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.
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 equations in light orange are the ones we will consider here. The other equations are also presented and consider in Nilsson & Riedel, 6 th 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.
Z 11 and z 22 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 z 22 . However, if the network contains dependent Sources, transformers, or op-amps, we must find V 1 /I 1 with I 2 = 0. Similarly for z 22 For z 12 and z 21 we still find ratios of port voltage To port current with the opposite port open. If you have only R, L, and C then z 12 = z 21 .
Similar to input and output impedance, y 11 and y 22 are determined by looking into the input and output ports with the opposite ports short circuited. When we look for y 12 and y 21 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.
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 Network 1 Network 2
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 h ij parameters which are given by the manufacturer. You will use this material in junior electronics.
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
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.
Remember that we are trying to solve for I 1 and I 2 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.
Two port networks unit ii
An Introduction To Two – Port Networks The University of Tennessee Electrical and Computer Engineering Knoxville, TNwlg
Two Port NetworksGeneralities: The standard configuration of a two port: I1 I2 + + Input Output V1 _ Port The Network Port V2 _ The network ? The voltage and current convention ?* notes
Two Port Networks Z parameters: V z11 is the impedance seen looking into port 1 z = 1 11 I I =0 when port 2 is open. 1 2 V z12 is a transfer impedance. It is the ratio of the z = 1 12 I I =0 voltage at port 1 to the current at port 2 when 2 1 port 1 is open. V z = 2 z21 is a transfer impedance. It is the ratio of the 21 I I =0 1 2 voltage at port 2 to the current at port 1 when port 2 is open. V z = 2 z22 is the impedance seen looking into port 2 22 I I =0 2 1 when port 1 is open.* notes
Two Port Networks Y parameters: I y11 is the admittance seen looking into port 1 y = 1 11 V V =0 when port 2 is shorted. 1 2 I y12 is a transfer admittance. It is the ratio of the y = 1 12 V V =0 current at port 1 to the voltage at port 2 when 2 1 port 1 is shorted. I y = 2 y21 is a transfer impedance. It is the ratio of the 21 V V =0 1 2 current at port 2 to the voltage at port 1 when port 2 is shorted. I y = 2 y22 is the admittance seen looking into port 2 22 V V =0 2 1 when port 1 is shorted.* notes
Two Port NetworksZ parameters: Example 1 Given the following circuit. Determine the Z parameters. I1 I2 8 Ω 10 Ω + + V 1 20Ω 20 Ω V 2 _ _ Find the Z parameters for the above network.
Two Port Networks Z parameters: Example 1 (cont 1)For z11: For z22: Z11 = 8 + 20||30 = 20 Ω Z22 = 20||30 = 12 Ω I1 I2 8 Ω 10 ΩFor z12: + + V 1 20Ω 20 Ω V 2 V z = 1 _ _ 12 I I =0 2 1 20 xI 2 x 20 Therefore: 8 xI 2V1 = = 8 xI 2 z 12 = =8 Ω = z 21 20 + 30 I2
Two Port NetworksZ parameters: Example 1 (cont 2) The Z parameter equations can be expressed in matrix form as follows. V1 z 11 z 12 I 1 V = z I z 22 2 2 21 V1 20 8 I 1 V = I 2 8 12 2
Two Port NetworksZ parameters: Example 2 (problem 18.7 Alexander & Sadiku) You are given the following circuit. Find the Z parameters. I1 I2 1Ω 4 Ω + + + 2 Ω 1Ω Vx V 1 V 2 - 2Vx _ _
Two Port Networks Z parameters: Example 2 (continue p2) I1 I2 V 1Ω 4 Ω z = 1 11 I I =0 + + 1 2 + 2 Ω 1Ω Vx V 1 V 2 - V V + 2V x 6V x + V x + 2V x 2VxI1 = x + x = _ _ 1 6 6 3V xI1 = ; but V x = V1 − I 1 Other Answers 2 Z21 = -0.667 ΩSubstituting gives; 3(V1 − I 1 ) V1 5 Z12 = 0.222 Ω I1 = or = z11 = Ω 2 I1 3 Z22 = 1.111 Ω
Two Port NetworksTransmission parameters (A,B,C,D):The defining equations are: V1 A B V2 I = C D − I 1 2 V1 V1 A= B= V2 I2 = 0 − I2 V2 = 0 I1 I1 C= D= V2 I2 = 0 − I2 V2 = 0
Two Port NetworksTransmission 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. I1 -I2 + + R 1 V 1 R 2 V 2 _ _ 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): Example (cont.) V1 = (R1 + R2)I1 + R2I2 V2 = R2I1 + R2I2 From these equations we can directly evaluate the A,B,C,D parameters. V1 R1 + R2 V1 A= = B= = R1 V2 I2 = 0 R2 − I2 V2 = 0 I1 I1 C= = 1 D= = 1 V2 I2 = 0 R2 − I2 V2 = 0Later we will see how to interconnect two of these networks together for a final answer* notes
Two Port NetworksHybrid Parameters: The equations for the hybrid parameters are: V1 h11 h12 I 1 I = h h22 V 2 2 21 V1 V1 h11 = h12 = I1 V2 = 0 V2 I1 = 0 I2 I2 h21 = h22 = I1 V2 = 0 V2 I1 = 0* notes
Two Port NetworksHybrid Parameters: The following is a popular model used to represent a particular variety of transistors. I1 K I2 1 + + + V K 2V 2 K 4 V 1 _ K 3V 1 2 _ _ We can write the following equations: V1 = AI 1 + BV 2 V2 I 2 = CI 1 + D* notes
Two Port NetworksHybrid Parameters: V1 = AI 1 + BV 2 V2 I 2 = CI 1 + D We want to evaluate the H parameters from the above set of equations. V1 V1 h11 = = K1 h12 = = K2 I1 V2 = 0 V2 I1 = 0 I2 h21 = = K3 I2 1 I1 V2 = 0 h22 = = K 4 V2 I1 = 0
Two Port NetworksHybrid Parameters: Another example with hybrid parameters. Given the circuit below. I1 -I2 The equations for the circuit are: + + R 1 V1 = (R1 + R2)I1 + R2I2 V 1 R 2 V 2 _ _ V2 = R2I1 + R2I2 The H parameters are as follows. V1 V1 h11 = h12 = I1 = R1 V2 = 1 V2=0 I1=0 I2 I2 1 h21 = = -1 h22 = = I1 V2 R2 V2=0 I1=0
Two Port Networks Modifying the two port network: Earlier we found the z parameters of the following network. I1 I2 8 Ω 10 Ω + + V 1 20Ω 20 Ω V 2 _ _ V1 20 8 I 1 V = 8 12 I 2 2* notes
Two Port NetworksModifying the two port network:We modify the network as shown be adding elements outside the two ports 6Ω I1 I2 8 Ω 10 Ω + + + 10 v V 1 20Ω 20 Ω V 2 4 Ω _ _ _ We now have: V1 = 10 - 6I1 V2 = - 4I2
Two Port Networks Modifying the two port network: We take a look at the original equations and the equations describing the new port conditions. V1 20 8 I 1 V1 = 10 - 6I1 V = 8 12 I 2 2 V2 = - 4I2 So we have, 10 – 6I1 = 20I1 + 8I2 -4I2 = 8I1 + 12I2* notes
Two Port NetworksModifying the two port network: Rearranging the equations gives, −1 I 1 26 8 10 I = 8 0 2 16 1 I 0.4545 = I 2 -0.2273
Two Port NetworksY Parameters and Beyond: Given the following network. I1 1 Ω I2 + + 1 s V 1 V s 2 _ _ 1 Ω (a) Find the Y parameters for the network. (b) From the Y parameters find the z parameters
Two Port Networks Y Parameter Example I I y = 1 y = 1 I1 = y11V1 + y12V2 11 V V =0 12 V 1 2 2 V =0 1 I2 = y21V1 + y22V2 I I I2 y = 2 y = 2 I1 1 Ω 21 V V =0 22 V V =0 1 2 2 1+ + 1 sV 1 s V 2 short_ _ We use the above equations to 1 Ω evaluate the parameters from the network. To find y11 2V1 = I 1 ( s )= I 2 so I y = 1 = s + 0.5 1 2+1 s 2s + 1 11 V 1 V =0 2
Two Port NetworksY Parameter Example I1 I2 1 Ω I y = 2 + + 21 V V =0 1 2 1 s V 1 s V 2 _ _ 1 Ω We see V1 = − 2I 2 I y = 2 = 0.5 S 21 V 1
Two Port Networks Y Parameter Example I1 I2 1 ΩTo find y12 and y21 we reverse short + + 1 s V things and short V1 1 s V 2 _ _ I y = 1 1 Ω 12 V V =0 2 1 I We have y = 2 22 V V =0 2 1 We have V2 = − 2I1 2s 1 V2 = I 2 y22 = 0.5 + ( s + 2) s I y = 1 = 0.5 S 12 V 2
Two Port NetworksY Parameter Example Summary: y11 y12 s + 0.5 − 0.5 Y = y = y22 − 0.5 0.5 + 1 s 21 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: I =Y V V =Z I −1 From above; V =Y I =Z I Therefore y −y 22 12 where z z ∆ Y ∆ Z =Y −1 = 11 12 = − y Y ∆Y = det Y z z y 21 22 21 11 ∆ ∆ Y Y
Two Port Parameter Conversions:To go from one set of parameters to another, locate the set of parametersyou are in, move along the vertical until you are in the row that containsthe parameters you want to convert to – then compare element for element ∆H z11 = h22
Interconnection Of Two Port NetworksThree ways that two ports are interconnected: ya Y parameters * Parallel yb [ y ] = [ ya ] [ ] + yb Z parameters [ z ] = [ za ] [ ] za * Series + zb zb ABCD parameters * Cascade Ta Tb [T ] = [Ta ] [Tb ]
Interconnection Of Two Port Networks Consider the following network: I1 I2 R 1 R 1 V2Find + + V1 V1 R 2 R V2 T1 2 T2 _ _ Referring to slide 13 we have; R1 + R2 R + R R1 1 2 R1 V V1 R R2 I = 2 2 1 1 1 1 − I 2 1 R2 R2
Interconnection Of Two Port Networks R1 + R2 R + R R1 1 2 R1 V V1 R R2 I = 2 2 1 −I 1 1 1 1 2 R2 R2 Multiply out the first row: R + R 2 R R + R V1 = 1 2 + 1 V + 1 2 R + R ( − I ) R R2 2 R2 1 1 2 2 Set I2 = 0 ( as in the diagram) V2 R2 2 Can be verified directly = V1 R1 + 3 R1 R2 R2 2 by solving the circuit 2
Basic Laws of Circuits End of Lesson Two-Port Networks