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

, inductance of the secondary coil

and the coupling factor

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
$\includegraphics[width=7.5cm]{nitrafo}$

$\displaystyle Y_{11} = Y_{44} = -Y_{41} = -Y_{14}$	$\displaystyle = \frac{1}{j\omega\cdot L_1\cdot (1-k^2)}$	(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)

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

(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

and

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)

Realizing this approach with the MNA matrix is straight forward: Every inductance

needs an additional matrix row. The corresponding element in the

matrix is $j\omega L$ . If two inductors are coupled the cross element in the

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.