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Transformers#

Transformers or positive immittance converters (PICs) are immittance converters which allow us to transform the behavior of an electronic component, whilst retaining both its voltage polarity and its current polarity.

Theoretical Model#

Definition: Transformer

A transformer is a time-invariant two-port for which there exists some \(n \in \mathbb{R}\) with the following property:

\[\left\vert\begin{aligned}v_1 &= n v_2 \\i_1 &= -\frac{1}{n} i_2\end{aligned}\right.\]

We call \(n\) the turn ratio and usually write it as \(n = \frac{N_1}{N_2}\) for some integers \(N_1, N_2\).

Notation

The following symbols are used for transformers:

Transformer Symbol

Theorem: Strict Linearity

Every transformer is strictly linear.

Proof

TODO

Theorem: Implicit Representation

If a transformer has turning ratio \(n\), then it has the following implicit representation:

\[\begin{bmatrix}1 & -n \\ 0 & 0\end{bmatrix}\boldsymbol{v} + \begin{bmatrix}0 & 0 \\ n & 1\end{bmatrix}\boldsymbol{i} = \boldsymbol{0}\]
Proof

TODO

Theorem: Explicit Representations

If a transformer has an admittance representation

\[\boldsymbol{i} = \boldsymbol{G}\boldsymbol{v},\]

then \(\boldsymbol{G}\) is zero.

If a transformer has an admittance representation

\[\boldsymbol{v} = \boldsymbol{R}\boldsymbol{i},\]

then \(\boldsymbol{R}\) is zero.

Every transformer has a hybrid representation:

\[\begin{bmatrix}v_1 \\ i_2\end{bmatrix} = \boldsymbol{H} \begin{bmatrix}i_1 \\ v_2\end{bmatrix} \qquad \boldsymbol{H} = \begin{bmatrix}0 & n \\ -n & 0\end{bmatrix}\]

Every transformer has an inverse hybrid representation:

\[\begin{bmatrix}i_1 \\ v_2\end{bmatrix} = \boldsymbol{H}' \begin{bmatrix}v_1 \\ i_2\end{bmatrix} \qquad \boldsymbol{H}' = \begin{bmatrix}0 & -\frac{1}{n} \\ \frac{1}{n} & 0\end{bmatrix}\]

Every transformer has a forwards transmission representation:

\[\begin{bmatrix}v_1 \\ i_1\end{bmatrix} = \boldsymbol{T} \begin{bmatrix}v_2 \\ -i_2\end{bmatrix} \qquad \boldsymbol{T} = \begin{bmatrix}n & 0 \\ 0 & \frac{1}{n}\end{bmatrix}\]

Every transformer has a backwards transmission representation:

\[\begin{bmatrix} v_2 \\ i_2 \end{bmatrix} = \boldsymbol{T}'\left(\begin{bmatrix}v_1 \\ -i_1\end{bmatrix}\right) \qquad \boldsymbol{T}' = \begin{bmatrix}\frac{1}{n} & 0 \\ 0 & n\end{bmatrix}\]
Proof

TODO

Theorem: Losslessness of the Transformer

Every transformer is lossless.

Proof

TODO

Theorem: Reciprocity of the Ideal Transformer

Every transformer is reciprocal.

Proof

TODO

Theorem: Symmetry Condition for the Transformer

A transformer is symmetrical if and only if its turn ratio \(n\) is \(1\) or \(-1\):

\[n = \pm 1\]
Proof

TODO

Theorem: Transformer as a Port Enforcer

Chaining a 4-terminal network \(E\) with two \(1:1\) transformers results in a two-port:

Transformer as Port-Enforcer

Proof

TODO

Multiport Transformer#