Prof. David R. Jackson Dept. of ECE Notes 14 ECE Microwave Engineering Fall 2015 Network Analysis Multiport Networks 1

A general circuit can be represented by a multi-port network, where the “ports” are defined as access terminals at which we can define voltages and currents. Examples: 1) One-port network 2) Two-port network 2 Note: Equal and opposite currents are assumed on the two wires of a port.

3) N -port network To represent multi-port networks we use:  Z (impedance) parameters  Y (admittance) parameters  ABCD parameters  S (scattering) parameters Not easily measurable at high frequency Measurable at high frequency Multiport Networks (cont.) 3

Poynting Theorem (Phasor Domain) 4 The last term is the VARS consumed by the region. The notation denotes time-average.

Consider a general one-port network Complex power delivered to network: Self Impedance ( Z in ) Average power dissipated in [W] Average magnetic energy (in [J]) stored inside V Average electric energy (in [J]) stored inside V + - 5

Self Impedance (cont.) Hence we have

We can show that for physically realizable networks the following apply: Self Impedance (cont.) Please see the Pozar book for a proof. Note: Frequency is usually defined as a positive quantity. However, we consider the analytic continuation of the functions into the complex frequency plane. 7

Consider a general 2-port linear network: In terms of Z -parameters, we have (from superposition): Impedance ( Z ) matrix Two-Port Networks Therefore

Elements of Z -Matrix: Z -Parameters (open-circuit parameters) Port 2 open circuited Port 1 open circuited

Z -Parameters (cont.) 10 We inject a current into port j and measure the voltage (with an ideal voltmeter) at port i. All ports are open-circuited except j. N -port network

Z -parameters are convenient for series connected networks. Z -Parameters (cont.) 11 Series:

Admittance ( Y ) Parameters Consider a linear 2-port network: Admittance matrix Short-circuit parameters or

Y -Parameters (cont.) 13 We apply a voltage across port j and measure the current (with an ideal current meter) at port i. All ports are short-circuited except j. N -port network

Y -parameters are convenient for parallel connected networks. 14 Y -Parameters (cont.) Parallel:

Relation Between Z and Y Parameters Relation between [ Z ] and [ Y ] matrices: Hence: Therefore 15 It follows that

Reciprocal Networks If a network does not contain non-reciprocal devices or materials* (i.e. ferrites, or active devices), then the network is “reciprocal.” * A reciprocal material is one that has symmetric permittivity and permeability tensors. A reciprocal device is one that is made from reciprocal materials. Note: The inverse of a symmetric matrix is symmetric. 16 Example of a nonreciprocal material: a biased ferrite (This is very useful for making isolators and circulators.)

Reciprocal Materials Ferrite: is not symmetric! Reciprocal: 17 (not a reciprocal material)

We can show that the equivalent circuits for reciprocal 2-port networks are: T-equivalent Pi-equivalent Reciprocal Networks (cont.) 18 “  network” “T network”

ABCD-Parameters There are defined only for 2-port networks. 19

Cascaded Networks 20 A nice property of the ABCD matrix is that it is easy to use with cascaded networks: you simply multiply the ABCD matrices together.

 At high frequencies, Z, Y, & ABCD parameters are difficult (if not impossible) to measure.  V and I are not always uniquely defined (e.g., microstrip, waveguides).  Even if defined, V and I are extremely difficult to measure (particularly I ).  Required open and short-circuit conditions are often difficult to achieve.  Scattering ( S ) parameters are often the best representation for multi-port networks at high frequency. Scattering Parameters 21 Note: We can always convert from S parameters to Z, Y, or ABCD parameters.

22 Scattering Parameters (cont.) Keysight (formerly Agilent) VNA shown performing a measurement. A Vector Network Analyzer (VNA) is usually used to measure S parameters. Port 1 Port 2

S -parameters are defined assuming transmission lines are connected to each port. On each transmission line: Scattering Parameters (cont.) 23 We define:

Why are the wave functions ( a and b ) defined as they are? Recall: Scattering Parameters (cont.) (assuming lossless lines) 24 Similarly,

For a One-Port Network 25 For a one-port network, S 11 is the same as  L.

Scattering matrix For a Two-Port Network 26 From linearity:

Scattering Parameters Output is matched Input is matched input reflection coef. w/ output matched reverse transmission coef. w/ input matched forward transmission coef. w/ output matched output reflection coef. w/ input matched Output is matched Input is matched 27

Scattering Parameters (cont.) 28 For a general multiport network: All ports except j are semi-infinite with no incident wave (or with matched load at ports). N -port network

Scattering Parameters (cont.) Illustration of a three-port network 29

A microwave system may have waveguides entering a device. In this case, the transmission lines are TEN models for the waveguides. Scattering Parameters (cont.) 30 Device TE 10 Waveguide

Shift in reference plane: Scattering Parameters (cont.) 31 The concept is illustrated for a two-port network, where i = 1 and j = 2 ).

For reciprocal networks, the S -matrix is symmetric. 32 Properties of the S Matrix Example of a nonreciprocal material: a biased ferrite

 For lossless networks, the S -matrix is unitary. Identity matrix Properties of the S Matrix (cont.) Equivalently, Notation: 33 N -port network (“Hermetian conjugate” or “Hermetian transpose”)

Properties of the S Matrix (cont.) 34 Interpretation: The inner product of columns i and j is zero unless i = j. S 1 vector S 3 vector The rows also form orthogonal sets (see the note on the previous slide).

Scattering Parameters (cont.) Note: If all lines entering the network have the same characteristic impedance, then 35 “unnormalized” scattering parameters “normalized” scattering parameters In general: Note: The unitary property of the scattering matrix requires normalized parameters. We use normalized parameters in these notes.

Example Find the S parameters for a series impedance Z. Note that two different coordinate systems are being used here! 36

Example (cont.) By symmetry: 37 S 11 Calculation:

Example (cont.) 38 S 21 Calculation: Voltage divider: We also have so

We then have Example (cont.) 39 Hence

Example Find the S parameters for a length L of transmission line. Note that three different coordinate systems are being used here! 40

Example (cont.) 41 S 11 Calculation:

Example (cont.) Hence 42

Example (cont.) We now try to put the numerator of the S 21 equation in terms of V 1 (0). Hence, for the denominator of the S 21 equation we have 43 S 21 Calculation: Total voltage at port 1:

Example (cont.) Next, we need to get V + (0) in terms of V 1 (0) : Hence, we have 44

Example (cont.) Therefore, we have so 45

Special cases: Example (cont.) 46 a)a) b)b)

Example Find the S parameters for a step-impedance discontinuity. 47 S 11 Calculation: S 21 Calculation:

Example (cont.) Because of continuity of the voltage across the junction, we have: Hence so 48

Example Not unitary  Not lossless 1)Find the input impedance looking into port 1 when ports 2 and 3 are terminated in 50 [  ] loads. 2)Find the input impedance looking into port 1 when port 2 is terminated in a 75 [  ] load and port 3 is terminated in a 50 [  ] load. 49 (For example, column 2 doted with the conjugate of column 3 is not zero.) These are the S parameters assuming 50  lines entering the device.

1) If ports 2 and 3 are terminated in 50 [Ω]: Example (cont.) (a 2 = a 3 = 0) 50

2) If port 2 is terminated in 75 [Ω] and port 3 in 50 [Ω]: Example (cont.) 51

Example (cont.) 52

Example 53 Find  in for the system shown below. Assume : so We also have: Two-port device

Example (cont.) 54 Hence so Two-port device

Transfer ( T ) Matrix For cascaded 2-port networks: T Matrix: 55 (derivation omitted)

Transfer ( T ) Matrix (cont.) Hence The T matrix of a cascaded set of networks is the product of the T matrices. 56 The T matrices can be multiplied together, just as for ABCD matrices. (definition of T matrix) But so that

57 Conversion Between Parameters (Two-Ports)

Example 58 Derive S ij from the Z parameters. (The result is given inside row 1, column 2, of the previous table.) Semi-infinite 1 2 S 11 Calculation:

Example (cont.) 59

Example (cont.) 60 so We have: We next simplify this.

Example (cont.) 61

Example (cont.) 62 Note: To get S 22, simply let Z 11  Z 22 in the previous result. Hence, we have: Hence This agrees with the table.

Example (cont.) 63 Assume Use voltage divider equation twice to get V 2 (0) : S 21 Calculation: Semi-infinite 1 2 Also,

Example (cont.) 64 Use voltage divider equation twice:

Example (cont.) 65 Hence where

Example (cont.) 66 After simplifying, we should get the result in the table: (You are welcome to check it!) where This is the result in the table. Our result:

Signal-Flow Graph 67 This is way to graphically represent S parameters. Please see the Pozar book for more details. The wave amplitudes are represented as nodes in the single-flow graph. Rule: The value at each node is the sum of the values coming into the node from the various other nodes. (wave amplitudes evaluated at z i = 0 )