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Model three coupled inductors for circuit envelope analysis

Elements

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

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

where:

*L*_{1},*L*_{2}, and*L*_{3}represent inductances.*M*represents the mutual inductance between the_{pq}*p*th and*q*th inductors, with coefficient of coupling*K*._{pq}*v*_{1}(*t*),*v*_{2}(*t*), and*v*_{3}(*t*) represent the voltage across the terminals of the inductors at time*t*.*i*_{1}(*t*),*i*_{2}(*t*), and*i*_{3}(*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
(*I _{k}*) and quadrature (

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

**Inductance L1**Specify the inductance of the first inductor,

*L*_{1}, 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,

*L*_{2}, 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,

*L*_{3}, 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,

*K*_{12}, 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,

*K*_{13}, 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,

*K*_{23}, as a scalar value between`0`

and`1`

, inclusive. The default value of this parameter is`0.9`

.

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`

.