# Jacob Shekel's method for ideal transformer

I would like to ask some questions relating to the thesis "Analysis of Linear Network" by Jacob Shekel.

From chaper III - "The Network", section 3.6 - Network Transformations, the author explains theory to deal with components such as an ideal transformer, ideal voltage divider.

Then in section 3.7 "How to Write the Y Matrix of a Given Network", he gives an example to show how to apply this theory to a circuit including two triodes and one ideal voltage divider.

I follow the example and understand how it is used with the ideal voltage divider but I am wondering how to apply the theory to an ideal transformer.

The ideal voltage divider draws no current so it has only voltage constraint. However, the ideal transformer is different. It has both voltage and current constraints. So how to use these constraints to create admittance matrix for a circuit including ideal transformer?

Hope someone could shed some light on how to apply this theory to a circuit containing an ideal transformer.

• google.ca/… – Sunnyskyguy EE75 Feb 23 '18 at 1:41
• @Tony Stewart.EE since'75: I read that long time ago. However the ideal transformer doesn't have admittance matrix. So the thesis have to use some tricks to include it into the circuit admittance matrix. – anhnha Feb 23 '18 at 2:01
• For those trying to answer the question, the pages on this thesis are from 27 to 35 when you're looking for Sections 3.6 and 3.7. – KingDuken Feb 23 '18 at 2:10
• I think embedding the section into your question would get more takers. – Andy aka Feb 23 '18 at 11:42

## 1 Answer

The ideal transformer model isn't going to do much for you here. What you really want is sometimes called the "perfect transformer model." How is this model different from an ideal transformer? It includes the effects of the inductances of the two coils and their coupling. This model is also known as the "coupled inductor model." For whatever reason, you'll be hard pressed to find the equations given below in such a succinct form in most discussions on transformers, even though the equations are really pretty simple and are much better than the ideal equations, since they capture the basic frequency behavior of a transformer without undue complication. The equations are also easy to remember.

In the "perfect" version of the coupled inductor model, there are no parasitics, (no series winding resistance, no core loss, etc.) just the coupled inductors. As such, there are only two simple equations needed to express this model:

$$\begin{array}{l}{V_1} = {Z_1}\,{i_1} - {Z_m}\,{i_2}\\{V_2} = {Z_m}\,{i_1} - {Z_2}\,{i_2}\end{array}$$

The "1" and "2" subscripts represent coil 1 (usually the primary) and coil 2 ( the secondary), respectively.

In matrix form, these become:

$$\left[ {\begin{array}{*{20}{c}}{{V_1}}\\{{V_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{Z_1}}&{ - {Z_m}}\\{{Z_m}}&{ - {Z_2}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{i_1}}\\{{i_2}}\end{array}} \right]$$

The signs on the currents as given are for when you have the current at the dotted terminal on the primary side flowing into the transformer, and the current at the dotted terminal on the secondary side flowing out of the transformer. These are in some sense the "natural" directions for the currents.

If instead you have both currents flowing into the transformer at the dotted terminals, then you can dispense with the minus signs in the above equations, although in simulation, you'll find one of the currents always coming out negative relative to the other.

The Z's are impedances as follows:

$$\begin{array}{l}{Z_1} = s\,{L_1} = j\omega {L_1}\\{Z_2} = s\,{L_2} = j\omega {L_2}\\{Z_m} = s\,M = j\omega M\end{array}$$

Here, L1 and L2 are the inductances of the primary and secondary coils, respectively, and M is the mutual inductance for the transformer. It's defined in terms of the other inductances as:

$$M = k\sqrt {L1\,L2}$$

where k is the coupling coefficient with value between 0 (no coupling) and 1 (perfect coupling).

Note: Even though we call the model "perfect" it actually allows for imperfect coupling. It's just "perfect" in the sense there are no parasitics. And it's not like parasitics can't be folded in, it just makes the equations more complicated.

In terms of turns ratio, in the perfect transformer model, there isn't a simple relationship between turns ratio and voltage and current ratios. Since the model includes inductances, that means the ratios vary with frequency. At high enough frequencies, though, the voltage and current ratios are related to the turns ratio to a good approximation.

You can relate the number of turns with the inductances by observing that the coil inductances will be proportional to the turns squared, or:

$$\begin{array}{l}{L_1} = {c_1}{N_1}^2\\{L_2} = {c_2}{N_2}^2\end{array}$$

where c1 and c2 are "coil constants" relating to the geometry of the coils (not including the number of turns which is spelled out separately above.) As an example, for an ideal solenoid (tightly wrapped with cross sectional radius much smaller than the length of the coil, the inductance formula is:

$$L = {\mu _0}{\mu _r}\frac{{{N^2}A}}{len}$$

where ${\mu_r}$ is the relative permeability of the coil's core, A is the cross section area, and len is the coil's length (not the wire length, the other length).