The University of Tennessee Electrical and Computer Engineering An Introduction To Two – Port Networks The University of Tennessee Electrical and Computer Engineering Knoxville, TN These notes were first prepared during the Spring Semester of 2002 in support of the ECE 202 course which will be changed to ECE 300 in the Fall Semester of 2002. The material on two Ports will be the same. wlg

Two Port Networks Generalities: The Network The standard configuration of a two port: I1 I2 The Network + + Input Port Output Port V1 V2 _ _ The two port network configuration shown with this slide is practically used universally in all text books. The same goes for the voltage and current assumed polarity and direction at the input and output ports. I think this is done because of how we define parameters with respect the voltages and currents at the input and out ports. With respect to the network, it may be configured with passive R, L, C, op-amps, transformers, dependent sources but not independent sources. The network ? The voltage and current convention ? * notes

Two Port Networks Network Equations: V1 = z11I1 + z12I2 V2 = b11V1 - b12I1 I2 = b21V1 – b22I1 Impedance Z parameters I1 = y11V1 + y12V2 I2 = y21V1 + y22V2 V1 = h11I1 + h12V2 I2 = h21I1 + h22V2 Admittance Y parameters Hybrid H parameters The equations in light orange are the ones we will consider here. The other equations are also presented and consider in Nilsson & Riedel, 6th ed. The parameters are defined in terms of open and short circuit conditions of the two ports. This will be illustrated and some examples presented. Transmission A, B, C, D parameters V1 = AV2 - BI2 I1 = CV2 - DI2 I1 = g11V1 + g12I2 V2 = g21V1 + g22I2 * notes

Two Port Networks Z parameters: z11 is the impedance seen looking into port 1 when port 2 is open. z12 is a transfer impedance. It is the ratio of the voltage at port 1 to the current at port 2 when port 1 is open. z21 is a transfer impedance. It is the ratio of the voltage at port 2 to the current at port 1 when port 2 is open. Z11 and z22 are easy to remember. We look into the input port and calculate the impedance with the output terminals open, if the circuit is contains only R, L. C elements. We do the same at the output port to find z22. However, if the network contains dependent Sources, transformers, or op-amps, we must find V1/I1 with I2 = 0. Similarly for z22 For z12 and z21 we still find ratios of port voltage To port current with the opposite port open. If you have only R, L, and C then z12 = z21. z22 is the impedance seen looking into port 2 when port 1 is open. * notes

Two Port Networks Y parameters: y11 is the admittance seen looking into port 1 when port 2 is shorted. y12 is a transfer admittance. It is the ratio of the current at port 1 to the voltage at port 2 when port 1 is shorted. y21 is a transfer impedance. It is the ratio of the current at port 2 to the voltage at port 1 when port 2 is shorted. Similar to input and output impedance, y11 and y22 are determined by looking into the input and output ports with the opposite ports short circuited. When we look for y12 and y21 we adhere to the above equations with respect to shorting the output terminals. If the circuit is not passive, we need to be careful and do exactly as the equations tell us to do. y22 is the admittance seen looking into port 2 when port 1 is shorted. * notes

Two Port Networks Z parameters: Example 1 Given the following circuit. Determine the Z parameters. Find the Z parameters for the above network.

Two Port Networks Z parameters: = Example 1 (cont 1) For z11: For z22: Therefore:  =

Two Port Networks Z parameters: Example 1 (cont 2) The Z parameter equations can be expressed in matrix form as follows.

Two Port Networks Z parameters: Example 2 (problem 18.7 Alexander & Sadiku) You are given the following circuit. Find the Z parameters.

Two Port Networks Z parameters: but Z21 = -0.667  Substituting gives; Example 2 (continue p2) ; but Other Answers Z21 = -0.667  Z12 = 0.222  Z22 = 1.111  Substituting gives; or

Two Port Networks Transmission parameters (A,B,C,D): The defining equations are: I2 = 0 V2 = 0 V2 = 0 I2 = 0

Two Port Networks Transmission parameters (A,B,C,D): Example Given the network below with assumed voltage polarities and Current directions compatible with the A,B,C,D parameters. We can write the following equations. V1 = (R1 + R2)I1 + R2I2 V2 = R2I1 + R2I2 It is not always possible to write 2 equations in terms of the V’s and I’s Of the parameter set.

Two Port Networks Transmission parameters (A,B,C,D): I2 = 0 V2 = 0 Example (cont.) V1 = (R1 + R2)I1 + R2I2 V2 = R2I1 + R2I2 From these equations we can directly evaluate the A,B,C,D parameters. = = I2 = 0 V2 = 0 The answers to the above are underneath the gray boxes. The A,B,C,D parameters are best used when we want to cascade two networks together such as follows = = V2 = 0 I2 = 0 Later we will see how to interconnect two of these networks together for a final answer Network 1 Network 2 * notes

Two Port Networks Hybrid Parameters: The equations for the hybrid parameters are: V2 = 0 I1 = 0 The H parameters are used almost solely in electronics. These parameters are used in the equivalent circuit Of a transistor. As you will see, the H parameter equations directly set-up so as to describe the device in terms Of the hij parameters which are given by the manufacturer. You will use this material in junior electronics. V2 = 0 I1 = 0 * notes

Two Port Networks Hybrid Parameters: The following is a popular model used to represent a particular variety of transistors. We can write the following equations: In this example we consider a very popular model of a transistor. We first assign some identifier, such as K1, K2, K3, K4, to the model. Next we write the equations at the input port and output port. In this case they are extremely simple to write. Then it turns out that we can use these equations to directly identify what will be called the H parameters. Note that one of the parameters is a voltage gain, one is a current gain, one is an resistance and one is a conductance. You will go into a fair amount of detail on this model later in the curriculum * notes

Two Port Networks Hybrid Parameters: We want to evaluate the H parameters from the above set of equations. = K1 = K2 V2 = 0 I1 = 0 = K3 = V2 = 0 I1 = 0

Two Port Networks Hybrid Parameters: Another example with hybrid parameters. Given the circuit below. The equations for the circuit are: V1 = (R1 + R2)I1 + R2I2 V2 = R2I1 + R2I2 The H parameters are as follows. = = 1 V2=0 I1=0 = - 1 = V2=0 I1=0

Two Port Networks Modifying the two port network: Earlier we found the z parameters of the following network. Equations from two port networks are not considered to be the end of the problem. We will practically always add components to the input and output. When we do this we must start with the two port parameter, in a certain form, and modify the equations to incorporate the changes at the two ports. For example, we may place a voltage source in series with a resistor at the input port and maybe a load resistor at the output port. The example here illustrates how to handle this problem. * notes

Two Port Networks Modifying the two port network: We now have: We modify the network as shown be adding elements outside the two ports We now have: V1 = 10 - 6I1 V2 = - 4I2

Two Port Networks Modifying the two port network: V1 = 10 - 6I1 We take a look at the original equations and the equations describing the new port conditions. V1 = 10 - 6I1 V2 = - 4I2 So we have, Remember that we are trying to solve for I1 and I2 after changing the port conditions. This involves doing some algebra but at a low level. We recombine terms and arrange the original equation in matrix form and we can easily take the inverse to find the solution or else we can use simultaneous equations with our hand calculators. 10 – 6I1 = 20I1 + 8I2 -4I2 = 8I1 + 12I2 * notes

Two Port Networks Modifying the two port network: Rearranging the equations gives, 26 8 10 8 16 0.4545 -0.2273

Two Port Networks Y Parameters and Beyond: Given the following network. Find the Y parameters for the network. From the Y parameters find the z parameters

Two Port Networks so s + 0.5 I1 = y11V1 + y12V2 I2 = y21V1 + y22V2 Y Parameter Example I1 = y11V1 + y12V2 I2 = y21V1 + y22V2 short We use the above equations to evaluate the parameters from the network. To find y11 so s + 0.5 =

Two Port Networks Y Parameter Example We see = 0.5 S

Two Port Networks To find y12 and y21 we reverse things and short V1 Y Parameter Example To find y12 and y21 we reverse things and short V1 short We have We have = 0.5 S

Two Port Networks Y Y Parameter Example Summary: = Now suppose you want the Z parameters for the same network.

Two Port Networks Going From Y to Z Parameters For the Y parameters we have: For the Z parameters we have: From above; Therefore where

Two Port Parameter Conversions:

Two Port Parameter Conversions: To go from one set of parameters to another, locate the set of parameters you are in, move along the vertical until you are in the row that contains the parameters you want to convert to – then compare element for element

Interconnection Of Two Port Networks Three ways that two ports are interconnected: ya * Parallel yb za * Series zb * Cascade Ta Tb

Interconnection Of Two Port Networks Consider the following network: I1 I2 + + V1 V2 T1 T2 _ _ Referring to slide 13 we have;

Interconnection Of Two Port Networks Multiply out the first row: Set I2 = 0 ( as in the diagram) Can be verified directly by solving the circuit

Basic Laws of Circuits End of Lesson Two-Port Networks