# Three-Winding Transformer

Model three coupled inductors for circuit envelope analysis

## Library

Elements

• ## Description

The Three-Winding Transformer block models three coupled inductors within the RF Blockset™ circuit envelope simulation environment. For an introduction to RF simulation, see the example, Simulate High Frequency Components.

The block implements the relations

`$\begin{array}{c}{v}_{1}\left(t\right)={L}_{1}\frac{d}{dt}\left[{i}_{1}\left(t\right)\right]+{M}_{12}\frac{d}{dt}\left[{i}_{2}\left(t\right)\right]+{M}_{13}\frac{d}{dt}\left[{i}_{3}\left(t\right)\right]\\ {v}_{2}\left(t\right)={M}_{12}\frac{d}{dt}\left[{i}_{1}\left(t\right)\right]+{L}_{2}\frac{d}{dt}\left[{i}_{2}\left(t\right)\right]+{M}_{23}\frac{d}{dt}\left[{i}_{3}\left(t\right)\right]\\ {v}_{3}\left(t\right)={M}_{13}\frac{d}{dt}\left[{i}_{1}\left(t\right)\right]+{M}_{23}\frac{d}{dt}\left[{i}_{2}\left(t\right)\right]+{L}_{3}\frac{d}{dt}\left[{i}_{3}\left(t\right)\right]\\ {M}_{pq}={K}_{pq}\sqrt{{L}_{p}{L}_{q}}\end{array}$`

where:

• L1, L2, and L3 represent inductances.

• Mpq represents the mutual inductance between the pth and qth inductors, with coefficient of coupling Kpq.

• v1(t), v2(t), and v3(t) represent the voltage across the terminals of the inductors at time t.

• i1(t), i2(t), and i3(t) represent the current through the inductors at time t. The block uses standard dot notation to indicate the direction of positive current flow relative to a positive voltage.

RF Blockset current and voltage signals consist of in-phase (Ik) and quadrature (Qk) components at each frequency fk specified in the Configuration block:

`$\begin{array}{c}i\left(t\right)=\sum _{\left\{{f}_{k}\right\}}\left({i}_{{I}_{k}}\left(t\right)+j\cdot {i}_{{Q}_{k}}\left(t\right)\right){e}^{j\left(2\pi {f}_{k}\right)t}\\ v\left(t\right)=\sum _{\left\{{f}_{k}\right\}}\left({v}_{{I}_{k}}\left(t\right)+j\cdot {v}_{{Q}_{k}}\left(t\right)\right){e}^{j\left(2\pi {f}_{k}\right)t}\end{array}$`

## Parameters

Inductance L1

Specify the inductance of the first inductor, L1, as a scalar value greater than or equal to `0`. Specify the units of the inductance from the corresponding drop-down list. The default value of this parameter is `1e-6` `H`.

Inductance L2

Specify the inductance of the second inductor, L2, as a scalar value greater than or equal to `0`. Specify the units of the inductance from the corresponding drop-down list. The default value of this parameter is `1e-6` `H`.

Inductance L3

Specify the inductance of the third inductor, L3, as a scalar value greater than or equal to `0`. Specify the units of the inductance from the corresponding drop-down list. The default value of this parameter is `1e-6` `H`.

Coefficient of coupling K12

Specify the coefficient of coupling for the mutual inductance of the first and second inductors, K12, as a scalar value between `0` and `1`, inclusive. The default value of this parameter is `0.9`.

Coefficient of coupling K13

Specify the coefficient of coupling for the mutual inductance of the first and third inductors, K13, as a scalar value between `0` and `1`, inclusive. The default value of this parameter is `0.9`.

Coefficient of coupling K23

Specify the coefficient of coupling for the mutual inductance of the second and third inductors, K23, as a scalar value between `0` and `1`, inclusive. The default value of this parameter is `0.9`.

### Note

The minimum nonzero inductance value that the RF Blockset environment recognizes is `1e-18` `H`. During simulation, the block uses a value of `1e-18` `H` for any inductance and mutual inductance values specified between `0` and `1e-18` `H`.