S-Parameters

664 14. S-Parameters

b1 S11 S12 a1 S11 S12 14 = ,S= (scattering matrix) (14.1.3) b2 S21 S22 a2 S21 S22

S-Parameters The matrix elements S11,S12,S21,S22 are referred to as the scattering parameters or the S-parameters. The parameters S11, S22 have the meaning of reflection coefficients, and S21, S12, the meaning of transmission coefficients. The many properties and uses of the S-parameters in applications are discussed in [1135–1174]. One particularly nice overview is the HP application note AN-95-1 by Anderson [1150] and is available on the web [1847]. We have already seen several examples of transfer, impedance, and scattering matrices. Eq. (11.7.6) or (11.7.7) is an example of a transfer matrix and (11.8.1) is the corresponding impedance matrix. The transfer and scattering matrices of multilayer structures, Eqs. (6.6.23) and (6.6.37), are more complicated examples. 14.1 Scattering Parameters The traveling wave variables a1,b1 at port 1 and a2,b2 at port 2 are defined in terms of V1,I1 and V2,I2 and a real-valued positive reference impedance Z0 as follows: Linear two-port (and multi-port) networks are characterized by a number of equivalent + − circuit parameters, such as their transfer matrix, impedance matrix, admittance matrix, V1 Z0I1 V2 Z0I2 a1 = a2 = and scattering matrix. Fig. 14.1.1 shows a typical two-port network. 2 Z0 2 Z0 (traveling waves) (14.1.4) − + V1 Z0I1 V2 Z0I2 b1 = b2 = 2 Z0 2 Z0

The deﬁnitions at port 2 appear different from those at port 1, but they are really

the same if expressed in terms of the incoming current −I2: − + − V2 Z0I2 V2 Z0( I2) a2 = = Fig. 14.1.1 Two-port network. 2 Z0 2 Z0 V + Z I V − Z (−I ) = 2 0 2 = 2 0 2 The transfer matrix, also known as the ABCD matrix, relates the voltage and current b2 2 Z0 2 Z0 at port 1 to those at port 2, whereas the impedance matrix relates the two voltages † The term traveling waves is justified below. Eqs. (14.1.4) may be inverted to express V1,V2 to the two currents I1,I2: the voltages and currents in terms of the wave variables: V1 AB V2 = (transfer matrix) I CD I 1 2 = + = + (14.1.1) V1 Z0(a1 b1) V2 Z0(a2 b2) (14.1.5) V1 Z11 Z12 I1 1 1 = (impedance matrix) = − = − V2 Z21 Z22 −I2 I1 (a1 b1) I2 (b2 a2) Z0 Z0 × Thus, the transfer and impedance matrices are the 2 2 matrices: = In practice, the reference impedance is chosen to be Z0 50 ohm. At lower fre- AB Z11 Z12 quencies the transfer and impedance matrices are commonly used, but at microwave T = ,Z= (14.1.2) CD Z21 Z22 frequencies they become difficult to measure and therefore, the scattering matrix description is preferred. −1 The admittance matrix is simply the inverse of the impedance matrix, Y = Z . The The S-parameters can be measured by embedding the two-port network (the device- scattering matrix relates the outgoing waves b1,b2 to the incoming waves a1,a2 that under-test, or, DUT) in a transmission line whose ends are connected to a network ana- are incident on the two-port: lyzer. Fig. 14.1.2 shows the experimental setup. † A typical network analyzer can measure S-parameters over a large frequency range, In the figure, I2 flows out of port 2, and hence −I2 flows into it. In the usual convention, both currents I1,I2 are taken to flow into their respective ports. for example, the HP 8720D vector network analyzer covers the range from 50 MHz to 14.1. Scattering Parameters 665 666 14. S-Parameters

40 GHz. Frequency resolution is typically 1 Hz and the results can be displayed either on a Smith chart or as a conventional gain versus frequency graph. −jδ1 −jδ2 a1 e 0 a1 a2 e 0 a2 = , = (14.1.7) jδ1 jδ2 b1 0 e b1 b2 0 e b2

where δ1 = βll and δ2 = βl2 are the phase lengths of the segments. Eqs. (14.1.7) can be rearranged into the forms: jδ1 b1 b1 a1 a1 e 0 = D , = D ,D= jδ2 b2 b2 a2 a2 0 e

The network analyzer measures the corresponding S-parameters of the primed variables, that is,

Fig. 14.1.2 Device under test connected to network analyzer. b S S a S S 1 = 11 12 1 = 11 12 Fig. 14.1.3 shows more details of the connection. The generator and load impedances ,S (measured S-matrix) (14.1.8) b2 S21 S22 a2 S21 S22 are conﬁgured by the network analyzer. The connections can be reversed, with the generator connected to port 2 and the load to port 1. The S-matrix of the two-port can be obtained then from: b1 b1 a1 a1 = D = DS = DS D ⇒ S = DS D b2 b2 a2 a2

or, more explicitly, jδ1 jδ1 S11 S12 e 0 S11 S12 e 0 = jδ2 jδ2 S21 S22 0 e S21 S22 0 e (14.1.9) + S e2jδ1 S ej(δ1 δ2) = 11 12 j(δ1+δ2) 2jδ2 S21e S22e

Fig. 14.1.3 Two-port network under test. Thus, changing the points along the transmission lines at which the S-parameters are measured introduces only phase changes in the parameters.

The two line segments of lengths l1,l2 are assumed to have characteristic impedance Without loss of generality, we may replace the extended circuit of Fig. 14.1.3 with the equal to the reference impedance Z0. Then, the wave variables a1,b1 and a2,b2 are one shown in Fig. 14.1.4 with the understanding that either we are using the extended recognized as normalized versions of forward and backward traveling waves. Indeed, two-port parameters S , or, equivalently, the generator and segment l1 have been re- according to Eq. (11.7.8), we have: placed by their Th´evenin equivalents, and the load impedance has been replaced by its propagated version to distance l . + − 2 V1 Z0I1 1 V2 Z0I2 1 a1 = = V1+ a2 = = V2− 2 Z0 Z0 2 Z0 Z0 (14.1.6) − + V1 Z0I1 1 V2 Z0I2 1 b1 = = V1− b2 = = V2+ 2 Z0 Z0 2 Z0 Z0

Thus, a1 is essentially the incident wave at port 1 and b1 the corresponding reﬂected wave. Similarly, a2 is incident from the right onto port 2 and b2 is the reﬂected wave from port 2. The network analyzer measures the waves a1,b1 and a2,b2 at the generator and load ends of the line segments, as shown in Fig. 14.1.3. From these, the waves at the Fig. 14.1.4 Two-port network connected to generator and load. inputs of the two-port can be determined. Assuming lossless segments and using the propagation matrices (11.7.7), we have: 14.2. Power Flow 667 668 14. S-Parameters

The actual measurements of the S-parameters are made by connecting to a matched For a lossy two-port, the power loss is positive, which implies that the matrix I −S†S load, ZL = Z0. Then, there will be no reflected waves from the load, a2 = 0, and the must be positive definite. If the two-port is lossless, Ploss = 0, the S-matrix will be S-matrix equations will give: unitary, that is, S†S = I. We already saw examples of such unitary scattering matrices in the cases of the equal b1 b1 = S11a1 + S12a2 = S11a1 ⇒ S11 = = reflection coefficient travel-time multilayer dielectric structures and their equivalent quarter wavelength mul- a = 1 ZL Z0 tisection transformers. b2 b2 = S21a1 + S22a2 = S21a1 ⇒ S21 = = transmission coefficient a = 1 ZL Z0 14.3 Parameter Conversions Reversing the roles of the generator and load, one can measure in the same way the It is straightforward to derive the relationships that allow one to pass from one param- parameters S12 and S22. eter set to another. For example, starting with the transfer matrix, we have:

1 D A AD − BC 14.2 Power Flow = + V1 = A I1 − I2 + BI2 = I1 − I2 V1 AV2 BI2 C C C C ⇒ Power ﬂow into and out of the two-port is expressed very simply in terms of the traveling I = CV + DI 1 D 1 2 2 V = I − I wave amplitudes. Using the inverse relationships (14.1.5), we ﬁnd: 2 C 1 C 2

The coefficients of I ,I are the impedance matrix elements. The steps are reversible, 1 ∗ 1 2 1 2 1 2 Re[V I1] = |a1| − |b1| 2 1 2 2 and we summarize the final relationships below: (14.2.1) 1 ∗ 1 2 1 2 − Re[V I2] = |a2| − |b2| Z11 Z12 1 AAD− BC 2 2 2 2 Z = = Z21 Z22 C 1 D The left-hand sides represent the power flow into ports 1 and 2. The right-hand sides (14.3.1) represent the difference between the power incident on a port and the power reflected AB 1 Z11 Z11Z22 − Z12Z21 ∗ T = = from it. The quantity Re[V2 I2]/2 represents the power transferred to the load. CD Z21 1 Z22 Another way of phrasing these is to say that part of the incident power on a port We note the determinants det(T)= AD − BC and det(Z)= Z11Z22 − Z12Z21. The gets reflected and part enters the port: relationship between the scattering and impedance matrices is also straightforward to derive. We define the 2×1 vectors: 1 2 1 2 1 ∗ |a1| = |b1| + Re[V I1] 2 2 2 1 (14.2.2) V1 I1 a1 b1 V = , I = , a = , b = (14.3.2) 1 2 1 2 1 ∗ − |a2| = |b2| + Re[V (−I2)] V2 I2 a2 b2 2 2 2 2 Then, the definitions (14.1.4) can be written compactly as: One of the reasons for normalizing the traveling wave amplitudes by Z0 in the definitions (14.1.4) was precisely this simple way of expressing the incident and reflected 1 1 = + = + powers from a port. a (V Z0I) (Z Z0I)I 2 Z0 2 Z0 If the two-port is lossy, the power lost in it will be the difference between the power (14.3.3) 1 1 entering port 1 and the power leaving port 2, that is, b = (V − Z0I)= (Z − Z0I)I 2 Z0 2 Z0 1 1 1 1 1 1 = ∗ − ∗ = | |2 + | |2 − | |2 − | |2 where we used the impedance matrix relationship V = ZI and defined the 2×2 unit Ploss Re[V1 I1] Re[V2 I2] a1 a2 b1 b2 2 2 2 2 2 2 matrix I. It follows then, † =| |2 +| |2 † =| |2 +| |2 † = † † Noting that a a a1 a2 and b b b1 b2 , and writing b b a S Sa, 1 1 = + −1 ⇒ = − = − + −1 we may express this relationship in terms of the scattering matrix: I (Z Z0I) a b (Z Z0I)I (Z Z0I)(Z Z0I) a 2 Z0 2 Z0

1 † 1 † 1 † 1 † † 1 † † Ploss = a a − b b = a a − a S Sa = a (I − S S)a (14.2.3) Thus, the scattering matrix S will be related to the impedance matrix Z by 2 2 2 2 2

−1 −1 S = (Z − Z0I)(Z + Z0I) Z = (I − S) (I + S)Z0 (14.3.4) 14.4. Input and Output Reﬂection Coefﬁcients 669 670 14. S-Parameters

Explicitly, we have: Z Z S S Γ − 12 21 12 21 L 1 Zin = Z11 − Γin = S11 + (14.4.3) Z11 − Z0 Z12 Z11 + Z0 Z12 Z + Z 1 − S Γ S = 22 L 22 L Z21 Z22 − Z0 Z21 Z22 + Z0 The equivalence of these two expressions can be shown by using the parameter Z − Z Z 1 Z + Z −Z conversion formulas of Eqs. (14.3.5) and (14.3.6), or they can be shown indirectly, as = 11 0 12 22 0 12 − − + = Z21 Z22 Z0 Dz Z21 Z11 Z0 follows. Starting with V2 ZLI2 and using the second impedance matrix equation, we can solve for I2 in terms of I1: where Dz = det(Z + Z0I)= (Z11 + Z0)(Z22 + Z0)−Z12Z21. Multiplying the matrix factors, we obtain: Z21 V2 = Z21I1 − Z22I2 = ZLI2 ⇒ I2 = I1 (14.4.4) Z22 + ZL Then, the first impedance matrix equation implies: 1 (Z − Z )(Z + Z )−Z Z 2Z Z = 11 0 22 0 12 21 12 0 S + − − (14.3.5) Z Z Dz 2Z21Z0 (Z11 Z0)(Z22 Z0) Z12Z21 12 21 V1 = Z11I1 − Z12I2 = Z11 − I1 = ZinI1 Z22 + ZL Similarly, the inverse relationship gives: Starting again with V2 = ZLI2 we find for the traveling waves at port 2: V2 − Z0I2 ZL − Z0 Z (1 + S )(1 − S )+S S 2S a = = I = 0 11 22 12 21 12 2 2 Z (14.3.6) 2 Z0 2 Z0 Z − Z D 2S21 (1 − S11)(1 + S22)+S12S21 L 0 s ⇒ a2 = b2 = ΓLb2 V + Z I Z + Z ZL + Z0 = 2 0 2 = L 0 = − = − − − b2 I2 where Ds det(I S) (1 S11)(1 S22) S12S21. Expressing the impedance param- 2 Z0 2 Z0 eters in terms of the transfer matrix parameters, we also find: = ⎡ ⎤ Using V1 ZinI1, a similar argument implies for the waves at port 1: B ⎢ A + − CZ0 − D 2(AD − BC) ⎥ V1 + Z0I1 Zin + Z0 1 ⎢ ⎥ = = S = Z0 (14.3.7) a1 I1 ⎣ B ⎦ 2 Z0 2 Z0 − Da − + − + Zin Z0 2 A CZ0 D ⇒ b1 = a1 = Γina1 Z0 − − Z + Z V1 Z0I1 Zin Z0 in 0 b1 = = I1 B 2 Z 2 Z where D = A + + CZ + D. 0 0 a Z 0 0 It follows then from the scattering matrix equations that: 14.4 Input and Output Reflection Coefficients S21 b2 = S21a1 + S22a2 = S22a1 + S22ΓLb2 ⇒ b2 = a1 (14.4.5) When the two-port is connected to a generator and load as in Fig. 14.1.4, the impedance 1 − S22ΓL and scattering matrix equations take the simpler forms:

which implies for b1: = = V1 ZinI1 b1 Γina1 S S Γ (14.4.1) b = S a + S a = S a + S Γ b = S + 12 21 L a = Γ a 1 11 1 12 2 11 1 12 L 2 11 − 1 in 1 V2 = ZLI2 a2 = ΓLb2 1 S22ΓL Reversing the roles of generator and load, we obtain the impedance and reflection where Z is the input impedance at port 1, and Γ , Γ are the reflection coefficients at in in L coefficients from the output side of the two-port: port 1 and at the load:

Z − Z Z − Z Z12Z21 S12S21ΓG = in 0 = L 0 Z = Z − Γ = S + (14.4.6) Γin ,ΓL (14.4.2) out 22 + out 22 − Zin + Z0 ZL + Z0 Z11 ZG 1 S11ΓG The input impedance and input reﬂection coefﬁcient can be expressed in terms of where the Z- and S-parameters, as follows: Zout − Z0 ZG − Z0 Γout = ,ΓG = (14.4.7) Zout + Z0 ZG + Z0 Appendix E

Two-Port Network Parameters

A microwave network with two terminals may be analyzed by considering the network as an equivalent two-port. The signals at the terminals or ports of the two-port may be described in terms of voltages and currents and the interaction between these signals in terms of impedances or admittances. Alternatively, the signals may be described in terms of incident and reﬂected waves and the interaction in terms of scattering.

An arbitrary two-port may be described in several ways. Here we will discuss the two- port in terms of impedance parameters (Z), admittance parameters (Y), chain or ABCD parameters and scattering parameters (S). For the description in terms of Z, Y or ABCD parameters we refer to Figure E.1(a). For the description in terms of S parameters we refer to Figure E.1(b). The relation between the voltages and currents in terms of impedance parameters is given by V Z Z I 1 = 11 12 1 . (E.1) V2 Z21 Z22 I2 The relation between the currents and voltages in terms of admittance parameters is given by I Y Y V 1 = 11 12 1 . (E.2) I2 Y21 Y22 V2

I1 I2 a1 a2

V1 12V2 b1 12b2

(a)(b)

Figure E.1 Two-port deﬁnition. (a) Using voltages and currents. (b) Using incident and reﬂected waves.

Table E.1 Two-port parameter conversion for voltage- and current-based matrices

Z Y ABCD

Y Y 22 − 12 A AD−BC Z11 Z12 |Y | |Y | C C Z Y Y D Z Z − 21 11 1 21 22 |Y | |Y | C C

Z Z 22 − 12 D BC−AD |Z| |Z| Y11 Y12 B B Y Z Z A − 21 11 Y Y − 1 |Z| |Z| 21 22 B B

Z |Z| Y 11 − 22 − 1 Z Z Y Y AB ABCD 21 21 21 21 Z |Y | Y CD 1 22 − − 11 Z21 Z21 Y21 Y21

Table E.2 Two-port parameter conversion for scattering and voltage- and current-based matrices

Z −Z Z +Z −Z Z ( 11 0)( 22 0) 12 21 2Z12Z0 S11 S12 Z Z S S S Z +Z Z −Z −Z Z 21 22 2Z21Z0 ( 11 0)( 22 0) 12 21 Z Z SY

(Y −Y )(Y +Y )+Y Y Y Y 0 11 0 22 12 21 − 2 12 0 S11 S12 Y Y S S S Y Y (Y +Y )(Y −Y )+Y Y 21 22 − 2 21 0 0 11 0 22 12 21 Y Y S ABCD

B A+ Z −CZ0−D 0 2(AD−BC) B B S S A+ Z +CZ0+D A+ Z +CZ0+D 11 12 0 0 S B S21 S22 −A+ Z −CZ0+D 2 0 B B A+ Z +CZ0+D A+ Z +CZ0+D 0 0

The relation between the voltages and currents in terms of chain parameters is given by V AB V 1 = 2 . (E.3) I1 CD −I2 Chain matrices are used when several two-ports are cascaded. The overall chain matrix is obtained by multiplying the chain matrices of the individual two-ports. Appendix E: Two-Port Network Parameters 247

Table E.3 Two-port parameter conversion for scattering and voltage- and current-based matrices

S11 S12 S S21 S22 ( +S )( −S )+S S S Z 1 11 1 22 12 21 Z 2 12 0 (1−S )(1−S )−S S 0 (1−S )(1−S )−S S Z 11 22 12 21 11 22 12 21 2S21 (1−S11)(1+S22)+S12S21 Z0 Z0 (1−S11)(1−S22)−S12S21 (1−S11)(1−S22)−S12S21 ( −S )( +S )+S S S Y 1 11 1 22 12 21 −Y 2 12 0 (1+S )(1+S )−S S 0 (1+S )(1+S )−S S Y 11 22 12 21 11 22 12 21 2S21 (1+S11)(1−S22)+S12S21 −Y0 Y0 (1+S11)(1+S22)−S12S21 (1+S11)(1+S22)−S12S21 ( +S )( −S )+S S ( +S )( +S )−S S 1 11 1 22 12 21 Z 1 11 1 22 12 21 2S 0 2S ABCD 21 21 1 (1−S11)(1−S22)−S12S21 (1−S11)(1+S22)+S12S21 Z0 2S21 2S21

The relation between the incident and reﬂected waves is given by b S S a 1 = 11 12 1 . (E.4) b2 S21 S22 a2 The Z, Y, ABCD and S matrices may be converted into each other. The interrelations between the voltage- and current-based matrices are given in Table E.1. The interrelations between the scattering and voltage- and current-based matrices are given in Tables E.2 and E.3.