Lecture 23 Date: 30.0.207 Multi-port networks mpedance and Admittance Matrix Lossless and Reciprocal Networks
ntroduction A pair of terminals through which a current may enter or leave a network is known as a port. Two-terminal devices or elements (such as resistors, capacitors, and inductors) result in oneport networks. Most of the circuits we have dealt with so far are two-terminal or one-port circuits. So far, considered the voltage across or current through a single pair of terminals such as the two terminals of a resistor, capacitor, inductor. We have also studied four-terminal or two-port circuits involving op amps, and transformers. A two-port network is an electrical network with two separate ports for input and output.
2-port Networks Requirement of Matrix Formulation Current/Voltage Device (more than one port) Current/Voltage NO!! These are called networks What is the way? mpedance or Admittance Matrix. Right? Can we characterie this using an impedance or admittance! n principle, N by N impedance matrix completely characteries a linear N- port device. Effectively, the impedance matrix defines a multi-port device the way a Z L describes a single port device (e.g., a resistor) Linear networks can be completely characteried by parameters measured at the network ports without knowing the content of the networks.
Multiport Networks Networks can have any number of ports however, analysis of a 2-port, 3-port or 4-port network is sufficient to explain the theory and the associated concepts. Port + V - 2 Port Network 2 + V 2 - Port 2 The ports can be characteried with many parameters (Z, Y, h, S, ABCD). Each has a specific advantage. For 2-port Network, each parameter set is related to 4 variables: o 2 independent variables for excitation o 2 dependent variables for response
The mpedance Matrix Let us consider the following 4-port network: This could be a simple linear device or a large/complex linear system ( ) Either way, the network can be fully described by its impedance matrix V( ) Port- P ( ) 2 2 Port-2 Port-4 ( ) 4 4 V ( ) 4-port Linear Microwave Network 2 2 V ( ) 4 4 2 2P Port-3 4 4P The arbitrary locations are known as ports of the network 3 3P Four identical cables used to connect this network to the outside world ( ) 3 3 V ( ) 3 3 Each cable has specific location that defines input impedances to the network
The mpedance Matrix (contd.) n principle, the current and voltages at the port-n of networks are given as: V ( ) ( ) n n np n n np the simplified formulations are: V V ( ) ( ) n n n np n n n np f we want to say that there exists a nonero current at port- and ero current at all other ports then we can write as: 0 2 3 4 0 n order to define the elements of impedance matrix, there will be need to measure/determine the V2 Z2 associated voltages and currents at the respective ports. Suppose, if we measure/determine current at port- and then voltage at port-2 then we can define: Trans-impedance Similarly, the trans-impedance parameters Z 3 and Z 4 are: Z V Z V 3 4 3 4 We can define other trans-impedance parameters such as Z 34 as the ratio between the complex values 4 (into port-4) and V 3 (at port-3), given that the currents at all other ports (, 2, and 3) are ero.
The mpedance Matrix (contd.) Therefore, the more generic form of trans-impedance is: Z V m n (given that k = 0 for all k n) Open the ports where the current needs to be ero How do we ensure that all but one port current is ero? The ports should be opened! not the cables connected to the ports V Port- We can then define the respective trans-impedances as: 2 0 4 0 Z 4-port Linear Microwave Network V V 2 V 4 m n Port-2 Port-4 3 0 V 3 Port-3 (given that all ports k n are open)
The mpedance Matrix (contd.) Once we have defined the trans-impedance terms by opening various ports, it is time to formulate the impedance matrix Since the network is linear, the voltage at any port due to all the port currents is simply the coherent sum of the voltage at that port due to each of the currents For example, the voltage at port-3 is: Therefore we can generalie the voltage for N-port network as: Where and V are N V Z m n n vectors given as: V= V, V, V,..., V 2 3 N V3 Z344 Z333 Z322 Z3 T = V = Z,,,..., 2 3 N T The term Z is matrix given by: Z Z Z Z Z2 Z Z Z 2 n m m2 mpedance Matrix
The Admittance Matrix Let us consider the 4-port network again: ( ) V( ) Port- P ( ) 2 2 Port-2 Port-4 ( ) 4 4 V ( ) 2 2 4-port Linear Microwave Network V ( ) 4 4 2 2P Port-3 4 4P 3 3P ( ) 3 3 V ( ) 3 3 This can be characteried using admittance matrix if currents are taken as dependent variables instead of voltages The elements of admittance matrix are called trans-admittance parameters Y
The Admittance Matrix (contd.) The trans-admittances Y are defined as: m Y (given that V V k = 0 for all k n) t is apparent that the voltage at all but one port must be equal to ero. This can be ensured by shortcircuiting the voltage ports. The ports should be short-circuited! not the cable connected to the ports n V Port- 2 4 mportant Y Z V2 0 4-port Linear Microwave Network V4 0 Port-2 Port-4 3 Port-3 V3 0 Now, since the network is linear, the current at any one port due to all the port voltages is simply the coherent sum of the currents at that port due to each of the port voltages.
The Admittance Matrix (contd.) For example, the current at port-3 is: 3 Y34V 4 Y33V 3 Y32V 2 Y3V N Therefore we can generalie the m current for N-port network as: YV n n Where and V are vectors given as: T V= V, V, V,..., V =,,,..., 2 3 N 2 3 N T = YV The term Y is matrix given by: Y Y Y Y2 Y Y Y Y 2 n m m2 Admittance Matrix The values of elements in the admittance matrix are frequency dependents and often it is advisable to describe admittance matrix as: Y ( ) Y2 ( ) Y n( ) Y2( ) Y( ) Ym ( ) Ym 2( ) Y ( )
The Admittance Matrix (contd.) You said that: Z s there any relationship between admittance and impedance matrix of a given device? Y Answer: Let us see if we can figure it out! Recall that we can determine the inverse of a matrix. Denoting the matrix inverse of the admittance matrix as Y, we find: Y = Y Y = Y YV YV Y = V = YV We also know: V = Z Z = Y OR Y Z
Reciprocal and Lossless Networks We can classify multi-port devices or networks as either lossless or lossy; reciprocal or non-reciprocal. Let s look at each classification individually. Lossless Network A lossless network or device is simply one that cannot absorb power. This does not mean that the delivered power at every port is ero; rather, it means the total power flowing into the device must equal the total power exiting the device. A lossless device exhibits an impedance matrix with an interesting property. Perhaps not surprisingly, we find for a lossless device that the elements of its impedance matrix will be purely reactive: Re( Z ) 0 For a lossless device f the device is lossy, then the elements of the impedance matrix must have at least one element with a real (i.e., resistive) component. Furthermore, we can similarly say that if the elements of an admittance matrix are all purely imaginary (i.e., Re{Y } =0), then the device is lossless.
Reciprocal and Lossless Networks (contd.) Reciprocal Network deally, most passive, linear components will turn out to be reciprocal regardless of whether the designer intended it to be or not! Reciprocity is a tremendously important characteristic, as it greatly simplifies an impedance or admittance matrix! Specifically, we find that a reciprocal device will result in a symmetric impedance and admittance matrix, meaning that: Z Z nm Y Y nm For a reciprocal device For example, we find for a reciprocal device that Z 23 =Z 32, and Y 2 =Y 2.
Reciprocal and Lossless Networks (contd.) Z= j2 0. j3 j 4 2 0.5 neither lossless nor reciprocal lossless, but not reciprocal Z= j2 j0. j3 j j j j4 j2 j0.5 Z= j2 j 4 j j2 4 j2 j0.5 reciprocal, but not lossless lossless and reciprocal Z= j2 j j4 j j j2 j4 j2 j0.5 Example determine the Y matrix of this twoport device. R β, Z 0 V 2R V2 β, Z 0 2
Example (contd.) Step-: Place a short at port 2 β, Z 0 2 V 2R R V2 0 Step-2: Determine currents and 2 Note that after the short was placed at port 2, both resistors are in parallel, with a potential V across each Therefore current is The current 2 equals the portion of current through R but with opposite sign V V 3V 2R R 2R 2 V R
Example (contd.) Step-3: Determine the trans-admittances Y and Y 2 Y 3 V 2 2 2R V Y R Note that Y 2 is real and negative This is still a valid physical result, although you will find that the diagonal terms of an impedance or admittance matrix (e.g., Y 22, Z, Y 44 ) will always have a real component that is positive To find the other two trans-admittance parameters, we must move the short and then repeat each of our previous steps!
Example (contd.) Step-: Place a short at port V 2 0 2R R V 2 Step-2: Determine currents and 2 Note that after a short was placed at port, resistor 2R has ero voltage across it and thus ero current through it! Therefore: V2 2 R Step-3: 2 V2 R Determine the trans-admittances Y 2 and Y 22 2 Therefore the admittance matrix is: Y= Y V Y 2 22 R V 2 3/2R /R /R /R 2 R s it lossless or reciprocal?
Example 2 2 Consider this circuit: + 6 V + V 2 Where the 3-port device is characteried by the impedance matrix: Z= 2 2 4 2 4 3 + V 3 determine all port voltages V, V 2, V 3 and all currents, 2, 3.