2. MICROWAVE CIRCUITS CHARACTERIZATION Introduction Z-Parameters

Home , Impedance parameters, Port (circuit theory), Scattering parameters

2. MICROWAVE CIRCUITS CHARACTERIZATION . Introduction . Z-Parameters [POZAR 5.3] [COLLIN 4.5] . Y-Parameters [POZAR 5.4] [COLLIN 4.5] . Transmission (ABCD-) Parameters [POZAR 5.5] [COLLIN 4.9] . Interconnection of two-port networks [POZAR 5.6] . S-Parameters [POZAR 5.4] [COLLIN 4.7] • Cons of [Z] and [Y] parameters • Definition • Properties of S-parameters • Pros of [Z] and [Y] parameters • Reference planes shift • Measurement of S-parameters • Conversions between two-port network parameters [POZAR 5.6] . Two-port networks [POZAR 5.6] [COLLIN 4.11] • Reflection coefficients • Power transfer

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 1 EETAC-UPC 2. MICROWAVE CIRCUITS CHARACTERIZATION . S-parameters of some two-port networks • Attenuators and amplifiers •Isolators[POZAR 10.4] • Filters . S-matrix of networks with more than two-ports [POZAR 8.1] • Power dividers [COLLIN 6.6] [POZAR 8.2, 8.3] • Circulators [COLLIN 6.10] [POZAR 10.6] • Directional Couplers [COLLIN 6.4, 6.5] [POZAR 8.5, 8.8]

[COLLIN] R.E. Collin, Foundations for Microwave Engineering, Wiley-Interscience, 2nd Edition, 2001 (New York) [POZAR] D.M. Pozar, Microwave Engineering, Addison-Wesley Publishing Company, 2nd Edition, 1993 (Reading, Massachusetts)

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 2 EETAC-UPC GLOSSARY -1/2 • an : normalized voltage input wave at port n [V ] •[a] : row vector of normalized voltage input waves at network ports [V-1/2] -1/2 • bn : normalized voltage output wave at port n [V ] •[b] : row vector of normalized voltage output waves at network ports [V-1/2] •  : phase constant [rad·m-1] • C : coupling [dB] • D : directivity [dB] • G : power gain [adim]

• GA : available gain [adim] • GT : transducer power gain [adim] • I : isolation [dB] • IL : insertion loss [dB] + • In : incident current phasor at port n [A] - • In : reflected current phasor at port n [A] • In : total current phasor at port n [A] •[I] : row vector of current at ports [A] • l : transmission line length [m] • L : attenuation [dB]

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 3 EETAC-UPC GLOSSARY

• Pavn : power available from a network (assuming conjugate matching) [W] • Pavs : power available from the source (assuming conjugate matching) [W] • Pi : incoming power at work [W] • PIN : power delivered to the input of a network [W] • PL : power delivered to the load [W] • Pincident : incoming power to the port [W] • Preflected : reflected power [W] • Ptransmitted : transmitted power [W] • R : resistance []

• RS : serial resistance [] • RP : shunt resistance [] • RL : return loss [dB]

• G : (voltage) source reflection coefficient [adim] • IN : (voltage) reflection coefficient of network input port [adim] • L : (voltage) load reflection coefficient [adim] • OUT : (voltage) reflection coefficient of network output port [adim] • Sij : scattering parameter between nodes i and j [adim] •[S] : scattering matrix [adim]

• tn : reference plane at port n Radiofrequency Engineering C. Collado, J.M. González-Arbesú 4 EETAC-UPC GLOSSARY • VCO : voltage-controlled oscillator

• VG : voltage at generator [V] + • Vn : incident voltage phasor at port n [V] - • Vn : reflected voltage phasor at port n [V] • Vn : total voltage phasor at port n [V] •[V] : row vector of voltages at ports [V] -1 • Yij : transfer admitance between nodes i and j [S= ] •[Y] : admitance matrix [S]

• Zij : transfer impedance between nodes i and j [] •[Z] : impedance matrix []

• ZG : generator impedance [] • ZIN : impedance at the input port of the network [] • ZL : load impedance [] • Z0 : characteristic impedance [] • Z0,n : characteristic impedance of port n []

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 5 EETAC-UPC INTRODUCTION Introduction • The low frequency analysis of circuits and networks (where dimensions are much lower than wavelength) can be analyzed by the already known Kirchhoff laws of voltage and current together with the impedance concepts of circuit theory. • High frequency regime requires different tools to account for the phase difference existing between one point and another in the circuits and networks. • Usually, the knowledge of the electromagnetics fields and/or currents and voltages in all points inside the network are not required. It uses to be enough knowing voltages or currents at a set of terminals. • A set of parameters will be defined in order to symplify the modelling of networks (or systems) accurately representing its performance.

microstrip black layout box

R R Parameters R /4

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 6 EETAC-UPC INTRODUCTION Introduction • This model is useful for analyzing modifications of the network (design), combination of networks (at system or subsystem level), or analyzing the performance of the network when combined with its neighbors.

• One has to be careful to not oversimplify the model of the circuit because it will lead to erroneous results.

black box

Parameters

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 7 EETAC-UPC INTRODUCTION Introduction • Let’s consider a microwave network (or system) having N terminals, also called ports. • The parameters modelling the network are usually defined at the ports. • These ports can be the physical ports of the network. However if a physical port supports more than one propagating mode, additional ports can be added to account for these modes. • Each port has a specific point called reference plane where parameters are measured. • The reference planes provide phase references for the voltage and current phasors.   V , I   t : reference plane; 3 3 V , I n N N V+ and I+: incident voltage   and current waves; - - V3 , I3   V and I : reflected voltage VN , I N and current waves; V and I: total voltage and t3 current waves: tN   Vn  Vn Vn   V1 , I1   I n  I n  I n     V2 , I 2 V1 , I1 t t2 1   V2 , I2 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 8 EETAC-UPC Z-PARAMETERS Z-Parameters • The impedance matrix [Z] relates the currents and the voltages in the terminals of the network: V  Z  I 

V1   Z11 Z12 ... Z1N   I1  total       total V2 Z21 Z22 ... I2 voltage at        current at each port  ...   ......   ...  each port       VN  Z N1 ...... Z NN  I N  transfer port impedances impedances

. Zij may be complex. . 2N2 degrees of freedom for an arbitrary network. . For a reciprocal network (not containing any nonreciprocal media such as ferrites t or plasmas, or active devices), the [Z] matrix is symmetric: Zij = Zji ([Z]=[Z] ). . If the network is lossless, the Zij elements are purely imaginary.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 9 EETAC-UPC Z-PARAMETERS Z-Parameters

• The total voltage at terminal i is: Vi  Zi1I1  Zi2 I 2 ... ZiN I N i 1,..., N

Vi • Each impedance in the [Z] matrix can be found as: Zij  I j Ik 0  k  j

• To find the impedances of the [Z] matrix port j has to be driven with the current Ij leaving all other ports open-circuited.

• A two-port network can be characterized by a 2x2 impedance matrix:

I1 I 2 V1  Z11I1  Z12 I 2  + + V2  Z21I1  Z22 I 2 V1 [Z] V2 - - V1 V1 V2 V2 Z11  Z12  Z21  Z22  I1 I 2 I1 I 2 I2 0 I1 0 I2 0 I1 0

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 10 EETAC-UPC Z-PARAMETERS Z-Parameters

Example: Lossy network. Find the impedance matrix of the lossy network of the figure.

R1 R2 I1 I 2 + +

V1 R3 V2 - -

V1 V1 Z11   R1  R3 Z12   R3 I1 I 2 I2 0 I1 0 R1  R3 R3  Z     R3 R2  R3  V2 V2 Z21   R3 Z22   R2  R3 I1 I 2 I2 0 I1 0

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 11 EETAC-UPC Y-PARAMETERS Y-Parameters • The admitance matrix [Y] relates the voltages and the currents in the terminals of the network: I  Y  V 

 I1  Y11 Y12 ... Y1N  V1  total       total I2 Y21 Y22 ... V2 current at        voltage at each port  ...   ......   ...  each port       I N  YN1 ...... YNN  VN  transfer port admitances admitances

. Yij may be complex. . 2N2 degrees of freedom for an arbitrary network. . For a reciprocal network (not containing any nonreciprocal media such as ferrites t or plasmas, or active devices), the [Y] matrix is symmetric: Yij = Yji ([Y]=[Y] ). . If the network is lossless, the Yij elements are purely imaginary.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 12 EETAC-UPC Y-PARAMETERS Y-Parameters

• The total current at terminal i is: Ii  Yi1V1 Yi2V2 ...YiNVN i 1,..., N

Ii • Each impedance in the [Y] matrix can be found as: Yij  V j Vk 0  k  j

• To find the impedances of the [Y] matrix port j has to be driven with the voltage Vj leaving all other ports short-circuited.

• Impedance and admitance matrices are the inverses of each other: Y  Z 1

• A two-port network can also be characterized by a 2x2 impedance matrix:

I1 I 2 I1  Y11V1 Y12V2  + + I 2  Y21V1 Y22V2 V1 [Y] V2 - - I1 I1 I 2 I 2 Y11  Y12  Y21  Y22  V1 V2 V1 V2 V2 0 V1 0 V2 0 V1 0

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 13 EETAC-UPC Y-PARAMETERS Y-Parameters

Example: Lossy network. According to the definition, find the admitance matrix of the lossy network of the figure. Assess the result by applying the fact that the Y-matrix is the inverse of the Z-matrix. R1 R2 I1 I 2 + +

V1 R3 V2 - -

I1 1 R2  R3 Y11    V1 R1  R2 // R3 R1R2  R1R3  R2 R3 V2 0

I2 1 R1  R3 Y22     R2  R3  R3  V2 R2  R1 // R3 R1R2  R2 R3  R1R3   V1 0 R R  R R  R R R R  R R  R R Y   1 2 1 3 2 3 1 2 2 3 1 3   R R  R I  I R R  R R  3 1 3  Y  2  1 3 2 3   3   21  R1R2  R1R3  R2 R3 R1R2  R2 R3  R1R3  V1 I1 R1  R2 // R3 R1R2  R1R3  R2 R3 V2 0

I1  I2 R3 R1  R3 R3 Y12     V2 I2 R2  R1 // R3 R1R2  R2 R3  R1R3 V1 0

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 14 EETAC-UPC TRANSMISSION (ABCD-) PARAMETERS Transmission (ABCD-) Parameters • Many two-port networks consist of a cascade of two or more two-port networks.

• These networks can be analyzed by defining the transmission parameters (or ABCD matrix) of each 2x2 network. output current I1 I 2 + A B + V1  A B V2  V1 V2  C D       -   - I1  C D I2 

V V I I • Parameters are found as: A  1 B  1 C  1 D  1 V2 I 2 V2 I2 I2 0 V2 0 I2 0 V2 0 • Parameters A and C are found by open-circuiting port 2 whereas parameters B and D are found by short-circuiting port 2.

• These parameters are useful to make libraries of parameters in order to design complex systems by cascading components.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 15 EETAC-UPC TRANSMISSION (ABCD-) PARAMETERS Transmission (ABCD-) Parameters • Using the transmission parameters a cascade of two two-port networks can be analyzed as follows: I1 I 2 I3

+ A B  + A B  + V 1 1 V 2 2 V 1 C D  2 C D  3 -  1 1  -  2 2  -

V1  A1 B1  V2  V2  A2 B2  V3                I1  C1 D1  I2  I2  C2 D2  I3 

V1  A1 B1  A2 B2  V3           I1  C1 D1  C2 D2  I3 

ABCD ABCD1  ABCD2 

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 16 EETAC-UPC TRANSMISSION (ABCD-) PARAMETERS Transmission (ABCD-) Parameters • Transmission parameters of some two-port circuits. [Image from: [POZAR]] [Image

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 17 EETAC-UPC INTERCONNECTION OF TWO-PORT NETWORKS Series and parallel connections • The matrix parameters of a two-port network resulting from the series or parallel connection of two-port networks can be found from, respectively, the impedance and admitance matrices of the networks.

+ + + + I1 a I 2 I1 a I 2 Z  V1 Y  V2 - -

V1 V2

Z b  Y b  - -

Z a  Z b Z a  Z b  Y a Y b Y a Y b  a b 11 11 12 12 a b 11 11 12 12 Z  Z  Z   a b a b  Y  Y  Y   a b a b  Z21  Z21 Z22  Z22  Y21 Y21 Y22 Y22 

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 18 EETAC-UPC S-PARAMETERS Cons of [Z] and [Y] parameters • Using impedance or admitance matrices is inconvenient when trying to measure these parameters at high frequencies. . In practice is rather difficult to design good open-circuits and good short-circuits. . By using open circuits or short circuits as terminals to measure the admitance and impedance parameters of a networks provokes the existance of reflected waves. Certain (active) networks can oscillate due to these waves. . To characterize the parameters is required to access the ports by using sections of transmission lines. Measurements are dependent from the length of these sections.

Using the scattering parameters is very easy to shift a reference plane tn.

ideal impedance simulated (considering radiation) input impedance (deembedded at Open-circuited Microstrip Line simulated short location) Short-circuited Microstrip Line (considering radiation) input impedance (deembedded at ideal impedance open location)

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 19 EETAC-UPC S-PARAMETERS Definition • A better representation in accordance with the existence of incident, reflected and transmitted waves involves using the scattering matrix. • This matrix is defined by measuring incident and reflected voltage normalized waves at each port.  tn: reference plane; V3  + - VN V and V : incident and reflected voltage waves;  a : incident voltage V3 n  normalized waves; VN bn : reflected voltage t3 normalized waves: tN Zo,n: reference impedance of  port: each port can have its V1 own impedance.

   V V V n n 2 an  bn   t1 Z Z t2 V1 0,n 0,n

 V2 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 20 EETAC-UPC S-PARAMETERS Definition • The scattering matrix is given by the relation between incoming and outcoming waves: reflected normalized incident normalized voltage at each port voltage at each port

 b1   S11 S12 ... S1N   a1        b2 S21 S22 ... a2 b b  S a        S  i       ij ...... a j       ak 0  k  j bN  S N1 ...... S NN  aN  transmission reflection coefficients coefficients • Matrix port j has to be driven with a voltage source whereas all the ports except the j port should be terminated in matched loads to avoid reflections. . The units of the S parameters are adimensional. . S parameters can be provided in dB: S  20log S ij dB ij • The incoming and outcoming 2 2 2 V  2 V  powers to port n are:  an n  bn n Pn   Pn   2 2Z0,n 2 2Z0,n Radiofrequency Engineering C. Collado, J.M. González-Arbesú 21 EETAC-UPC S-PARAMETERS Properties of S-parameters • If the network is reciprocal: S St

• If the network is passive (PincidentPreflected and PincidentPtransmitted other ports): Sij 1

• If the network is passive and lossless (Pincident=Preflected + Ptransmitted other ports) t * the scattering matrix is unitary: SS U P • It a two-port network is symmetrical Z incident,i (in general, changing the numbering 0,n Preflected,i of the ports means no change in the matrix): S  S S  S 11 22 21 12 Ptransmitted,n P • It a two-port network is unilateral transmitted,j (waves flow only in one direction): Z0,j S12  0

Z0,k Ptransmitted,k

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 22 EETAC-UPC S-PARAMETERS Properties of S-parameters

• For a two-port network the properties of a symmetric, passive and lossless network translate into the following equations:

passive and lossless symmetry S t S*  U  S S S11  S22 * * S11 S12 S11 S21  1 0 S21  S12   * *     S21 S22 S12 S22  0 1 S11  S22  k

2 2 2 S21  S12   1 k S11  S12 1 2 2 S21  S22 1 * * S21S11  S22S12  0 * * S11S21 S12S22  0

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 23 EETAC-UPC S-PARAMETERS Properties of S-parameters

Example: Resistor. Find the scattering matrix of the lossy network of the figure. a a 1 R 2

b1 b2

Terminating port 2 with its matched load: R a1

b1 R  Z0  Z0 R S11    a1 R  Z0  Z0 R  2Z0 a2 0 R   R    R  b b Z     V 1   S V 1  1 2 0  V1 V1  V1 V1 1   11 1   b2 V2 Z0  Z0   Z0  2Z0 S21         a1 V1 V1 V1 R  2Z0 a2 0 a2 0 a 0 2 a 0 Given the symmetry of the network:  R 2Z 2  0 does not  R  2Z R  2Z  S   0 0  S22  S11 S21  S12 correspond with  2Z0 R    voltage divider  R  2Z0 R  2Z0 

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 24 EETAC-UPC S-PARAMETERS Properties of S-parameters

Example: Transmission line. Find the scattering matrix of the transmission line of the figure.

a1 a2

Z0 , 

b1 b2 l

Terminating port 2 with its matched load:

b1 S11   0 a1 a2 0   j l  0 e      j l S    jl  b2 V2 V1 e  jl e 0 S21       e   a1 V1 V1 a2 0

Given the symmetry of the network: S22  S11 S21  S12

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 25 EETAC-UPC Take your time…

Example: Half-wavelength transmission line. Find the scattering matrix of the transmission line

of the figure having a characteristic impedance Z’0 and a length of /2. Reference impedance is Z0.

a1 a2

Z’0 , 

b1 b2 /2

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 26 EETAC-UPC Take your time…

Solution: Half-wavelength transmission line. By terminating port 2 with a matched load Z we get: 0 a1   a1 V3 V4 b2 b1 Z   0 b1 V3 V4 Z0 Z’0 ,  Z0 Z0

  0  l=/2  IN ' 2 jl 2 j l IN  Le IN  Le b    jl 1 V  V e S11   0 4 3 a1 a2 0   '   0 1 b V4 V4 Z0 V 1 S  2   4 L  1 S    21   ' 1 0 a1 V V Z   1    a2 0 3 3 0   V3  1  IN  Given the symmetry of the network: S22  S11 S21  S12 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 27 EETAC-UPC S-PARAMETERS Pros of [Z] and [Y] parameters • The advantages of characterizing microwave networks using the scattering parameters are: . Fabricating a good matched load is easiest than to fabricate an open-circuit or a short-circuit. . By loading with a matched load there are no reflected waves on the terminated ports. Networks are characterized in the same Z environmental conditions they will 0,n operate. There is no risk that t ' n tn active circuits oscillate during their characterization. ln . Shifting the reference planes when characterizing the network does not affect the measurement. Matched ports still remain matched.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 28 EETAC-UPC S-PARAMETERS Reference planes shift • If a reference plane is moved from its original location (which is common when characterizing microwave networks) the waves’ phase references also change.

 jln ' an  e an an bn ' ' an bn '  jln bn  e bn

' 2 jln Snn  e Snn t ' n tn 2 jln ' Snn  e Snn ln • The phase in the terminal n is shifted by twice the electrical length of the shift in

terminal plane tn.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 29 EETAC-UPC  S-PARAMETERS Reference planes shift  • Generalizing the reference plane shift to the N-ports of the network.  b Sa

   e j l1 0 ... 0  e j l1 0 ... 0      0 e j l2 ... 0 0 e j l2 ... 0   b'  S  a'  ......   ......       0 0 ... e j lN   0 0 ... e jlN         j l  j l e 1 0 ... 0  e 1 0 ... 0      0 e j l2 ... 0 0 e j l2 ... 0  b'    S  a'  ......   ......     j l    jl   0 0 ... e N   0 0 ... e N 

  j l1  j l1 e 0 ... 0  e 0 ... 0  Note:   j l2    j l2  ' 0 e ... 0 0 e ... 0 ' 2 jln S    S   Snn  e Snn  ......   ......  '  j ln lm    j l    jl  Snm  e Snn  0 0 ... e N   0 0 ... e N 

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 30 EETAC-UPC S-PARAMETERS Reference planes shift

Example: Resistor. Find the scattering matrix of the lossy network of the figure. a a 1 R 2

 b1 60º 60º b2



  R 2Z0    j    j  3   3 e 0  R  2Z0 R  2Z0 e 0  S       j  2Z R   j  0 e 3  0  0 e 3        R  2Z0 R  2Z0 

2  j 3 e  R 2Z0  S    R  2Z0 2Z0 R 

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 31 EETAC-UPC S-PARAMETERS Reference planes shift

Example: Characterizing a SMD resistor. Find the scattering matrix of a SMD resistor using microstrip technology. SMD substrate connector

microstrip lines

l1 l2

L=1 L=1

Previously to mounting the SMD, S11 and S22 of the network with the open-circuited microstrip lines should be

measured using a vector network analyzer: S11, oc and S22, oc S11,oc S22,oc Assuming that there is no coupling between the  S 0  e2 j l1 0  ports, measured two port matrix should be [S]open: S  11,oc  open    2 jl 2   0 S22,oc   0 e  Using a vector network analyzer the [S] matrix of 1/ 2 1/ 2 the network including the SMD is measured S11,oc 0  S22,oc 0  S measured   1/ 2  S transistor   1/ 2  between the connectors: [S]measured .  0 S11,oc   0 S22,oc  Then, the transistor scattering S 1 2 0  S 1 2 0  parameters can be extracted from the 11,oc 22,oc S transistor   1 2  S measured  1 2  measured scattering parameters:  0 S11,oc   0 S22,oc  Radiofrequency Engineering C. Collado, J.M. González-Arbesú 32 EETAC-UPC S-PARAMETERS Measurement of S-parameters • The magnitude of the S parameters can be found by terminating the ports of the network and measuring incoming, reflected, and transmitted powers. 2 V  a 2 P P  i  i Z incident,i . Powers at ports: incident,i 0,n 2Z0i 2 Preflected,i 2 V  b 2 P  i  i reflected,i 2Z 2 P 0i transmitted,n 2 V  2 Ptransmitted,j j bi Ptransmitted,j   2Z0 j 2 Z0,j . By measuring the ratio between the reflected and the incoming power at each port the matching of each port is found: 2 Z0,k P b Ptransmitted,k reflected,i i 2   Sii P a 2 . By measuring the ratio between the incident,i i 2 transmitted power at port j with respect Ptransmitted,j bj 2 to the incoming power at port i, the   S P 2 ji scattering parameter ji can be found: incident,i ai

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 33 EETAC-UPC S-PARAMETERS Measurement of S-parameters • Some set-ups used to measure the magnitude of a two-port network (DUT, Device Under Test).

. To measure the magnitude of S21: a RF signal generator and a spectrum analyzer are required. The equipment (and transmission lines used) have to be calibrated previously. CALIBRATION MEASUREMENT Spectrum Analyzer Spectrum Analyzer RF Signal Generator RF Signal Generator

P Pincident Ptransmitted incident DUT

. To measure the magnitude of S11 a circulator is required. The equipment used in the measurement also has to be calibrated previously. Spectrum Analyzer CALIBRATION Spectrum Analyzer MEASUREMENT RF Signal Generator

Preflected RF Signal Generator DUT Pincident P Radiofrequency Engineering incident C. Collado, J.M. González-Arbesú 34 EETAC-UPC S-PARAMETERS Measurement of S-parameters • To measure magnitude and phase of a network, a vector network analyzer is used:

. Previous to the measurement, the equipment (and transmission lines Vector Network Analyzer used) has to be calibrated...

CALIBRATION

port #2

control E-Cal (electronic port #1 calibration kit)

. ... and then the network can be Vector Network Analyzer measured applying the calibration. MEASUREMENT

port #2

port #1 DUT

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 35 EETAC-UPC S-PARAMETERS Conversions between two-port network parameters • The following table summarizes the conversions between two-port network parameters. [Image from: [POZAR]] [Image

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 36 EETAC-UPC TWO-PORT NETWORKS Reflection coefficients • Reflection coefficients of an arbitrary two-port network can be calculated from the knowledge of the scattering parameters of the network. Z a a G 1 2 Z  Z   L 0 b b L 1 2 Z L  Z0 VG [S] ZL ZG  Z0 G  ZG  Z0

G  IN OUT  L

b1 a2 Similarly: IN   S11  S12 a1 a1 S12S21L S12S21G IN  S11  OUT  S22  1 b2 a1 1 S22 L 1 S11G   S21  S22 L a2 a2

. Return losses: RL  20 log IN

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 37 EETAC-UPC TWO-PORT NETWORKS Power transfer •Also, power transfer characteristics of an arbitrary two-port network can be calculated from the knowledge of the scattering parameters of the network.

ZG a1 a2

b1 b2 VG [S] ZL

G  IN OUT  L . Power gain: is the ratio of the power dissipated in the load PL ZL G  to the power delivered to the input of the network. PIN P . Available gain: is the ratio of the power available from the G  avn two-port network to the power available from the source. A Pavs . Transducer power gain: is the ratio of the power dissipated P G  L in the load ZL to the power available from the source. T Pavs Radiofrequency Engineering C. Collado, J.M. González-Arbesú 38 EETAC-UPC TWO-PORT NETWORKS Power transfer • Power delivered to the input: ZG a1 a2 Z V  V   V  1   V IN b1 b2 1 1 1 IN G Z ZG  Z IN  VG 1 G VG [S] L V1  1  2 1   Z  Z IN  G IN IN 0 1  IN  G  IN OUT  L  2 2 2 1 2 2 V1 2 V 1 2 P   V   V    1  G G 1  IN 2Z  1 1  2Z IN 8Z 2 IN 0 0 0 1 G IN       V2 V1 V2 V1 V2 L   S21 • Power delivered to the load:  S21  S22  S21  S22 V2  V1 Z Z Z Z Z  1 S  0 0 0 0 0  22 L  2 2 2 2 1 2 2 V2 2 V S 1  2 P   V   V    1  G 21 G  1  L 2Z  2 2  2Z L 8Z 2 2 L 0 0 0 1 S22 L 1 G IN 2  2 • Power available from the source (conjugate matching VG 1 G  Pavs  PIN *  2 between source and network input impedance): IN G 8Z 0 1 G 2 2  2 • Power available from the network (conjugate matching VG S21 1 G    Pavn  PL *  2 2 between network output impedance and load): L OUT 8Z 0 1 S11 G 1 OUT  2 . Where the following ratio has been 1 S 2 1 2  1 2  11 G  OUT previously calculated: G IN  * 2 L OUT * 1 S22 OUT Radiofrequency Engineering C. Collado, J.M. González-Arbesú 39 EETAC-UPC TWO-PORT NETWORKS Power transfer

ZG a1 a2

b1 b2

VG [S] ZL

G  IN OUT  L

2 2 P S 1 independent of ZG . Power gain: G  L  21  L P 2 2 IN 1 S22 L 1 IN

2 2 independent of P S 1 ZL . Available gain: G  avn  21 G (well... in fact, it is assumed that A P 2 2 the load is matched to the port) avs 1 S11 G 1 OUT  2  2  2  dependent PL S21 1 G 1 L . Transducer power gain: G   of ZL and ZG T P 2  2 avs 1 S22 L 1 GIN

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 40 EETAC-UPC TWO-PORT NETWORKS Power transfer

ZG a1 a2

b1 b2

VG [S] ZL

G  IN OUT  L • Gain units are adimensional or decibels (10 log).

• Specific situations for the transducer power gain GT:  2 2 1 L  . Matched generator (G=0): GT  S21 2 1 S22 L

2 . Matched generator and load (G=0 and L=0): GT  S21  2 2 1 G  1 L  . Short transmission line (S =S =1 and S =S =0): G  21 12 11 22 T  2 1 G L Radiofrequency Engineering C. Collado, J.M. González-Arbesú 41 EETAC-UPC S-PARAMETERS OF SOME TWO-PORT NETWORKS Attenuators and amplifiers • The purpose of an attenuator is to reduce the power of a signal, whereas the purpose of a power amplifier is to increase significantly its power. • The ideal scattering matrix of attenuators and amplifiers is like the following: matched ports unidirectional performance 0 0 [S]  k 0 attenuation (01)

G L

. The network is neither reciprocal nor symmetric. . The scattering matrix is not unitary: network is not passive (amplifier) or is not lossless (attenuator). . Amplifier: G=k [adim] or G=20 log k [dB] . Insertion losses: L=k [adim] or L=-20 log k [dB]

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 42 EETAC-UPC S-PARAMETERS OF SOME TWO-PORT NETWORKS Attenuators and amplifiers • Important parameters for attenuators are: frequency range, attenuation accuracy, attenuation variation versus frequency, input and output matching, power-handling capability, and phase linearity.

• Some attenuators:

 R 2Z0  6 dB attenuator: R  2Z0 R  R  2Z R  2Z  S   0 0  1 1 2Z R S21  S11   0  2 2    R  2Z0 R  2Z0  mismatched attenuator

Matching condition:  Z  R  0 0 S RS RS 2 2   Z0  RS Z0  RS RP  S    RP 2R Z  R S  0 S 0    Z0  RS  Attenuation [dB]=20 log |(Z0+RS)/(Z0-RS)|

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 43 EETAC-UPC S-PARAMETERS OF SOME TWO-PORT NETWORKS Attenuators and amplifiers

Matching condition:  R  Z  0 P 0 RS 2   2RP Z0 RP  Z0 RS  2 2 S    RP RP R  Z R  Z P 0  P 0 0     RP  Z0  Attenuation [dB]=20 log |(RP+Z0)/(RP-Z0)|

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 44 EETAC-UPC S-PARAMETERS OF SOME TWO-PORT NETWORKS Attenuators and amplifiers

Example: Mismatched attenuator. Compute the transducter gain of a 50 -attenuator when loaded with an impedance of 75 . Calculate the reflection coefficient of the load, and the reflection coefficient at the input port of the attenuador when loaded. 50 

75  VG

L=10 dB

. Given the structure of the attenuator considered  0 0.316 S    (matched ports): 0.316 0  . The transducer gain in case of a matched load after 2 the attenuator is (S22=G=L=0): GT  S21  0.1  10 dB

. The transducer gain in case of a mismatched 2 2 GT  S21 1 L  0.096  10.2 dB load after the attenuator is (S22=G=0): 75  50 . Reflection coefficient of the load:    0.2 L 75  50 . Reflection coefficient at the input port: IN  S12S21L  0.02

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 45 EETAC-UPC S-PARAMETERS OF SOME TWO-PORT NETWORKS Isolators • An isolator is a two-port network that transmits power in one direction only. It prevents the input port impedance depending from the output port impedance.

Frequency Matching to VCO G depends on IN reference impedance can not be good L=0

 IN  L • The ideal scattering matrix of an isolator is:

matched ports unidirectional performance 0 0 S12S21L [S]    IN  S11   0 neither attenuation 1 0 1 S22 L nor gain . The network is non-reciprocal, non-symmetric, and non-unitary.

. Actual devices are characterized by insertion losses (|S21|<1), matching at input and output ports (|S11|>0 and |S22|>0), and isolation (|S12|>0). . Isolation [dB]: I=-20 log |S12| . Insertion losses [dB]: IL=-20 log |S21| Radiofrequency Engineering C. Collado, J.M. González-Arbesú 46 EETAC-UPC S-PARAMETERS OF SOME TWO-PORT NETWORKS Filters • Filters are two-port networks designed to reject or select frequency bands among the radiofrequency spectrum. • The ideal scattering matrix of a filter intentionally changes with frequency. PASS-BAND REJECTED-BAND

matched ports unmatched ports bidirectional 0 1 performance 1 0 [S]  [S]  1 0 0 1 neither attenuation   total   nor gain attenuation ] . The S-matrix is reciprocal and symmetric, and unitary.

. Actual devices have insertion losses (|S21|<1) and are not ideally matched (|S11|>0) in the pass-band. [Image from: http://www.bscfilters.com/

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 47 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Power dividers • Power dividers (or power splitters) couple a defined quantity of power from the input port to the outer ports. When used in reverse mode they act as a power combiners.

P2=  P1 P2 P1 P1=P2+P3 P3= (1-) P1 P3

• Power dividers are 3- or 4-ports networks. • It is impossible to construct a 3-port lossless reciprocal network matched to all ports. • Directional couplers and hybrid junctions can also be used as passive microwave dividers/combiners.

• Some power dividers follow: . T-junction power dividers: it is a simple (lossless) power divider that cannot be matched simultaneously at all ports, outputs are not isolated. . Wilkinson power dividers: it is a lossy power divider with matched ports, isolation between output ports is achieved.

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 48 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Power dividers • T-junction power divider. . It is the junction of three transmission lines. . The impedances of the output transmission lines can be matched to provide various power division ratios (asymmetric dividers). . To help matching the impedances seen from the output ports quarter-wave transformers can be made. /4 Z 0 2 Z Z0 . Lossless divider: it is a Z port 2 0 0 2Z0 port 2 Z lossless power divider (IL=3 dB) 0 Z0 port 1 port 1 /4 Z mismatched 0 2Z0 with mismatched output ports Z0 Z port 3 2 Z0 0 and matched input port. input port 3 /4 Z  0  j 2  j 2 0 2 Z port 2   0 Z [S]   j 2 1 2 1 2  port 1 0 2 Z   0 Z  j 2 1 2 1 2 matched 0 port 3   input /4

. Resistive divider: it is a lossy 0 1 2 1 2 Z 3 Z   0 0 port 2 Z0 3 (IL=6 dB) power divider but with   Z [S]  1 2 0 1 2 port 1 0   Z 3 matched ports. Unequal power 0 Z 1 2 1 2 0  0 port 3 divider ratios are also possible.   Radiofrequency Engineering C. Collado, J.M. González-Arbesú 49 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Power dividers • Wilkinson power divider. . It is a lossy power divider with insertion losses of 3 dB (IL=3 dB). . The input impedance of its ports are simultaneously matched. . Output ports are isolated. . By changing the characteristic impedances of the output transmission lines the ratio of the power outputs can be changed. . The scattering matrix of this structure can be calculated by and means of an even-odd analysis.

/4 Z0 port 2  0  j 2  j 2 2 Z   0 [S]   j 2 0 0 Z0 2Z0   port 1  j 2 0 0    2 Z0 Z 0 port 3 isolation /4

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 50 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Power dividers

Example: T-junction lossless power divider. Find the scattering matrix of the power divider of the figure. /4 Z 0 port 2 2 Z0 Z port 1 0 2 Z 0 Z 0 port 3 /4

S11 S12 S12  . Given the symmetry of the structure: S22= S33 and S21= S31 S  S S S  . To fully characterize the scattering matrix of the network S11,  12 22 23  S22, S21 and S23 have to be calculated. S12 S23 S22 

. S11 is computed by terminating ports 2 and 3: . S22 is computed by terminating ports 1 and 3: 2  2Z  /4 port 2 2 /4 0 Z ZC   2Z0 ZD Z D  ZC // 2Z0  Z0 port 1 0 port 1 Z0 port 2 2 3 2 Z0 2 Z0  2Z0  Z0 Z0 Z Z0 Z IN  ZC // ZC  Z0 0 ZC   3Z0 Z D 2 Z0 2 Z0 Z0 2Z0 2Z0 Z0 Z0 2Z0 Z 3Z0  Z0 1 C S  0 ZC S   /4 11 /4 22 3Z0  Z0 2 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 51 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Power dividers

. S21 is computed by terminating ports 2 and 3:  /4  V2 V4 Z0 V   port 2 3 V4 2 Z0 /4   Z V1 V3 Z port 1 0 0 2 Z Z0 2 Z0 0 port 1 Z Z 2Z 0 0 0  2 Z 0 2 Z Z 0 Z port 1 0 port 3 0 ’ ZC /4 /4 /4    Z  2Z 1 2 j2  1 2   0 0  '  e 4     . To calculate S is easier to 23 Z0  2Z0 1 2  1 2 consider that the network is    j  passive and lossless: b V  V  V  V  1 V e 4 1 j S  2  2  4 4  4  3    21     '  t * a1 V1 V3 V3 V3 1 V3 1  2 S S U a2 a3 0   j  j  0 S * S  S * S  S * S  0   13 11 23 21 33 31  2 2   j 1 1 * . Finally: S      S * S  1  2 2 S   33 31   S   2 23  S  33 2  j 1 1   21      2 2 2 

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 52 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Power dividers

Example: T-junction lossless asymmetric power divider. Find the characteristic impedances of the lines to get an asymetric power division between the two branches of the divider. Assume that the ratio between the output power of the lower branch with respect to the upper one is P /P =N. 3 2 /4 Z 0 port 2 Z0,2 Z port 1 0 Z 0,3 Z 0 port 3 /4

 2  2  2  2 V1 V1 Z V1 V1 Z . Power at output ports: 0 0 P2   2 P3   2 2Z IN ,2 2 Z0,2 2Z IN ,3 2 Z0,3 P  P  P  1 N P . Considering that the input port is 1 2 3 2 2 2 Z  1 N Z matched and that the divider is lossless: V  V  0,2 0 1 1 Z0  1 N 2 2Z0 2 Z0,2 . Given the ratio between the 2 P3 Z0,2 1 1 N power at the output ports:  2  N Z0,3  Z0,2  Z0 P2 Z0,3 N N Radiofrequency Engineering C. Collado, J.M. González-Arbesú 53 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Circulators • A circulator is a passive-nonreciprocal lossless three-port device matched to all ports.

• The ideal scattering matrix of a circulator is:

0 0 1 port 2 0 1 0 port 2   port 1   port 1 [S]  1 0 0 [S]  0 0 1 0 1 0 port 3 1 0 0 port 3

• Assuming circular symmetry, a lossless mismatched circulator has a scattering matrix like the following:  isolation  return  losses insertion port 2      losses port 1    2 2 2 [S]      1    port 3   

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 54 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Circulators

Protection of High Power Amplifiers • Circulators can be used as isolators by Z0 terminating one of the ports with a matched load. G

• Circulators can also be used as duplexer routing signals from a transmitter to an antenna and from the antenna to the receiver, not allowing signals to pass directly from transmitter to receiver. ,

Block diagrams of a monostatic radar Radar Handbook

Isolation: Warning

[Image from: [POZAR]] [Image from: M.I. [Image Skolnik, McGraw-Hill, New 1990] Edition, York, 2nd Radiofrequency Engineering C. Collado, J.M. González-Arbesú 55 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Couplers

• Directional couplers are microwave devices that can be used for power division and power combining. • Directional couplers may be three or four-port networks. • Couplers are reciprocal, lossless and passive. • Directional couplers can have arbitrary power ratios, whereas hybrid junctions are designed to have equal power division.

port 1: input port 3: through Directional Coupler

port 2: isolated port 4: coupled

port 1: input port 3: through Hybrid Junction port 2: isolated port 4: coupled

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 56 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Couplers

• Directional couplers. Scattering port 1: input port 3: through matrices of ideal couplers look like the following. port 2: isolated port 4: coupled

0 0   0 0 j        0 0   0 0 j [S]   [S]       0  0   j 0  0  Antisymmetrical   Symmetrical    coupler   0 0  coupler  j  0 0  . Lossless network: | |2 +| |2 = 1 . The parameters that characterize directional couplers are coupling, isolation, return losses, and directivity.

. Return losses [dB]: RL = -20 log |S11| . Coupling [dB]: C = -20 log |S | 14 I  C  D dB . Isolation [dB]: I = -20 log |S12| . Directivity [dB]: D = 20 log |S14/S12 |

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 57 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Couplers

port 1: input port 3: through • Hybrid junctions. Scattering matrices of hybrid junctions look like the following (directional couplers having  port 2: isolated port 4: coupled equall power division:    1 2 ).

0 0 1 1  0 0 1 j   1 0 0 1 1   [S]    1 0 0 j 1 2 1 1 0 0  [S]   180º Hybrid   Quadrature 2 1 j 0 0 (antisymmetrical 1 1 0 0  (90º) coupler   coupler) (symmetrical  j 1 0 0 0 coupler) 0 0 0 90 90 180 0

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 58 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Couplers • Some applications of couplers follow: . Measurement of reflection coefficients.

0 0 0 DUT   DUT 0 0 0 0 0 0 180 180180 180 Because directivity Preflected Pincident is not infinite, it’s Pincident Preflected better using two couplers.

. Adding and substracting signals . Stabilizing amplifiers. (e.g. tracking radars).

0 V 0 0 A G 180 VB

V=VA+VB V=VA-VB

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 59 EETAC-UPC S-MATRIX OF SOME NETWORKS WITH MORE THAN TWO-PORTS Couplers

0 . Baluns: useful to connect balanced to 0 0 unbalanced (or viceversa) networks. 180 V V V e j180º e j0º 2 2 . Analyzing the spectrum of high power signals.

Spectrum Analyzer

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 60 EETAC-UPC Take your time… Example: Isolator. Analyze the isolator from the brochure provided by the manufacturer. ] [Images from: [Images http://www.fairviewmicrowave.com

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 61 EETAC-UPC Take your time…

Solution: Isolator.

Insertion loss: IL  1 dB  |S12|0.89 [adim] Isolation: I  14 dB  |S21|0.2 [adim] Matching: VSWR: 1.5:1 (worst case)  ||0.2 [adim]  ||-14 dB

IL  1dB -14 dB -14 dB   | |

 I  14 dB  | |

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 62 EETAC-UPC Take your time… Example: Resistive power divider. Analyze the power divider from the data provided by the manufacturer. ] [Images from: [Images http://www.fairviewmicrowave.com

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 63 EETAC-UPC Take your time…

Solution: Resistive power divider.

Insertion loss: IL  8.5 dB  |S21|0.37 and |S21|0.37 [adim] Matching: VSWR: 1.7:1 (worst case)  ||0.26 [adim]  ||-11.7 dB

||-11.7 dB

IL  8.5 dB -11.7 dB -11.7 dB -11.7   | |   | |

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 64 EETAC-UPC Take your time… Example: 8-Way divider. Analyze the divider from the data provided by the manufacturer. ] [Images from: [Images http://www.fairviewmicrowave.com

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 65 EETAC-UPC Take your time…

Solution: Resistive power divider.

Insertion loss: IL  10.5 dB  |S12|0.30 [adim] Isolation: I  19 dB  |S21|0.11 [adim] Input matching: VSWR: 1.4:1 (worst case)  ||0.167 [adim]  ||-15.5 dB Input matching: VSWR: 1.3:1 (worst case)  ||0.13 [adim]  ||-17.7 dB -17.7 dB  |  | -15.5 dB  |  |

I  19 dB

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 66 EETAC-UPC Take your time… Example: Circulator. Analyze the circulator from the brochure provided by the manufacturer. ] [Images from: [Images http://www.fairviewmicrowave.com

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 67 EETAC-UPC Take your time…

Solution: Circulator.

Insertion loss: IL  0.5 dB  |S12|0.94 [adim] Isolation: Ityp = 17 dB  |S21|=0.14 [adim] Matching: VSWR: 1.3:1 (worst case)  ||0.13 [adim]  ||-17.7 dB

||-17.7 dB

Ityp = 17 dB -17.7 dB -17.7 dB

 IL  0.5 dB  | |   | |

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 68 EETAC-UPC Take your time… Example: Directional coupler. Analyze the directional coupler from the brochure provided by the manufacturer. ] [Images from: [Images http://www.fairviewmicrowave.com

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 69 EETAC-UPC Take your time…

Solution: Directional coupler.

Insertion loss: IL  0.5 dB (for device having a coupling of 20 dB)  |S31|0.94 [adim] Isolation: I = 30 dB  |S21|=0.032 [adim] Matching: VSWR: 1.2:1 (worst case)  ||0.091 [adim]  ||-20.8 dB

Directivity: D  25 dB  |S21|/|S41|  17.7 [adim]

50  load ||-20.8 dB D  25 dB (4) 2

IL  0.5 dB -20.8 dB -20.8 dB   | 1 3 |   | SMA connectors |

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 70 EETAC-UPC Take your time… Example: N-connector 90º hybrid coupler. Analyze the hybrid from the brochure provided by the manufacturer. ] [Images from: [Images http://www.fairviewmicrowave.com

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 71 EETAC-UPC Take your time…

Solution: N-connector 90º hybrid coupler. Coupling: C = 3.1  0.5 dB  0.74 – 0.66 [adim] Matching: VSWR: 1.2:1 (worst case)  ||0.091 [adim]  ||-20.8 dB Isolation: I  25 dB  I  0.056 [adim] worst matching in-band ||-20.8 dB N connectors I  25 dB

C =3.1  0.5 dB

||-20.8 dB ||-20.8 dB

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 72 EETAC-UPC Take your time… Example: SMA-connector 90º hybrid coupler. Analyze the hybrid from the brochure provided by the manufacturer. ] [Images from: [Images http://www.fairviewmicrowave.com

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 73 EETAC-UPC Take your time…

Solution: SMA-connector of a 90º hybrid coupler. Coupling: C = 3.1  0.8 dB  0.77 – 0.64 [adim] Matching: VSWR: 1.3:1 (worst case)  ||0.13 [adim]  ||-17.7 dB Isolation: I  20 dB  I  0.1 [adim] worst matching in-band SMA connectors

||-17.7 dB I  20 dB IN ISO

C =3.1  0.8 dB

90º 0º ||-17.7 dB ||-17.7 dB

Radiofrequency Engineering C. Collado, J.M. González-Arbesú 74 EETAC-UPC