Single- and Multiport Networks. RF Electronics Spring, 2018 Robert R. Krchnavek Rowan University

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1 Single- and Multiport Networks RF Electronics Spring, 208 Robert R. Krchnavek Rowan University

2 Objectives Generate an understanding of the common network representations of Z, Y, h, and ABCD. To be able to obtain matrix parameters by suitable tests on the networks. To be able to convert from one network representation to another. Understand the implications of interconnecting networks. Develop a working knowledge of S-parameters.

3 Single- and Multiport Networks Single- and multiport network representations consist of input/output relationships without knowing the internal network. The relationships can be determined experimentally. Provides a means of designing/analyzing networks with overall system performance in mind. Very important in RF/MW work because detailed modeling of individual devices requires complex EM work.

4 Single- and Multiport Networks Voltage and Current Definitions + v i One-port Network + v i i 2 Two-port Network + v 2 Port Port 2 + v i Port Port 2 i 2 + v 2 Multiport Network i + N- v N- Port N- Port N i N + v N

5 Z-Matrix Representation The voltage at each port is given by v = Z i + Z 2 i Z N i N v 2 = Z 2 i + Z 22 i Z 2N i N.. v N = Z N i + Z N2 i Z NN i N This is easily represented as a matrix v Z Z 2 Z N v 2. = Z 2 Z 22 Z 2N i v N Z N Z N2 Z NN i i N

6 Z-Matrix Representation Each element in the Z-matrix is determined as follows Z nm = v n i m i k =0 (for k=m) Note: This involves setting the current on all the other ports equal to 0. This is done using an open circuit. Recall, true open circuits are difficult to create at RF/MW frequencies because of capacitance.

7 Y-Matrix Representation An alternative representation is the admittance or Y-matrix representation. And, the individual elements are determined by i i 2. i N = Y Y 2 Y N Y 2 Y 22 Y 2N Y N Y N2 Y NN v v 2. v N Y nm = i n v m v k =0 (for k=m)

8 Z- and Y-Matrix [v] = [Z] [i] and [i] = [Y ] [v] ] [ ] [v] = [Z] [Y ] [v] [Z] [Y ] = = []

9 ABCD and h-matrix Representation The hybrid, or h-matrix for a two-port network is given by v h h = 2 i i 2 h 2 h 22 v 2 The definition for the individual h parameters is h = v The ABCD matrix for a two-port network is given by v A B = C D i The h-matrix is often used to characterize a transistor (at low frequencies) and the importance of the ABCD matrix will be seen shortly. i v 2 =0 v2 i 2

10 Interconnecting Networks Networks can be connected to other networks to create larger networks. The larger network representations can often be determined by the interconnection pattern of the sub-networks. Series, parallel and cascading interconnections are common. Cascading networks are the most common in RF/ MW design.

11 Cascading Networks and the ABCD Representation + + v' i' i' 2 - " i v 2 ' v" i 2 " + v" 2 i ' i' 2 + i" i 2 " + + port A' B' " v 2 ' v A " B" v' C ' D' v 2 " port 2 C" D" - v i v i = = v i A C = B D A C A C B D v 2 B D i 2 v 2 i 2 = A C B D v i

12 Circuit i Z i 2 v v 2 A= C= 0 i i 2 v Y v 2 A= C= Y Z A Z C Z B i i 2 A= v v 2 C= ABCD-Parameters Z A Z C Z C B= Z D= B= 0 D= Z B Z A Z A Z = + B B Z B Z C D= Z C ABCD- Representation for Common Two-Port Networks Y A Y B i i 2 A= v v 2 Y C Y B Y C Y C Y A Y A Y = + B B Y C B= Y C Y A Y C D= i l i 2 v Z 0, β v 2 A= cosβl C= jsinβl Z 0 B= jz 0 sinβl D= cosβl d i N: i 2 A= N v v 2 C= 0 B= 0 D= --- N

13 Conversions Between Matrix Representations [Z] [Y] [h] [ABCD] [Z] Z Z 2 Z 2 Z 22 Z ΔZ Z ΔZ Z ΔZ Z ΔZ ΔZ Z 22 Z Z 22 Z Z Z 22 Z Z 2 ΔZ Z Z Z 2 Z 2 [Y] Y ΔY Y ΔY Y ΔY Y ΔY Y Y 2 Y 2 Y Y Y Y Y Y ΔY Y Y Y 2 ΔY Y Y 2 Y Y 2 [h] Δh h 22 h h 22 h h h h h h h h Δh h h h 2 h 2 h 22 Δh h 2 h h 2 h h h 2 [ABCD] A --- C --- C ΔABCD C D --- C D --- B -- B ΔABCD B A --- B B --- D --- D ΔABCD D C --- D A B C D

14 Conclusion # For Z, Y, h, and ABCD matrix representations, the individual matrix elements are easily determined by selectively shorting or opening the various ports of the network. PROBLEM: In RF/MW design, short-circuits and open-circuits are difficult to achieve so it is correspondingly difficult to obtain the matrix representation. SOLUTION: The scattering or S-parameters provide a method of solving this problem.

15 Scattering or S-parameter Representation The Scattering or S-parameter representation solves the problem of not being able to achieve the perfect open- or short-circuits required to determine the matrix elements for the previously defined representations (Z, Y, h, ABCD). It does this by considering how a wave acts when it is incident on the Device Under Test (DUT). Since we are now dealing with wave phenomena, terminating transmission lines (attached to the DUT) with their characteristic impedance can be used to determine the S-parameters. We begin by defining the terminal parameters. Previously, this was either a voltage or a current. Now, it is a normalized power.

16 S-parameters Definitions a a 2 [ ] S b b 2 Consider the network above. We define a n as follows and we want to determine the significance of this definition. a n = 2 Z 0 (V n + Z 0 I n ) Note: The subscript n can take on any port number and in the two-port network above, we would have a and a 2.

17 S-parameters Definitions For port, we would have a = 2 (V + Z 0 I ) Z 0 V = V + I = I + + I = V + V Z 0 a = 2 Z 0 a = V + Z0 Z 0 V + + V + Z 0 V + Z 0 V Z 0 We see that a is the incident voltage (V + ) normalized to the square root of the characteristic impedance. Since power is V 2 /R, we call this an incident normalized power wave.

18 S-parameters Definitions In a similar fashion, we can define a reflected normalized power wave as follows b n = 2 Z 0 (V n Z 0 I n ) The S-parameters for a two-port network are then defined as [ b b 2 ] = [ S S 2 S 2 S 22 ][ a a 2 ] where.....

19 S = b a S 2 = b a 2 S 2 = b 2 a S 22 = b 2 a 2 a2 =0 a =0 a2 =0 a =0 reflected power wave at port incident power wave at port reflected power wave at port incident power wave at port 2 reflected power wave at port 2 incident power wave at port reflected power wave at port 2 incident power wave at port 2 NO incident power at port 2 NO incident power at port NO incident power at port 2 NO incident power at port The requirement that no incident power appears at either port or port 2 is achieved by terminating the transmission line in its characteristic impedance.

20 S and S 2 Z 0 a a 2 = 0 V G Z 0 [S] Z 0 Z L b b 2 No incident power is seen at port 2 because the transmission line load, Z L, is equal to Z 0. S = b a S 2 = b 2 a a2 =0 = a2 =0 = V V + =Γ in = Z in Z 0 Z in + Z 0 V 2 Z0 2 (V Z + Z 0 I ) = 2V 0 S is the reflection coefficient and S 2 is the forward voltage gain. 2 V G = 2V 2 V G

21 S 2 and S 22 a = 0 a 2 Z 0 Z G Z 0 [ ] S Z 0 V G2 b b 2 No incident power is seen at port because Z G is set to Z 0. S 2 = b a 2 S 22 = b 2 a 2 = a =0 a =0 = V 2 V 2 + V Z0 2 (V Z 2 + Z 0 I 2 ) = 2V 0 =Γ out = Z out Z 0 Z out + Z 0 V G2 = 2V V G2

22 S-parameters and Cascading Networks S-parameters do not directly apply to cascaded networks. Cascaded networks are common in RF systems. The chain scattering matrix works with cascading networks. The T-parameters are obtained from S-parameters.

23 S-parameters and Cascading Networks [ b b 2 ] [ a b ] = = [ S S 2 S 2 S 22 ][ a a 2 ] [ T T 2 T 2 T 22 ][ b2 a 2 ] a a 2 [ ] S b b 2

24 b A Chain Scattering Matrix A A a a b 2 B a B 2 port [ T] [ T] port 2 A B b A b A 2 a B b B 2 [ a b ] = [ T T 2 T 2 T 22 ][ b2 a 2 ] This matrix is similar to the ABCD matrix and is useful for cascading networks.

25 Chain Scattering Matrix [ a b ] = [ T T 2 T 2 T 22 ][ b2 a 2 ] T = a b 2 a 2 =0 = S 2

26 Summary Traditional network representations (Z, Y, h, ABCD) do not work at RF/MW frequencies because we cannot experimentally determine the matrix elements. This is because we cannot create perfect opens or shorts which is required to determine the different elements. S-parameters solve this problem by using transmission line segments terminated with a matched load to prevent reflections. For cascaded networks, T-parameters should be used. T-parameters are easily derived from S- parameters and vice-versa.

A two-port network is an electrical network with two separate ports

5.1 Introduction A two-port network is an electrical network with two separate ports for input and output. Fig(a) Single Port Network Fig(b) Two Port Network There are several reasons why we should study