9/4/2007 Microwave Network Theory 1/3 Jim Stiles The Univ. of Kansas Dept. of EECS E. Microwave Network Theory Note that a passive load is a one-port device—a device that can be characterized (at one frequency) by impedance ZL or load reflection coefficient ΓL . However, many microwave devices have multiple ports! Most common are two-port devices (e.g., amplifiers and filters), devices with both a gozenta and a gozouta. Note that a transmission line is also two-port device! Q: Are there any known ways to characterize a multi-port device? A: Yes! Two methods are: 1. The impedance matrix—a multi-port equivalent of ZL 2. The scattering matrix—a multi-port equivalent of ΓL HO: The Impedance Matrix gozenta gozouta9/4/2007 Microwave Network Theory 2/3 Jim Stiles The Univ. of Kansas Dept. of EECS Q: You say that the impedance matrix characterizes a multi-port device. But is this characterization helpful? Can we actually use it to solve real problems? A: Example: Using the Impedance Matrix Q: The impedance matrix relates the quantities ()Iz and ()Vz , is there an equivalent matrix that relates ()Vz+ and ()Vz−? A: Yes! The scattering matrix relates the t.l. waves entering and exiting a multi-port device! HO: The Scattering Matrix Q: Can the scattering matrix likewise be used to solve real problems? A: Of course! Example: The Scattering Matrix Example: Scattering Parameters Q: But, can the scattering matrix by itself tell us anything about the device it characterizes? A: Yes! It can tell us if the device is matched, or lossless, or reciprocal.9/4/2007 Microwave Network Theory 3/3 Jim Stiles The Univ. of Kansas Dept. of EECS HO: Matched, Lossless, Reciprocal09/04/07 The Impedance Matrix 1/7 Jim Stiles The Univ. of Kansas Dept. of EECS The Impedance Matrix Consider the 4-port microwave device shown below: Note in this example, there are four identical transmission lines connected to the same “box”. Inside this box there may be a very simple linear device/circuit, or it might contain a very large and complex linear microwave system. ()44Iz ()22Iz port 1 ()11Vz+− ()44Vz+− ()22Vz+− port 3 port 4 port 2 4-port microwave device Z0 Z0 Z0 Z0 33Pzz= 22Pzz= 11Pzz= 44Pzz= ()33Vz+− ()33Iz ()11Iz09/04/07 The Impedance Matrix 2/7 Jim Stiles The Univ. of Kansas Dept. of EECS Æ Either way, the “box” can be fully characterized by its impedance matrix! First, note that each transmission line has a specific location that effectively defines the input to the device (i.e., z1P, z2P, z3P, z4P). These often arbitrary positions are known as the port locations, or port planes of the device. Thus, the voltage and current at port n is: ()nn nPVzz= ()nn nPIz z= We can simplify this cumbersome notation by simply defining port n current and voltage as In and Vn : ()nnn nPVVz z== ()nnnnPIIzz== For example, the current at port 3 would be ()3333PIIzz== . Now, say there exists a non-zero current at port 1 (i.e., 10I≠), while the current at all other ports are known to be zero (i.e., 2340III===). Say we measure/determine the current at port 1 (i.e., determine 1I), and we then measure/determine the voltage at the port 2 plane (i.e., determine 2V).09/04/07 The Impedance Matrix 3/7 Jim Stiles The Univ. of Kansas Dept. of EECS The complex ratio between 21 and VI is know as the trans-impedance parameter Z21: 2211VZI= Likewise, the trans-impedance parameters Z31 and Z41 are: 34314111 and VVZZII== We of course could also define, say, trans-impedance parameter Z34 as the ratio between the complex values 4I (the current into port 4) and 3V(the voltage at port 3), given that the current at all other ports (1, 2, and 3) are zero. Thus, more generally, the ratio of the current into port n and the voltage at port m is: (given that 0 for all )mmnknVZIknI==≠09/04/07 The Impedance Matrix 4/7 Jim Stiles The Univ. of Kansas Dept. of EECS A: Place an open circuit at those ports! Placing an open at a port (and it must be at the port!) enforces the condition that 0I=. 1I 40I= 3V+− 20I= 1V+− 4V+− 30I= 2V+− 4-port microwave device Z0 Z0 Z0 Z0 Q: But how do we ensure that all but one port current is zero ?09/04/07 The Impedance Matrix 5/7 Jim Stiles The Univ. of Kansas Dept. of EECS Now, we can thus equivalently state the definition of trans-impedance as: (given that all ports are )mmnnVZknI=≠open A: OK, say that none of our ports are open-circuited, such that we have currents simultaneously on each of the four ports of our device. Since the device is linear, the voltage at any one 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 can be determined by: 333332231134 4VZI ZI ZI ZI=+++ Q: As impossible as it sounds, this handout is even more boring and pointless than any of your previous efforts. Why are we studying this? After all, what is the likelihood that a device will have an open circuit on all but one of its ports?!09/04/07 The Impedance Matrix 6/7 Jim Stiles The Univ. of Kansas Dept. of EECS More generally, the voltage at port m of an N-port device is: 1NmmnnnVZI==∑ This expression can be written in matrix form as: =VIZ Where I is the vector: []123TNI,I,I, ,I=I  and V is the vector: 123TNV,V,V, ,V⎡⎤=⎣⎦V … And the matrix Z is called the impedance matrix: 11 11nmmnZZZZ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦Z… The impedance matrix is a N by N matrix that completely characterizes a linear, N -port device. Effectively, the impedance matrix describes a multi-port device the way that LZ describes a single-port device (e.g., a load)!09/04/07 The Impedance Matrix 7/7 Jim Stiles The Univ. of Kansas Dept. of EECS But beware! The values of the impedance matrix for a particular device or network, just like LZ, are frequency dependent! Thus, it may be more instructive to explicitly write: ()()()() ()11 11nmmnZZZZωωωωω⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦Z…9/4/2007 Example Using the Impedance Matrix 1/3 Jim Stiles The Univ. of Kansas Dept. of EECS Example: Using the Impedance Matrix Consider the following circuit: Where the 3-port device is characterized by the impedance matrix: 212114241⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦Z