An ideal transformer is a pair of 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 inductors 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 on 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 enters a dotted terminal while the other leaves its dotted terminal, then $n_1 i_1 = n_2 i_2$. If both currents either enter or both leave the dotted terminals, then $n_1 i_1 = -n_2 i_2$.
• If both voltages are taken positive at the dotted terminals (or both at the 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 notation in Fig. 1, the sign on 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 AC circuits that contain ideal transformers. In addition, when the primary and secondary windings are not electrically connected, an alternative technique called the transformer elimination method can be used to simplify the analysis.

The nodal analysis method follows the same algorithm as in DC and AC circuits. The only changes for circuits with ideal transformers are highlighted 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 note that in circuits with transformers whose primary and secondary windings are unconnected, you need to select multiple reference nodes in order to uniquely determine 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, $c$ control-variable equations, and $2t$ transformer equations). It is good practice to write the $n+c+2t$ equations in the order specified below:

Step 4. Solve the system of 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 outer 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 outer 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 to write the $n+c$ equations in the order specified below:

Step 4. Solve the system of 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 when there are no electrical connections between the primary and secondary windings. In this case, the transformer can be removed and the values of components on one side of the transformer can be scaled to account for the turns ratio. When using the transformer elimination method, follow these steps:

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 on the right-hand side of the previous equation is positive if the dotted terminals are in the same direction and negative if the dotted terminals are in opposite directions. For instance, in the network shown in Fig. 3, the scaling factor is negative because the dotted terminals are in opposite directions.
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 scaled 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 is to use the Norton and Thévenin theorems to simplify the circuit. Deriving equivalent circuits can include the ideal transformer or not, and may help produce a simpler circuit that is easier to analyze. If the equivalent circuits include the ideal transformer, pay attention to correct scaling of the different components (done the same way as in the transformer elimination method).