Hybrid two port network models

Hybrid two port network modelling, as described by the texts, is a topic which enables one to reduce a complicated network into a simple two port network. I guess this is something which must have originated from mathematicians. I would be really grateful if someone provides me a link , introducing this topic, and endowed with mathematical rigor. I hope the demand of my question is going to be clarified in the following lines. In a two port network, the input affects the output by a Norton current source and the output affects the input by a Thevenin voltage source. Is it so that in any case the input can only affect the output by current source and the output can only affect by a voltage source , why not the other way round, the input affecting by voltage and the output affecting by current ? In case of a transistor it is clear and it holds good, but can we develop a formal and general proof for it? Definitely, I need a mathematically proven way to construct a model for a network. It will be even more appreciable if anyone can illustrate the benevolence of this network model in case of even more intricate networks.

• Replace effect by affect Aug 24 '12 at 16:46
• @stevenvh Done. Wanna answer? Aug 24 '12 at 16:53
• Some other time. I've gotta go. :-) Aug 24 '12 at 17:00

AS photon describes the 2 port parameters are determined by the Laplace parameters S11 thru S22.

There also exists an inverse relationship called y parameters.

Both have determinates D in the matrix calculations and may be used to convert from s to y and visa versa.

s parameters are more useful for cascaded networks.

y parameters are more useful for parallel networks.

Full explanation are in the links above.

There are many ways to form two-port network models.

In general, you have a voltage and current at port 1 and a voltage and current at port 2. So you have 4 variables. In general if you know two of these variables you can solve for the other two. Depending which two you know and which two you don't, you are talking about using a different type of model.

If you know both currents and want to solve for the voltages you need an impedance model:

V1 = Z11 I1 + Z12 I2

V2 = Z21 I1 + Z22 I2

If you know both voltages and want to solve for the currents you need an admittance model:

I1 = Y11 V1 + Y12 V2

I2 = Y21 V1 + Y22 V2

If you know the voltage on one port and the current on the other, and want to solve for the remaining variables, then you need a hybrid model:

V1 = H11 I1 + H12 V2

I2 = H21 I1 + H22 V2

Except for degenerate cases (where one of the model parameters goes to 0 or infinity) you can convert any of these models to any other using linear algebra.

You can even work out how to solve for both variables on one of the ports given you know both variables on the other port, for example:

V1 = A V2 + B (-I2)

I1 = C V2 + D (-I2)

Or you can invent new variables such as $a_1 = \frac{1}{2}\frac{V_1 + Z_0 I_1}{\sqrt{Z_0}}$ and $b_1 = \frac{1}{2}\frac{V_1 - Z_0 I_1}{\sqrt{Z_0}}$ for some convenient Z0. With these definitions you can use the scattering parameters or S-parameter model that is often used in rf work:

b1 = S11 a1 + S12 a2

b2 = S21 a1 + S22 a2

Different models might be more convenient for solving different types of problems, or might be more amenable to computation without loss of precision depending on the properties of the network being studied.

Is it so that in any case the input can only affect the output by current source and the output can only affect by a voltage source?

Remember that for every Norton-equivalent current source there is an equivalent Thevenin voltage source that provides the exact same output to every load, and vice versa. Only if the output admittance goes to zero in the Norton model is it impossible to find an equivalent Thevenin model. This is true whether the source in question is a fixed current source or a controlled source (like in a Z-network model).

So the hybrid model you described is just a convenient way to look at certain networks (like a BJT in common emitter configuration).

I need a mathematically proven way to construct a model for a network.

Generally you need to use some kind of behavioral model (like a SPICE model) to find what is the response of the network to each of the input variables with proper terminations on each of the ports.

For example, for a Z-network, you'd model the internals of the network with open circuits on both outputs. First you'd apply I1 with I2 equal to zero, and find the response at V1 and V2 to get two of the network parameters. Then you'd apply I2 with I1 at zero to get the other two.

• We can only do so much in an answer here. For more details you might want a book like this one (unfortunately probably out of print, I don't know the current equivalent): amazon.com/Linear-Nonlinear-Circuits-Leon-Chua/dp/0070108986 Aug 24 '12 at 17:29
• Why exactly do we need this parameters? Obviously, just doing all this calculation and equation gymnastics does not make sense in any way unless and until it is of practical interest. Could you provide situations where this parameters offer simplicity? Oct 2 '12 at 15:25
• @Primeczar, I already stated in my answer, you use the hybrid parameters "if you know the voltage on one port and the current on the other" and want to solve for the unknown variables. Oct 2 '12 at 15:57