Subsections

Non-ideal transformer

Many simulators support non-ideal transformers (e.g. mutual inductor in SPICE). An often used model consists of finite inductances and an imperfect coupling (straw inductance). This model has three parameters: Inductance of the primary coil $ L_1$, inductance of the secondary coil $ L_2$ and the coupling factor $ k=0...1$.

Mutual inductors with two or three of inductors

This model can be replaced by the equivalent circuit depicted in figure 9.4. The values are calculated as follows.

$\displaystyle \textrm{turn ratio:}$ $\displaystyle \qquad T = \sqrt{\frac{L_1}{L_2}}$ (9.42)
$\displaystyle \textrm{mutual inductance:}$ $\displaystyle \qquad M = k\cdot L_1$ (9.43)
$\displaystyle \textrm{primary inductance:}$ $\displaystyle \qquad L_{1,new} = L_1 - M = L_1\cdot (1-k)$ (9.44)
$\displaystyle \textrm{secondary inductance:}$ $\displaystyle \qquad L_{2,new} = L_2 - \frac{M}{T^2} = L_2\cdot (1-k)$ (9.45)

Figure 9.4: equivalent circuit of non-ideal transformer
\includegraphics[width=7.5cm]{nitrafo}

The Y-parameters of this component are:

$\displaystyle Y_{11} = Y_{44} = -Y_{41} = -Y_{14}$ $\displaystyle = \frac{1}{j\omega\cdot L_1\cdot (1-k^2)}$ (9.46)
$\displaystyle Y_{22} = Y_{33} = -Y_{23} = -Y_{32}$ $\displaystyle = \frac{1}{j\omega\cdot L_2\cdot (1-k^2)}$ (9.47)
$\displaystyle Y_{13} = Y_{31} = Y_{24} = Y_{42} = -Y_{12} = -Y_{21} = -Y_{34} = -Y_{43}$ $\displaystyle = \frac{k}{j\omega\cdot\sqrt{L_1\cdot L_2}\cdot (1-k^2)}$ (9.48)

Furthermore, its S-parameters are:

$\displaystyle D = (k^2 - 1) \cdot \dfrac{\omega^2\cdot L_1\cdot L_2}{2\cdot Z_0} + j\omega L_1 + j\omega L_2 + 2\cdot Z_0$ (9.49)

$\displaystyle S_{14} = S_{41} = \frac{j\omega L_2 + 2\cdot Z_0}{D}$ (9.50)

$\displaystyle S_{11} = S_{44} = 1 - S_{14}$ (9.51)

$\displaystyle S_{23} = S_{32} = \frac{j\omega L_1 + 2\cdot Z_0}{D}$ (9.52)

$\displaystyle S_{22} = S_{33} = 1 - S_{23}$ (9.53)

$\displaystyle S_{12} = -S_{13} = S_{21} = -S_{24} = -S_{31} = S_{34} = -S_{42} = S_{43} = \dfrac{j\omega\cdot k \cdot\sqrt{L_1\cdot L_2}}{D}$ (9.54)

Also including an ohmic resistance $ R_1$ and $ R_2$ for each coil, leads to the following Y-parameters:

$\displaystyle Y_{11} = Y_{44} = -Y_{41} = -Y_{14}$ $\displaystyle = \dfrac{1}{j\omega\cdot L_1\cdot \left(1-k^2\cdot\dfrac{j\omega L_2}{j\omega L_2 + R_2}\right) + R_1}$ (9.55)
$\displaystyle Y_{22} = Y_{33} = -Y_{23} = -Y_{32}$ $\displaystyle = \dfrac{1}{j\omega\cdot L_2\cdot \left(1-k^2\cdot\dfrac{j\omega L_1}{j\omega L_1 + R_1}\right) + R_2}$ (9.56)
$\displaystyle Y_{13} = Y_{31} = Y_{24} = Y_{42} = -Y_{12}$ $\displaystyle = -Y_{21} = -Y_{34} = -Y_{43} = k\cdot\frac{j\omega\sqrt{L_1\cdot L_2}}{j\omega\cdot L_2 + R_2}\cdot Y_{11}$ (9.57)

Building the S-parameters leads to too large equations. Numerically converting the Y-parameters into S-parameters is therefore recommended.

The MNA matrix entries during DC analysis and the noise correlation matrices of this transformer are:

$\displaystyle (\underline{Y}) = \begin{pmatrix}1/R_1 & 0 & 0 & -1/R_1 \\ 0 & 1/...
...-1/R_2 & 0 \\ 0 & -1/R_2 & 1/R_2 & 0 \\ -1/R_1 & 0 & 0 & 1/R_1 \\ \end{pmatrix}$ (9.58)

$\displaystyle (\underline{C}_Y) = 4\cdot k\cdot T\cdot \begin{pmatrix}1/R_1 & 0...
...-1/R_2 & 0 \\ 0 & -1/R_2 & 1/R_2 & 0 \\ -1/R_1 & 0 & 0 & 1/R_1 \\ \end{pmatrix}$ (9.59)

$\displaystyle (\underline{C}_S) = 4\cdot k\cdot T\cdot Z_0\cdot \begin{pmatrix}...
...cdot Z_0 + R_1)^2} & 0 & 0 & \tfrac{R_1}{(2\cdot Z_0 + R_1)^2} \\ \end{pmatrix}$ (9.60)

A transformer with three coupled inductors has three coupling factors $ k_{12}$, $ k_{13}$ and $ k_{23}$. Its Y-parameters write as follows (port numbers are according to figure 9.3).

$\displaystyle A = j\omega\cdot (1 - k_{12}^2 - k_{13}^2 - k_{23}^2 + 2\cdot k_{12}\cdot k_{13}\cdot k_{23} )$ (9.61)
$\displaystyle Y_{11} = Y_{66} = -Y_{16} = -Y_{61} = \dfrac{1-k_{23}^2}{L_1\cdot A}$ (9.62)
$\displaystyle Y_{22} = Y_{33} = -Y_{23} = -Y_{32} = \dfrac{1-k_{12}^2}{L_3\cdot A}$ (9.63)
$\displaystyle Y_{44} = Y_{55} = -Y_{45} = -Y_{54} = \dfrac{1-k_{13}^2}{L_2\cdot A}$ (9.64)
$\displaystyle Y_{12} = Y_{21} = Y_{36} = Y_{63} = -Y_{13} = -Y_{31} = -Y_{26} = -Y_{62} = \dfrac{k_{12}\cdot k_{23} - k_{13}}{\sqrt{L_1\cdot L_3}\cdot A}$ (9.65)
$\displaystyle Y_{15} = Y_{51} = Y_{46} = Y_{64} = -Y_{14} = -Y_{41} = -Y_{56} = -Y_{65} = \dfrac{k_{13}\cdot k_{23} - k_{12}}{\sqrt{L_1\cdot L_2}\cdot A}$ (9.66)
$\displaystyle Y_{25} = Y_{52} = Y_{43} = Y_{34} = -Y_{24} = -Y_{42} = -Y_{53} = -Y_{35} = \dfrac{k_{12}\cdot k_{13} - k_{23}}{\sqrt{L_2\cdot L_3}\cdot A}$ (9.67)

Mutual inductors with any number of inductors

A more general approach for coupled inductors can be obtained by using the induction law:

$\displaystyle V_L = j\omega L\cdot I_L + j\omega\cdot \sum_{n=1}^N k_n\cdot\sqrt{L\cdot L_n}\cdot I_{L,n}$ (9.68)

where $ V_L$ and $ I_L$ is the voltage across and the current through the inductor, respectively. $ L$ is its inductance. The inductor is coupled with $ N$ other inductances $ L_n$. The corresponding coupling factors are $ k_n$ and $ I_{L,n}$ are the currents through the inductors.

Realizing this approach with the MNA matrix is straight forward: Every inductance $ L$ needs an additional matrix row. The corresponding element in the $ D$ matrix is $ j\omega L$. If two inductors are coupled the cross element in the $ D$ matrix is $ j\omega
k\cdot\sqrt{L_1\cdot L_2}$. For two coupled inductors this yields:

$\displaystyle \begin{bmatrix}. & . & . & . & +1 & 0\\ . & . & . & . & -1 & 0\\ ...
...\\ 0\\ 0\\ \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix}$ (9.69)

Obviously, this approach has an advantage: It also works for zero inductances and for unity coupling factors and is extendible for any number of inductors. It has the disadvantage that it enlarges the MNA matrix.

The S-parameter matrix of this component is obtained by converting the Z-parameter matrix of the component. The Z-parameter matrix can be constructed using the following scheme: The self-inductances on the main diagonal and the mutual inductances in the off-diagonal entries.

$\displaystyle \left(\underline{Z'}\right) = j\omega\cdot \begin{bmatrix}L_1 & k\cdot\sqrt{L_1\cdot L_2} \\ k\cdot\sqrt{L_1\cdot L_2} & L_2\\ \end{bmatrix}$ (9.70)

This matrix representation does not contain the second terminals of the inductances. That's why the Z-parameter matrix must be converted into the Y-parameter matrix representation which is then extended to contain the additional terminals.

$\displaystyle \left(\underline{Z'}\right) \rightarrow \left(\underline{Y'}\righ...
...} & +y_{22} & -y_{22} \\ -y_{21} & +y_{21} & -y_{22} & +y_{22} \\ \end{bmatrix}$ (9.71)

The resulting Y-parameter matrix can be converted into the appropriate S-parameters numerically by eqn. (15.7).


This document was generated by Stefan Jahn on 2007-12-30 using latex2html.