Frequency-Domain Models for Nonlinear

Volume 109, Number 4, July-August 2004

Journal of Research of the National Institute of Standards and Technology [J. Res. Natl. Inst. Stand. Technol. 109, 407-427 (2004)]

Frequency-Domain Models for Nonlinear Microwave Devices Based on Large-Signal Measurements

Volume 109

Jeffrey A. Jargon and Donald C. DeGroot National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305 and K. C. Gupta Center for Advanced Manufacturing and Packaging of Microwave, Optical, and Digital Electronics (CAMPmode) University of Colorado at Boulder, Boulder, CO 80309

Number 4

In this paper, we introduce nonlinear largesignal scattering (S) parameters, a new type of frequency-domain mapping that relates incident and reflected signals. We present a general form of nonlinear largesignal S-parameters and show that they reduce to classic S-parameters in the absence of nonlinearities. Nonlinear largesignal impedance (Z) and admittance (Y) parameters are also introduced, and equations relating the different representations are derived. We illustrate how nonlinear large-signal S-parameters can be used as a tool in the design process of a nonlinear circuit, specifically a single-diode 1 GHz frequency-doubler. For the case where a nonlinear model is not readily available, we developed a method of extracting nonlinear large-signal S-parameters obtained with artificial neural network models trained with multiple measurements made

July-August 2004

by a nonlinear vector network analyzer equipped with two sources. Finally, nonlinear large-signal S-parameters are compared to another form of nonlinear mapping, known as nonlinear scattering functions. The nonlinear large-signal S-parameters are shown to be more general.

Key words: frequency-domain; large-signal; measurement; microwave; model; network analyzer; nonlinear; scattering parameter.

Accepted: June 21, 2004

Available online: http://www.nist.gov/jres

[email protected] [email protected]

1.

Introduction

frequency at a time, and detecting the response of the DUT at its signal ports. Since the DUT is linear, the input and output signal frequencies are the same as the source; these signals can be described by complex numbers that account for the signals’ amplitudes and phases. The input-output relationships are described by ratios of complex numbers, known as S-parameters. For a two-port network, four S-parameters completely describe the behavior of a linear DUT when excited by a sine wave at a particular frequency. Although the

Vector network analyzers (VNAs) are one of the most versatile instruments available for RF and microwave measurements. They are used to measure complex scattering parameters (S-parameters) of linear devices or circuits. RF engineers use them to verify their designs, confirm proper performance, and diagnose failures. A VNA works by exciting a linear device under test (DUT) with a series of sine wave signals, one 407

Volume 109, Number 4, July-August 2004

Journal of Research of the National Institute of Standards and Technology measurement of S-parameters by VNAs is invaluable to the microwave designer for modeling and measuring linear circuits, these measurements are oftentimes inadequate for nonlinear circuits operating at large-signal conditions, since nonlinearities transfer energy from the stimulus frequency to products at new frequencies. Thus, conventional linear network analysis, which relies on the assumption of superposition, must be replaced by a more general type of analysis, which we refer to as nonlinear network analysis. Nonlinear network analysis involves characterizing a nonlinear device under realistic, large-signal operating conditions. To do this, complex traveling waves (rather than ratios) are measured at the ports of a DUT not only at the stimulus frequency (or frequencies), but also at other frequencies where energy may be created. Assuming the input signals are sine-waves and the DUT exhibits neither sub-harmonic nor chaotic behavior, the input and output signals will be combinations of sine-wave signals, caused by the nonlinearity of the DUT in conjunction with impedance mismatches between the measuring system and the DUT. If a single excitation frequency is present, new frequency components will appear at harmonics of the excitation frequency, and if multiple excitation frequencies are present, new frequency components will appear at the intermodulation products as well as at harmonics of each of the excitation frequencies. In practice, there will be a limited number of significant harmonics and intermodulation products. The set of frequencies at which energy is present and must be measured is known as the frequency grid. A class of instruments known as nonlinear vector network analyzers (NVNA) are capable of providing accurate waveform vectors by acquiring and correcting the magnitude and phase relationships between the fundamental and harmonic components in the periodic signals [1-5]. An NVNA excites a nonlinear DUT with one or more sine wave signals and detects the response of the DUT at its signal ports. Assuming the DUT does not exhibit any sub-harmonic or chaotic behavior, the input and output signals will be combinations of sine wave signals due to the nonlinearity of the DUT in conjunction with mismatches between the system and the DUT. With these facts in mind, the major difference between a linear VNA and an NVNA is that a VNA measures ratios between input and output waves one frequency at a time while an NVNA measures the actual input and output waves simultaneously over a broad band of frequencies. Even though S-parameters cannot adequately represent nonlinear circuits, some type of parameters relat-

ing incident and reflected signals are beneficial so that the designers can “see” application-specific engineering figures of merit that are similar to what they are accustomed to. In first part of this paper, we propose definitions of such ratios that we refer to as nonlinear large-signal scattering (S) parameters. We also introduce nonlinear large-signal impedance (Z) and admittance (Y) parameters, and present equations relating the different representations. Next, we make two simplifications when considering the cases of a one-port network with a single-tone excitation and a two-port network with a single-tone excitation. For existing nonlinear models, we can readily generate nonlinear large-signal S-parameters by performing a harmonic balance simulation. For devices, with no model available, we can extract these parameters from artificial neural network (ANN) models that are trained with multiple frequency-domain measurements made on a nonlinear DUT with an NVNA. To illustrate applications and generation of nonlinear large-signal Sparameters, we present two examples. First, we illustrate how nonlinear large-signal S-parameters can be used as a tool in the process of designing a simple nonlinear circuit, specifically a single-diode 1 GHz frequency-doubler circuit. And secondly, we describe a method for generating nonlinear large-signal S-parameters based upon ANN models trained on frequencydomain data measured using an NVNA. We compare a diode circuit model, generated using this method, to a harmonic balance simulation of a commercial device model. Finally, we compare our nonlinear large-signal Sparameters to another form of nonlinear mapping, known as nonlinear scattering functions [6-7]. Specifically, we show that the two formulations are not equivalent. Nonlinear large-signal S-parameters are more general than the nonlinear scattering functions, which are useful in approximating a specific class of nonlinearity in a more compact form.

2.

Nonlinear Large-Signal Scattering Parameters

In this section, we introduce the concept of nonlinear large-signal scattering parameters. Like commonly used linear S-parameters, nonlinear large-signal scattering (S) parameters can also be expressed as ratios of incident and reflected wave variables. However, unlike linear S-parameters, nonlinear large-signal S-parameters depend upon the signal magnitude and must account for the harmonic content of the input and out408

Volume 109, Number 4, July-August 2004

Journal of Research of the National Institute of Standards and Technology put signals since energy can be transferred to other frequencies in a nonlinear device. After presenting the general form of nonlinear largesignal S-parameters, we also introduce nonlinear large-signal impedance (Z) and admittance (Y) parameters, and present equations for relating the different representations. Next, we make two simplifications in which we consider the cases of a one-port network with a single-tone excitation and a two-port network with a single-tone excitation.

For each nonlinear large-signal scattering parameter Sijkl the index i refers to the port number of the b wave, the index j refers to the port number of the a wave, k is the harmonic index of the b wave, and l is the harmonic index of the a wave. The vectors a j and bi are (M=3)-element vectors given by

2.1

Equation (3) can be expanded as follows

 a j1   bi1    a j =  a j 2  ; bi =  bi 2  .  a j3   bi 3   

General Form

 b11   S1111 b   S  12   1121  b13   S1131  =  b21   S2111 b22   S2121    b23   S2131

Consider an N-port network. Normalized wave variables ajl and bjl at the jth port and lth harmonic are proportional to the incoming and outgoing waves, respectively, and may be defined in terms of the voltages associated with these waves as follows: a jl =

V jl+ Z oj

; bjl =

Vjl− Zoj

,

(1)

S1112 S1122 S1132 S2112 S2122 S2132

S1113 S1123 S1133 S2113 S2123 S2133

S1211 S1221 S1231 S2211 S2221 S2231

S1212 S1213 S1222 S1223 S1232 S1233 S2212 S2213 S2222 S2223 S2232 S2233

(5)

  a11   a    12   a13    .  a21   a22     a23 

(6) where V +jl and V –jl represent voltages associated with the incoming and outgoing waves in the transmission lines connected to the jth port and containing frequencies of the lth harmonic; Zoj represents the characteristic impedance of the line at the jth port. The nonlinear large-signal scattering matrix S of the network expresses the relationship between a’s and b’s at various ports and harmonics through the matrix equation b = S a,

Note that in each of the four sub-matrices, the diagonal elements contain the same-frequency scattering parameters, the upper right elements contain the frequency down-conversion scattering parameters, and the lower left elements contain the frequency up-conversion scattering parameters. If the device under consideration contains no nonlinearities (i.e., no power is transferred to other frequencies), then Eq. (6) reduces to 0 0 S1211 0 0  b11   S1111 b   0 S 0 0 S1222 0 1122  12    b13   0 0 S1133 0 0 S1233  = 0 0 S2211 0 0  b21   S2111 b22   0 S2122 0 0 S2222 0    0 S2133 0 0 S2233 b23   0

(2)

where b and a are (N × M)-element column vectors. Here N refers to the number of ports and M refers to the number of harmonics being considered. Matrix S is an (N × M)2-element square matrix. We assume all a’s and b’s are phase referenced to a11 to enforce time invariance [8]. As an example, consider a two-port network with 3 harmonics; Eq. (2) then becomes  b1  [S11 ] [ S12 ]   a1   =   , b2  [S21 ] [S22 ]  a2 

which is the matrix representation for the well-known linear S-parameters involving three excitation frequencies.

(3)

2.2

Sij12 Sij 22 Sij 32

Sij13   Sij 23  . Sij 33 

    ,    

(7)

where  Sij11  [Sij ] =  Sij 21  Sij 31 

  a11 a   12   a13    a21   a22    a23

Nonlinear Large-Signal Impedance Parameters

Rather than expressing the relationship between a’s and b’s in terms of a nonlinear large-signal scattering matrix S, we can alternatively express the relationship

(4)

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Volume 109, Number 4, July-August 2004

Journal of Research of the National Institute of Standards and Technology Vik = Vik+ + Vik− = Zoi ( aik + bik ),

between voltages (V’s) and currents (I’s) in terms of a nonlinear large-signal impedance matrix Z, as follows V = Z I,

where the subscripts refer to the ith port and the kth harmonic. We can similarly express Ijl as

(8)

I jl = I +jl + I jl− =

where V and I are (N×M)-element column vectors. Once again N refers to the number of ports and M refers to the number of harmonics being considered. Z is an (N×M)2-element square matrix. For a two-port network with 3 harmonics, Eq. (8) becomes V1  [Z 11 ] [ Z12 ]   I1   =   , V2  [Z 21 ] [ Z 22 ]   I2 

(9)

 Z ij11 Z ij12 Zij13    [Z ij ] =  Z ij 21 Zij 22 Zij 23  .  Z ij 31 Z ij 32 Zij 33   

(10)

[0]   I1  [U ] / Zo1  = U [0] [ ] / Zo 2  I  2 

For each nonlinear large-signal impedance parameter Zijkl, the index i refers to the port number of the voltage V, the index j refers to the port number of the current I, k is the harmonic index of V, and l is the harmonic index of I. The vectors Vi and I j are (M=3)-element vectors given by

Z1112 Z1122 Z1132 Z2112 Z2122 Z2132

Z1113 Z1123 Z1133 Z2113 Z2123 Z 2133

Z1211 Z1221 Z1231 Z2211 Z2221 Z 2231

 V1+  V1−     +  −  −   ,  V2  V2  

(15)

where [U] is the identity matrix. Equation (9) can be expressed as V1+  V1−  [Z 11 ] [ Z12 ]   I1   ++ − =    . V2  V2  [Z 21 ] [ Z 22 ]   I 2 

(16)

Combining Eqs. (15) and (16) gives (11) V1+  V1−   ++ − = V2  V2  [0]   V1+  V1−   [Z 11 ] [ Z12 ]  [U ] / Zo1   −   [Z ] [ Z ]   [0] [U ] / Z o 2   V2+  V2−    21 22  

Equation (9) can be expanded to V11   Z1111 V   Z  12   1121 V13   Z1131  = V21   Z 2111 V22   Z 2121    V23   Z 2131

1 1 Vjl+ − Vjl− ) = ( ajl − bjl ), (14) ( Z oj Z oj

where the subscripts refer to the jth port and at the lth harmonic. For simplicity, we will assume for now that the network under consideration consists of two ports. Later, we can easily generalize the equations relating the S and Z matrices for any N-port network. If we allow the two transmission lines or waveguides connecting the two ports to have different characteristic impedances, Zo1 and Zo2, Eq. (14) can be expressed in matrix form as

where

 I j1  Vi1    Vi = Vi 2  ; I j =  I j 2   I j3  Vi 3   

(13)

Z1212 Z1213 Z1222 Z1223 Z1232 Z1233 Z2212 Z2213 Z2222 Z2223 Z2232 Z2233

  I11   I    12    I13    .  I 21   I 22     I23 

(17) or V1+  V1−  [Z 11′ ] [ Z12′ ]   V1+  V1−    ++ − =  ′    +  −  −   , (18) V2  V2  [Z 21 ] [ Z 22′ ]   V2  V2  

(12)

where 2.3

Relating S and Z Matrices [0]  [Z 11′ ] [ Z12′ ]  [ Z11 ] [ Z12 ]  [U ] / Zo1 [Z ′ ] [ Z ′ ]  = [ Z ] [ Z ]   [0] [U ] / Zo 2  22  22    21  21

The S and Z matrices can be expressed in terms of one another, if we know how a and b relate to V and I. From Eq. (1), we can express Vik in terms of ajl and bik as follows:

(19)

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Journal of Research of the National Institute of Standards and Technology [ Z 11′ ] [ Z12′ ] [ Z ′ ] [ Z ′ ] =  21 22 

is the normalized impedance matrix. Equation (18) can be rewritten as  [Z 11′ ] [ Z12′ ]  [U ] [0]  V1−   +   −  =  [Z 21′ ] [ Z 22′ ]   [0] [U ]   V2   [Z 11′ ] [ Z12′ ]  [U ] [0]  V1+   −   +   [Z 21′ ] [ Z 22′ ]   [0] [U ]   V2 

(20)

and Eq.(3) can be rewritten as [U ] / Z o1  V −  [0]    1−  = [U ] / Z o1  V2   [0]  V +  [0] [S11 ] [ S12 ]  [U ] / Z o1   1+  . [S ] [S ]   [U ] / Zo1  V2   21 22    [0]

  [U ]    [U ] [0]   Z o1   [0] [U ] +     [0]   

  [U ]   Z  [S11 ] [S12 ]  o1 [U ]  [S21 ] [S22 ]    [0] Z o 2  

  [U ]    [U ] [0]   Z o1 −   [0] [U ]     [0]   

  [U ] [0]   Z  [S11 ] [S12 ]  o1 [U ]  [S21 ] [S22 ]    [0] Z o 2  

−1

[0]

−1

   [U ]    Zo 2    [0]

−1

 [0]    . [U ]    Z o 2    (24)

(21) If Zo1 = Zo2, Eq. (24) reduces to [Z 11′ ] [ Z12′ ]   [U ] [0]  [ S11 ] [ S12 ]   [Z ′ ] [ Z ′ ]  =   [0] [U ]  + [ S ] [ S ]     21  21 22  22   

Combining Eqs. (20) and (21) allows us to solve for S in terms of Z:

−1

 [U ] [0]  [S11 ] [ S12 ]      . (25) −   [0] [U ]  [S21 ] [ S22 ]  

[S11 ] [ S12 ]  [S ] [ S ] =  21 22  [U ] / Z o1   [Z 11′ ] [ Z12′ ]  [U ] [0]  −1 [0]    +  [U ] / Z o 2   [Z 21′ ] [ Z 22′ ]  [0] [U ]   [0]

2.4

−1

 [0]  [Z 11′ ] [ Z12′ ]  [U ] [0]   [U ] / Z o1 − .       [U ] / Z o 2   [Z 21′ ] [ Z 22′ ]  [0] [U ]   [0] (22)

Nonlinear Large-Signal Admittance Parameters

We can also express the relationship between voltages (V’s) and currents (I’s) in terms of a nonlinear large-signal admittance matrix Y, as follows I = Y V,

If Zo1 = Zo2, Eq. (22) reduces to −1

[S11 ] [ S12 ]   [ Z11′ ] [ Z12′ ]  [U ] [0]   [S ] [S ]  =  [ Z ′ ] [ Z ′ ]  +  [0] [U ]      21 22  22    21  [Z 11′ ] [ Z12′ ]  [U ] [0]    −  .  [Z 21′ ] [ Z 22′ ]   [0] [U ]  

(26)

where Y is an (N×M)2-element square matrix. For a two-port network with three harmonics, for example, Eq. (26) becomes (23)

Alternatively, we can combine Eqs. (20) and (21) to solve for Z in terms of S:

 I1  [Y11 ] [ Y12 ]   V1    = [Y ] [ Y ]    , 22  V2   I 2   21

(27)

 Y ij11 Yij12 Yij13    [Y ij ] =  Yij 21 Yij 22 Yij 23  .  Y ij 31 Yij 32 Yij 33   

(28)

where

For each nonlinear large-signal admittance parameter Yijkl, the index i refers to the port number of the current I, the index j refers to the port number of the voltage V,

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Volume 109, Number 4, July-August 2004

Journal of Research of the National Institute of Standards and Technology k is the harmonic index of I, and l is the harmonic index of V. The vectors V j and Ii are, once again, (M=3)-element vectors, defined in Eq. (11). Equation (27) can be expanded as follows  I11   Y1111  I  Y  12   1121  I13   Y1131  =  I 21  Y 2111  I 22  Y 2121     I 23  Y 2131

Y1112 Y1122 Y1132 Y2112 Y2122 Y2132

Y1113 Y1123 Y1133 Y2113 Y2123 Y2133

Y1211 Y1221 Y1231 Y2211 Y2221 Y2231

Y1212 Y1222 Y1232 Y2212 Y2222 Y2232

Y1213 Y1223 Y1233 Y2213 Y2223 Y2233

where −1

[0]  [ Y11 ] [ Y12 ]  [Y11′ ] [ Y12′ ]  [U ] / Zo1 [Y ′ ] [ Y ′ ]  =    [U ] / Z o 2  [ Y21 ] [ Y22 ]  22   [0]  21

 V11     V12   V13   .  V21   V    22   V23 

(34) is the normalized admittance matrix. Equation (33) can be rewritten as  [U ] [0]  [Y11′ ]   + ′   [0] [U ]  [Y 21 ]  [U ] [0]  [Y11′ ]   − ′   [0] [U ]  [Y 21 ]

(29) 2.5 Relating S and Y Matrices

[ Y12′ ]   V1−  =  [ Y22′ ]   V2−  [ Y12′ ]   V1+   [ Y22′ ]   V2+ 

(35)

and Eq.(3) can be rewritten as The S and Y matrices can also be expressed in terms of one another, using Eqs. (13) and (14) which show how a and b relate to V and I. For simplicity, we will again assume the network under consideration consists of two ports. If we allow the two transmission lines or waveguides connecting the two ports to have different characteristic impedances Zo1 and Zo2, Eq. (14) can be expressed in matrix form as [0]   V1+  V1−    I1  [U ] / Zo1    −   ,  = [U ] / Z o 2   V2+  V2−    I 2   [0]

[U ] / Z o1  V −  [0]    1−  = [U ] / Z o1  V2   [0]  V +  [0] [S11 ] [ S12 ]  [U ] / Z o1   1+  . [S ] [S ]   [U ] / Zo1  V2   [0]  21 22  

Combining Eqs. (35) and (36) allows us to solve for S in terms of Y : [S11 ] [ S12 ]  [S ] [ S ] = 22   21

(30)

[U ] / Z o1  [0]   [U ] / Z o 2    [0]

where [U] is the identity matrix. Equation (27) can be expressed as  I1  [Y11 ] [ Y12 ]   V1+  V1−     = [Y ] [ Y ]    +  +  −   . 22   V2   I 2   21 V2  

(36)

−1

[U ] [0]  [Y11′ ] [Y12′ ]    [0] [U ] + [Y ′ ] [ Y ′ ]  22      21

−1

  [U ] [0]  [Y11′ ] [Y12′ ]   [U ] / Z o1 [0] −  .       ′ ′ [U ] / Z o 2    [0] [U ] [Y21 ] [ Y22 ]   [0]

(31)

(37)

Combining Eqs. (30) and (31) gives If Zo1 = Zo2, Eq. (37) reduces to: V1+  V1−   +− − = V2  V2 

−1

[S11 ] [ S12 ]   [U ] [0]  [Y11′ ] [ Y12′ ]   [S ] [S ]  =   [0] [U ]  + [Y ′ ] [ Y ′ ]   22     21  21 22  

−1

[0]  [Y11 ] [ Y12 ]   V1+  V1−   [U ] / Zo1     +    [0] [U ] / Z o 2  [Y 21 ] [ Y22 ]   V2+  V2−   

 [U ] [0]  [Y11′ ] [ Y12′ ]     . − ′ ′    [0] [U ]  [Y 21 ] [ Y22 ]  

(32) or

(38)

Alternatively, we can combine Eqs. (35) and (36) to solve for Y in terms of S: V1+  V1−  [Y11′ ] [ Y12′ ]   V1+  V1−    +  −  −  = [Y ′ ] [ Y ′ ]    +  +  −   , 22   V2  V2  V2   21 V2  

(33) 412

Volume 109, Number 4, July-August 2004

Journal of Research of the National Institute of Standards and Technology [Y11′ ] [Y12′ ] [Y ′ ] [Y ′ ] = 22   21   [U ]    [U ] [0]   Z o1 −  [0] [U ]   [0]     

 [U ]   Z o 2 

  [U ]    [U ] [0]   Z o1 +  [0] [U ]   [0]     

  [U ] [0]   Z  [S11 ] [S12 ]  o1 [U ]  [S21 ] [S22 ]    [0] Z o 2  



−1

[0] 

 [U ]  [S11 ] [S12 ]  Zo1 [S ] [S ]   21 22   [0] 

 b11  S1111 S1112 S1113 b  =  S  12   1121 S1122 S1123 b13  S1131 S1132 S1133



[0]  

  [U ]   Zo 2   

−1

  a11  0    0

 ,  

(42)

for a one-port network with a single-tone excitation a11. This matrix can be rewritten as a set of three equations: b11 = S1111 a11 ; b12 = S1121 a11 ; b13 = S1131 a11 .

−1

 [0]    . [U ]    Z o 2    (39)

(43)

Likewise, Eq. (12) reduces to V11   Z1111 Z1112 Z1113 V  =  Z  12   1121 Z1122 Z1123 V13   Z1131 Z1132 Z1133

If Zo1 = Zo2, Eq. (39) reduces to

  I11 I   12   I13

 ,  

(44)

where the voltage V11 at the first harmonic can be expressed as

[Y11′ ] [ Y12′ ]   [U ] [0]  [S11 ] [ S12 ]   [Y ′ ] [ Y ′ ]  =    − 22   21   [0] [U ]  [S21 ] [ S22 ]  

V11 = Z1111 I11 + Z1112 I12 + Z1113 I13 .

(45)

−1

 [U ] [0]  [S11 ] [S12 ]      . (40) +   [0] [U ]  [S21 ] [ S22 ]  

2.6

From Eqs. (13) and (14), we know that V11 = Zo1 (a11 + b11 ),

One-Port Network With Single-Tone Excitation

I11 =

For a one-port network with a single-tone excitation at the fundamental frequency, we can extract a reflection coefficient given by

S11k1 =

(

b1k ∠ φb1k − kφa11 a11

)

I12 = I13 =

. a1m = 0 for all m (m ≠ 1)

1 Z o1 1 Z o1 1 Z o1

(a11 − b11 ), (a12 − b12 ) = −

b12

(a13 − b13 ) = −

b13

Zo1 Zo1

,

(46)

.

Combining Eqs. (45) and (46) gives

(41)

Z o1 (a11 + b11 ) =

The limitation imposed on the equation is that all other incident waves other than a11 equal zero. Instead of simply taking the ratio of b1k to a11, we reference the phase of b1k to that of a11. To do this, we must subtract k times the phase of a11 from b1k [8]. For a one-port network with a single-tone excitation at the fundamental frequency, we can show that the equation relating S and Z reduces to the same wellknown equation for the linear case if we assume that no energy is redistributed into the form of frequency down-conversion. To illustrate this, we will consider only M=3 harmonics, for the sake of simplicity. Equation (6) reduces to

1 Z o1

 Z1111 (a11 − b11 ) − Z1112 b12 − Z1113 b13 .

(47)

Substituting Eq. (43) into Eq. (47) and solving for Z1111 gives

Z1111 =

Z o1 (1 + S1111 ) + Z1112 S1121 + Z1113 S1131

(1 − S1111 )

.

(48)

If no energy is redistributed into the form of frequency down-conversion (i.e., Z1112 = Z1113 = 0), then Eq. (48) reduces to the same equation as in the linear case:

413

Volume 109, Number 4, July-August 2004

Journal of Research of the National Institute of Standards and Technology

Z11 = Zo1

(1 + S11 ) . (1 − S11 )

DUT, one of the equations in the matrix defined by Eq. (6) reduces to

(49)

b2 k = S21k1 a11 + S22 kk a2 k .

A similar derivation can be performed to show that

(1 − S1111 )/ Z o1 − Y1112 S1121 Y1111 = (1 + S1111 )

− Y1113 S1131

If we solve Eq. (54) for S22kk, we obtain . (50)

S22 kk =

Once again, if no energy is transferred to frequency down-conversion (i.e., Y1112 = Y1113 = 0), then Eq. (50) reduces to the same equation as in the linear case: Y11 =

2.7

1 1 (1 − S11 ) = . Z11 Z o1 (1 + S11 )

(51)

b2 k + ∆b2 k = S21k1 a11 + S22 kk ( a2 k + ∆ a2 k ),

S11 k1 =

1k

11

a11

)

∆b2 k = S22 kk ∆a2 k , . amn = 0 for all m, n [(m ≠1)∧ (n ≠1)]

S22 kk =

As with Eq. (41), instead of simply taking the ratio of b1k to a11, we phase reference to a11. To do this we must subtract k times the phase of a11 from b1k. The limitation once again imposed on the equation is that all other incident waves other than a11 equal zero. Another valuable parameter, the forward transmission coefficient, is similarly extracted as follows

(

b2 k ∠ φb − kφa 2k

a11

11

)

(56)

(57)

which does not depend on S21k1. If we solve Eq. (57) for S22kk, we obtain

(52)

S21k1 =

(55)

where ∆a2k