An ideal transformer are two coupled inductors with zero internal resistance, in which the magnetic flux generated by one inductor is entirely linked to the other inductor. One of the inductor is usually called the primary inductor and the other is called the secondary inductor. However, since an ideal transformer is a perfectly symmetric device (remember that the mutual inductance of two magnetically coupled inductors satisfies $M=L_{12}=L_{21}$!), this labeling is somewhat arbitrary. In practice, the primary inductor is usually the one connected to the input voltage source, while the secondary inductor is the one connected to the load (like in Fig. 1).

The turns ratio of the transformer is the ratio of the number of turns of the primary to secondary coils and is usually denoted by $n_1:n_2$. This ratio is an important parameter of the transformer because it determines the ratio of the AC voltages and AC currents between the primary and secondary inductors. For any ideal transformer, it can be shown that currents and the voltages in the primary and secondary windings satisfy the following transformer equations $$\begin{equation}n_1 \cdot i_1=\pm n_2 \cdot i_2\end{equation}$$ $$\begin{equation}\frac{v_1}{n_1}=\pm \frac{v_2}{n_2}\end{equation}$$ where the sign in the right-hand-side of the above equations is positive if the windings are in the same direction and negative if the windings are in opposite directions. Therefore, similar to our discussion from mutually coupled inductors, it is important to use the dot convention to indicate the direction of two windings when representing ideal transformers in electric circuits:

• If one current is defined as entering a dotted terminal and the other current is defined as leaving a dotted terminal, then $n_1 i_1=n_2 i_2$. Otherwise (i.e. if the currents are either both entering or both exiting the dotted terminals), $n_1 i_1=-n_2 i_2$.
• If both voltages are referenced positive at the dotted terminals or undotted terminals, then $\frac{v_1}{n_1}=\frac{v_2}{n_2}$. Otherwise, $\frac{v_1}{n_1}=-\frac{v_2}{n_2}$.

For instance, using the dot notations in Fig. 1, the sign in the right-hand-side of the transformer equations is negative because both currents $i_1$ and $i_2$ enter the dotted terminals.

The nodal and mesh analysis methods described in the previous chapters can be applied to the analysis of AC circuits with ideal transformers, as described in this section. Moreover, if there are no electrical connections between the primary and secondary windings of the transformers, an additional method called the transformer elimination method can be used to analyze the circuit.

The nodal analysis method follows the same algorithm and in the case of DC and AC circuits. The only changes that appear in the case of circuits with ideal transformers are highligheted below.

Assume we have a circuit with $n$ nodes (excluding the ground nodes), $m$ voltage sources, $c$ control variables, and $t$ transformers. Denote the currents going through the primary and secondary windings by $i_1$, $i_2$, $i_3$,...,$i_{2t}$ (indexes $1$ and $2$ refer to the first transformer, $3$ and $4$ refer to the second transformer, etc.)

Step 1. Identify the nodes in the circuit and select the reference nodes. The reference nodes are treated as ground nodes. Please notice that in circuits transformers in which the primary and seoncary windings are unconnected, you need to select multiple nodes in order to uniquely compute the potentials in the circuit.

Step 2. Label the potentials at each of the $n$ nodes with $v_1$, $v_2$, ..., $v_n$.

Step 3. Write the system of nodal analysis equations, which will have $n+c+2t$ equations ($m$ voltage constrained equations, $n-m$ KCL equations, and $c$ equations for the control variables, and $2t$ equations for transformers). It is a good practice two write the $n+c$ equations in the order specified below:

Step 4. Solve the system on nodal analysis equations to compute the $n$ nodal potentials, $c$ control variables, and $2t$ transformer variables (currents).

Step 5. Compute the sought variables.

The mesh analysis method can be applied by using the same algorithm and in the case of DC and AC circuits.

Assume we have a circuit with $n$ meshes (excluding the outside mesh), $m$ current sources, $c$ control variables, and $t$ transformers. Denote the voltages across the primary and secondary windings by $v_1$, $v_2$, $v_3$,...,$v_{2t}$ (indexes $1$ and $2$ refer to the first transformer, $3$ and $4$ refer to the second transformer, etc.)

Step 1. Identify the meshes in the circuit. The outside mesh is selected as a reference mesh.

Step 2. Label the mesh current of the $n$ meshes with $i_1$, $i_2$, ..., $i_n$.

Step 3. Write the system of mesh analysis equations, which will have $n+c+2t$ equations ($m$ current constrained equations, $n-m$ KCL equations, and $c$ equations for the control variables, and $2t$ equations for transformers). It is a good practice two write the $n+c$ equations in the order specified below:

Step 4. Solve the system on mesh analysis equations to compute the $n$ mesh currents, $c$ control variables, and $2t$ transformer variables (voltages).

Step 5. Compute the sought variables.

The transformer elimination method can be used only when there are no electrical connections between the primary and secondary windings of the transformers. In this case, the transformer can be eliminated from the circuit by removing it and scalling the values of all the components in either the primary side or the secondary side of the transformer. When using the transformer elimination method we:

1. Select one side of the network that will be modified (let's call it the modified region). We usually want to leave the side of the transformer that contains the sought variables untouched and modify only the other side.
2. Compute the scaling factor $$\begin{equation}k=\pm \frac{n_2}{n_1}\end{equation}$$ where the sign in the right-hand-side of the previous equation is positive if the dotted terminals are on the same direction and negative if the dotted terminals are on oposite directions. For instance, in the network shown in Fig. 3, the sclaling factor is negative because the dotted terminals are in oposite direction.
3. Remove the transformer and modify the values of the components in the modified region as follows:
   • If the modified region is the primary side of the transformer, the values of the impedances are multiplied by $k^2$, the values of the voltage sources are multiplied by $k$, while the values of the current sources are divided by $k$.
   • If the modified region is the secondary side of the transformer, the values of the impedances are divided by $k^2$, the values of the voltage sources are divided by $k$, while the values of the current sources are multiplied by $k$.
4. Use any technique such as nodal and mesh analysis, superposition, circuits simplification, etc. to compute the values of the sought variables. Note that since the sought variables appear in the region that is left unmodified, the computed values are the real ones. If the sought variables had appeared in the modified region, the computed values had to be sclaed back to their original values.

For instance, let us consider the circuit shown in Fig. 3 - for simplicity we have not indicated any sought variables. The bottom two circuits are equivalent to the first circuit. The devices that were changed are highlighted with red.

An alternative approach would be to use the Norton and Thévenin theorems to simplify the circuit. The calculation of the equivalent circuits can include or not the ideal transformer and help deriving a simpler circuit that can be analyzed easier. If the equivalent circuits include the ideal transformer, special emphasis needs to be paid to the correct scaling of the differenet components in the circuit (which is done in the same way as when using the Transformer Elimination Method).