzed as follows:
From the transmission line theory, the input impedance seen by the source is given by
44
Zin=Z0
RL+ jZ0 tan( βℓ )Z0+ jR L tan( βℓ (2.2)
and the voltage and current at the source end of the line is given by
I S=V 0
RS+ZinV S=
V 0
RS+Z inZ in
(2.3)The voltage and current can be translated to the load side of the line by using (1.20)
[V S
I S ]=[cos( βℓ ) jZ0 sin( βℓ )j sin ( βℓ )/Z0 cos ( βℓ ) ][V L
I L ](2.4)
It yields
V L=V 0 Z in
RS+Zin(cos( βℓ )− jsin ( βℓ)
Z0
Z in)
(2.5)
This expression yields V L=
V 0
2e− jβℓ
for RS=RL=Z0 as expected. The voltage V L as a
function of frequency is plotted in Fig. 2.2 for , , RS=20 ohm ℓ=2cm ,
and V S=2volt . It is seen that V L=V 0 /2 at low frequencies and starts varying as the frequency is increased.
The effect of interconnect on the circuit performance can be included if the line is divided into a number of segments of sufficiently small lengths so that the voltage or current over each segment can be assumed to be constant as shown in Fig. 2.2. Each line segment of
length Δz can be modeled by a series resistance R1 , series inductance L1 , and shunt
capacitance C1 . This equivalent circuit will account for voltage and current variation over the segment length Δz . We can build the complete lumped circuit model for the interconnect by combining the effect of all the segments into which the interconnect is divided. An integrated approach is to consider the line as a single segment, characterize the line in terms of distributed parameters R, L, C, and G where these line constants are given in terms of unit length, and the sinusoidal voltage variation given by (2.1) is built into the analysis. The propagation of voltage and current waves on the line is discussed in Sec 1.2 and the voltage and current distributions are given by (1.20).
45
