Presentation on theme: "The University of Tennessee Electrical and Computer Engineering"— Presentation transcript:
1 The University of Tennessee Electrical and Computer Engineering An Introduction ToTwo – Port NetworksThe University of TennesseeElectrical and Computer EngineeringKnoxville, TNThese notes were first prepared during the Spring Semester of2002 in support of the ECE 202 course which will be changedto ECE 300 in the Fall Semester of The material on twoPorts will be the same.wlg
2 Two Port Networks Generalities: The Network The standard configuration of a two port:I1I2The Network++InputPortOutputPortV1V2__The two port network configuration shown withthis slide is practically used universally in all textbooks. The same goes for the voltage and currentassumed polarity and direction at the input andoutput ports. I think this is done because of howwe define parameters with respect the voltagesand currents at the input and out ports.With respect to the network, it may be configuredwith passive R, L, C, op-amps, transformers, dependent sources but not independent sources.The network ?The voltage and current convention ?* notes
3 Two Port Networks Network Equations: V1 = z11I1 + z12I2 V2 = b11V1 - b12I1I2 = b21V1 – b22I1ImpedanceZ parametersI1 = y11V1 + y12V2I2 = y21V1 + y22V2V1 = h11I1 + h12V2I2 = h21I1 + h22V2AdmittanceY parametersHybridH parametersThe 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.TransmissionA, B, C, DparametersV1 = AV2 - BI2I1 = CV2 - DI2I1 = g11V1 + g12I2V2 = g21V1 + g22I2* notes
4 Two Port Networks Z parameters: z11 is the impedance seen looking into port 1when port 2 is open.z12 is a transfer impedance. It is the ratio of thevoltage at port 1 to the current at port 2 whenport 1 is open.z21 is a transfer impedance. It is the ratio of thevoltage at port 2 to the current at port 1 whenport 2 is open.Z11 and z22 are easy to remember. We look intothe input port and calculate the impedance withthe output terminals open, if the circuit iscontains only R, L. C elements.We do the same at the output port to find z22.However, if the network contains dependentSources, transformers, or op-amps, we mustfind V1/I1 with I2 = 0. Similarly for z22For z12 and z21 we still find ratios of port voltageTo 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 2when port 1 is open.* notes
5 Two Port Networks Y parameters: y11 is the admittance seen looking into port 1when port 2 is shorted.y12 is a transfer admittance. It is the ratio of thecurrent at port 1 to the voltage at port 2 whenport 1 is shorted.y21 is a transfer impedance. It is the ratio of thecurrent at port 2 to the voltage at port 1 whenport 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 theoutput terminals.If the circuit is not passive, we need to be carefuland do exactly as the equations tell us to do.y22 is the admittance seen looking into port 2when port 1 is shorted.* notes
6 Two Port Networks Z parameters: Example 1 Given the following circuit. Determine the Z parameters.Find the Z parameters for the above network.
7 Two Port Networks Z parameters: = Example 1 (cont 1) For z11: For z22: Therefore:=
8 Two Port Networks Z parameters: Example 1 (cont 2)The Z parameter equations can be expressed inmatrix form as follows.
9 Two Port Networks Z parameters: Example 2 (problem 18.7 Alexander & Sadiku)You are given the following circuit. Find the Z parameters.
10 Two Port Networks Z parameters: but Z21 = -0.667 Substituting gives; Example 2 (continue p2);butOther AnswersZ21 = Z12 = Z22 = Substituting gives;or
11 Two Port Networks Transmission parameters (A,B,C,D): The defining equations are:I2 = 0V2 = 0V2 = 0I2 = 0
12 Two Port Networks Transmission parameters (A,B,C,D): Example Given the network below with assumed voltage polarities andCurrent directions compatible with the A,B,C,D parameters.We can write the following equations.V1 = (R1 + R2)I1 + R2I2V2 = R2I R2I2It is not always possible to write 2 equations in terms of the V’s and I’sOf the parameter set.
13 Two Port Networks Transmission parameters (A,B,C,D): I2 = 0 V2 = 0 Example (cont.)V1 = (R1 + R2)I1 + R2I2V2 = R2I R2I2From these equations we can directly evaluate the A,B,C,D parameters.==I2 = 0V2 = 0The 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 = 0I2 = 0Later we will see how to interconnect two of these networks together for a final answerNetwork 1Network 2* notes
14 Two Port Networks Hybrid Parameters: The equations for the hybrid parameters are:V2 = 0I1 = 0The H parameters are used almost solely in electronics. These parameters are used in the equivalent circuitOf a transistor. As you will see, the H parameter equations directly set-up so as to describe the device in termsOf the hij parameters which are given by the manufacturer. You will use this material in junior electronics.V2 = 0I1 = 0* notes
15 Two Port Networks Hybrid Parameters: The following is a popular model used to representa particular variety of transistors.We can write the following equations:In this example we consider a very popular model ofa transistor. We first assign some identifier, suchas K1, K2, K3, K4, to the model. Next we write theequations at the input port and output port. In thiscase they are extremely simple to write.Then it turns out that we can use these equations todirectly identify what will be called the H parameters.Note that one of the parameters is a voltage gain, oneis a current gain, one is an resistance and one isa conductance. You will go into a fair amount ofdetail on this model later in the curriculum* notes
16 Two Port Networks Hybrid Parameters: We want to evaluate the H parameters from the above set of equations.=K1=K2V2 = 0I1 = 0=K3=V2 = 0I1 = 0
17 Two Port Networks Hybrid Parameters: Another example with hybrid parameters.Given the circuit below.The equations for the circuit are:V1 = (R1 + R2)I1 + R2I2V2 = R2I R2I2The H parameters are as follows.==1V2=0I1=0=- 1=V2=0I1=0
18 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 consideredto be the end of the problem. We will practicallyalways add components to the input and output. Whenwe do this we must start with the two port parameter,in a certain form, and modify the equations toincorporate the changes at the two ports. For example,we may place a voltage source in series with a resistorat the input port and maybe a load resistor at the output port.The example here illustrates how to handle this problem.* notes
19 Two Port Networks Modifying the two port network: We now have: We modify the network as shown be adding elements outside the two portsWe now have:V1 = I1V2 = - 4I2
20 Two Port Networks Modifying the two port network: V1 = 10 - 6I1 We take a look at the original equations and the equations describingthe new port conditions.V1 = I1V2 = - 4I2So we have,Remember that we are trying to solve for I1 andI2 after changing the port conditions. Thisinvolves doing some algebra but at a low level.We recombine terms and arrange the originalequation in matrix form and we can easilytake the inverse to find the solution or elsewe can use simultaneous equations withour hand calculators.10 – 6I1 = 20I1 + 8I2-4I2 = 8I1 + 12I2* notes
21 Two Port Networks Modifying the two port network: Rearranging the equations gives,268108160.4545
22 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
23 Two Port Networks so s + 0.5 I1 = y11V1 + y12V2 I2 = y21V1 + y22V2 Y Parameter ExampleI1 = y11V1 + y12V2I2 = y21V1 + y22V2shortWe use the above equations toevaluate the parameters from thenetwork.To find y11sos + 0.5=
24 Two Port NetworksY Parameter ExampleWe see= 0.5 S
25 Two Port Networks To find y12 and y21 we reverse things and short V1 Y Parameter ExampleTo find y12 and y21 we reversethings and short V1shortWe haveWe have= 0.5 S
26 Two Port Networks Y Y Parameter Example Summary: =Now suppose you want the Z parameters for the same network.
27 Two Port Networks Going From Y to Z Parameters For the Y parameters we have:For the Z parameters we have:From above;Thereforewhere
29 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
30 Interconnection Of Two Port Networks Three ways that two ports are interconnected:ya* Parallelybza* Serieszb* CascadeTaTb
31 Interconnection Of Two Port Networks Consider the following network:I1I2++V1V2T1T2__Referring to slide 13 we have;
32 Interconnection Of Two Port Networks Multiply out the first row:Set I2 = 0 ( as in the diagram)Can be verified directlyby solving the circuit