1. Two-port networks Adam Wąż
lecture based on J.D. Irwin, R.M. Nelms Basic Engineering Circuit Analysis

2. Introduction The learning goals:
Know how to calculate admittance, impedance, hybrid, and transmission parameters for two-port networks
Be able to convert between admittance, impedance, hybrid, and transmission parameters
Understand the interconnection of two-port networks to form more complicated networks
Knowing the typical configuration two-port networks
Know how to calculate the parameters of terminated two-port networks

3. Introduction
Two port networks – any linear network which has one input port and one output port.
Two port networks – there is not any coupling between input and output outside of network!
Two port networks – is customary to label the voltages and currents; the upper terminals are positive with respect to the lower terminals, the currents are into the two-port at the upper terminals, and, because KCL must be satisfied it at each port, the current is out of the two-port at the lower terminals.

4. Network equations
If the network is linear and contains no independent sources, the principle of superposition can be applied to determine the current I1:
y11 and y12 constants of proportionality with units of Siemens [S].
The equations that describe the two-port network:
or in a matrix:

5. Network equations Impedance Z parameters Admittance Y parameters
Transmission
T (A,B,C,D)
parameters
Hybrid H parameters

6. Network equations – admittance parameters
Determination the Y parameters:
For example, the y11 is equal to I1 divided by V1 with the output short-circuited (V2 = 0).
The group of Y parameters are referred to as the short-circuit admittance parameters
Applying the preceding definitions, these parameters could be determined experimentally for a two-port network whose actual configuration is unknown.

7. Admittance parameters - example
Determine the Y parameters for the two-port network. Calculate the current in a 4 W load, which is connected to the output port when a 2 A current source is applied at the input port.

8. Admittance parameters - example
These equations are simply the nodal equations for the network.

9. Network equations – impedance parameters
Determination the Z parameters:
Z parameters are called open-circuit impedance parameters.
Example
Find the Z parameters for the network. Find the current in a 4 W resistor that is connected to the output terminals when a 12ej0 source with an internal impedance 1+j0 W of is connected to the input.

10. Impedance parameters - example
These equations are the mesh equations for the network!

11. Network equations – hybrid parameters
Determination the H parameters:
H parameters are especially important in transistor’s amplifier analysis.

12. Hybrid parameters - example
Determine the hybrid parameters for equivalent circuit for the op-amp in non-inverter configuration

13. Network equations – transmission parameters
Determination the T parameters:
A - open-circuit voltage ratio
B – negative short-circuit transfer impedance
C - open-circuit transfer admittance
D – negative short-circuit current ratio
The transmission parameters are commonly referred to as the ABCD or T parameters.

14. Transmission parameters - example
Determine the transmission parameters for the network

15. Parameter conversions
If all the two-port parameters for a network exist, it is possible to relate one set of parameters to another since the parameters interrelate the variables V1, I1, V2 and I2. Z Y T H 1 2

16. Parameter conversions - example
Determine the Y parameters for a two-port if the Z parameters are:

17. Interconnection of two-ports
Interconnected two-port circuits are important because when designing complex systems it is generally much easier to design a number of simpler subsystems that can then be interconnected to form the complete system.
Series interconnection
The connection is regular if:
Then:

18. Series interconnection – checking the regularity
V=0 V1 V2
Regular
series connection
Irregular
series
connection

19. Interconnection of two-ports
Parallel interconnection
The connection is regular if:
Then:
V=0 V1 V2

20. Interconnection of two-ports
Regular
parallel
connection
Irregular
parallel
connection
Not
exist!
Examples of regular and irregular interconection was taken form lecture: Cz. Michalik „Elementy teorii czwórników”

21. Interconnection of two-ports
Cascade interconnection
Be careful with multiplication matrix. It is not reversible
Cascade interconnection is always regular

22. Each two-ports network consist of R, L, C, M elements is reciprocal!
Properties of two-ports network
Reciprocal of two-ports network
A two-ports is reciprocal if the response to excitation ratio is unchanged when the ports are interchanged:
Each two-ports network consist of R, L, C, M elements is reciprocal!
Symmetrical-reciprocal two-ports
A reciprocal two-ports can be also symmetrical if after rotation along the vertical axis (ports interchanging) the currents and voltages (on the input and output) do not change:

23. Typical configurations of two port network
Two-ports network – G configuration
That type of two port network is also called voltage divider.

24. Typical configurations of two port network
Two-ports network– T configuration

25. Typical configurations of two port network
Two-ports network– P configuration

26. Typical configurations of two port network
Two-ports network– brigde configuration

27. Typical configurations of two port network
Two-ports network– ideal transformer
Z and Y matrix not exist.
n – transformer ratio
Ideal transformer has a important property, according to T matrix:

28. Typical configurations of two port network
Two-ports network– gyrator
R – gyrator’s constant
Because DT= -1 the gyrator is not reciprocal two-ports network.
Gyrator has a property, according to T matrix:
If the capicitor is connected to the gyrator’s output:
R =Rg
U. Tietze, Ch. Schenk, „Electronic circuits ”

29. Typical configurations of two port network
Two-ports network– dependences sources
Current source voltage controled
Current source current controled
Voltage source voltage controled
Voltage source current controled

30. Equivalence of two port network
Two, with different structures two-ports network are equivalent if they have the same matrix.
In practise, the most important is designing the equivalent two-port circuit from given matrix Z or Y
Let us assume that we have a Z matrix:
Equivalent two-port circuit with one current controlled voltage source.

31. Terminated two port network
Input impedance
Z-parameter equations
From Crammer’s rules:

32. Terminated two port network
Effective voltage gain
Voltage gain

33. Terminated two port network
Zin_0 – input impedance when I2=0
Zin_c - input impedance when V2=0

34. Summary
A port is a terminal pair where energy can be supplied or extracted. A two-port network is a four-terminal circuit with the terminals paired to form an input port and an output port
Two port method applies to linear circuit with no independent sources, no initial energy storage, and net current of zero at both ports
The only accessible variables are the port voltages V1 and V2 and the port currents I1 and I2. Two-port parameters are defined by expressing two of these four port variables in terms of other two variables
The most often used two-port parameters are the impedance, admittance, hybrid and transmission parameters. Each set of two-port parameters defines two simultaneous linear equations in the port variables
In general, two-port parameters are network functions of the complex frequency variable s. Setting s=jw yields the sinusoidal steady state values of two-port parameters. Setting s=0 yields the dc values of two port parameters.
The two-port is reciprocal and symmetrical if the voltage response observed at the port due to a current excitation at the other port is unchanged when response and excitation ports are interchanged
Every two-port parameter can be calculated or measured by applying excitation at one of the ports and connecting a short circuit or an open circuit at the other port.

35. Thank You