An Introduction ToTwo Port NetworksThe University of TennesseeElectrical and Computer EngineeringKnoxville, TNwlg

Two Port NetworksGeneralities:The standard configuration of a two port:The Network

InputPortOutput Port+

__+V1V2I1I2The network ?The voltage and current convention ?* notes

Two Port NetworksNetwork Equations:V1 = z11I1 + z12I2

V2 = z21I1 + z22I2I1 = y11V1 + y12V2

I2 = y21V1 + y22V2V1 = AV2 - BI2

I1 = CV2 - DI2V2 = b11V1 - b12I1

I2 = b21V1 b22I1V1 = h11I1 + h12V2

I2 = h21I1 + h22V2I1 = g11V1 + g12I2

V2 = g21V1 + g22I2ImpedanceZ parametersAdmittanceY parametersTransmissionA, B, C, DparametersHybridH parameters * notes

Two Port NetworksZ 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.z22 is the impedance seen looking into port 2 when port 1 is open.

* notes

Two Port NetworksY 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.y22 is the admittance seen looking into port 2 when port 1 is shorted.

* notes

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

Two Port NetworksZ parameters:Example 1 (cont 1)For z11: Z11 = 8 + 20||30 = 20 For z22: For z12: Z22 = 20||30 = 12 Therefore:=

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

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

Two Port NetworksZ parameters:Example 2 (continue p2);butSubstituting gives;orZ21 = -0.667

Z12 = 0.222

Z22 = 1.111 Other Answers

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

Two Port NetworksTransmission parameters (A,B,C,D):ExampleGiven 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 + R2I2

V2 = R2I1 + R2I2It is not always possible to write 2 equations in terms of the Vs and Is Of the parameter set.

Two Port NetworksTransmission parameters (A,B,C,D):Example (cont.)V1 = (R1 + R2)I1 + R2I2

V2 = R2I1 + R2I2From these equations we can directly evaluate the A,B,C,D parameters.I2 = 0V2 = 0I2 = 0V2 = 0= ===Later 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:V2 = 0I1 = 0V2 = 0I1 = 0* notes

Two Port NetworksHybrid Parameters:The following is a popular model used to representa particular variety of transistors.We can write the following equations:* notes

K2K3K1Two Port NetworksHybrid Parameters:We want to evaluate the H parameters from the above set of equations.V2 = 0I1 = 0V2 = 0I1 = 0====

1- 1Two Port NetworksHybrid Parameters:Another example with hybrid parameters.Given the circuit below.The equations for the circuit are:V1 = (R1 + R2)I1 + R2I2

V2 = R2I1 + R2I2The H parameters are as follows.V2=0V2=0I1=0I1=0====

Two Port NetworksModifying the two port network:Earlier we found the z parameters of the following network.* notes

Two Port NetworksModifying the two port network:We modify the network as shown be adding elements outside the two portsWe now have:V1 = 10 - 6I1V2 = - 4I2

Two Port NetworksModifying the two port network:We take a look at the original equations and the equations describingthe new port conditions.V1 = 10 - 6I1V2 = - 4I2So we have,10 6I1 = 20I1 + 8I2

-4I2 = 8I1 + 12I2* notes

-0.22730.4545 010168826Two Port NetworksModifying the two port network:Rearranging the equations gives,

Two Port NetworksY Parameters and Beyond:Given the following network.Find the Y parameters for the network.

From the Y parameters find the z parameters

Two Port NetworksI1 = y11V1 + y12V2

I2 = y21V1 + y22V2To find y11so= s + 0.5We use the above equations toevaluate the parameters from thenetwork.Y Parameter Exampleshort

Two Port NetworksY Parameter ExampleWe see= 0.5 S

Two Port NetworksY Parameter ExampleWe have= 0.5 SshortWe haveTo find y12 and y21 we reverse things and short V1

Two Port NetworksY Parameter ExampleSummary:Y=Now suppose you want the Z parameters for the same network.

Two Port NetworksGoing From Y to Z ParametersFor the Y parameters we have:For the Z parameters we have:From above; Thereforewhere

Two Port Parameter Conversions:

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

Interconnection Of Two Port NetworksThree ways that two ports are interconnected:* Parallel* Series * CascadeyaybzazbTaTb

Interconnection Of Two Port NetworksConsider the following network:T1T2Referring to slide 13 we have;+_+_V1V2I1I2

Interconnection Of Two Port NetworksMultiply out the first row:Set I2 = 0 ( as in the diagram)Can be verified directlyby solving the circuit

End of Lesson Basic Laws of CircuitsTwo-Port Networks

These notes were first prepared during the Spring Semester of 2002 in support of the ECE 202 course which will be changedto ECE 300 in the Fall Semester of 2002. The material on twoPorts 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 configuredwith 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, 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.Z11 and z22 are easy to remember. We look intothe input port and calculate the impedance withthe 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 dependentSources, transformers, or op-amps, we mustfind V1/I1 with I2 = 0. Similarly for z22

For 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.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 carefuland 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 1Network 2

The 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.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 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 I1 and I2 after changing the port conditions. This involves 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.