# Model a nonideal transformer in Spice

-June 05, 2000

Designers often use transformers as voltage, current, and impedance adapters. Transformers usually comprise two inductively coupled coils, wound around a ferrite core. The coupling between the windings is never perfect. Spice provides a model (Figure 1a) of the coupled inductors using the k parameter, which is the coefficient of coupling between the windings. The model takes into account self and mutual inductances. With nonideal transformers, the problem is to determine k. Figure 1b shows a proposed equivalent circuit of a nonideal transformer, in which the conduction losses in the windings and the core losses are assumed to be negligible. LS is the equivalent leakage inductance of the transformer, LP is its magnetization inductance, and T is an ideal transformer (k=1) with transformation ratio equal to n. To obtain equivalence between the two circuits in Figure 1, we consider the equations describing these circuits. For the circuit in Figure 1a, the expressions are

For the circuit in Figure 1b, the equations are

Comparing the two systems and considering M=k(L1L2)½, we obtain

Then, if you know the LP and LS values, you also know the coupling factor, k. L1 is the inductance measured at the operating frequency between terminals In1 and In2 with no load connected between Out1 and Out2. Similarly, L2 is the inductance measured at the operating frequency between terminals Out1 and Out2 with no load connected between In1 and In2. LS is the inductance measured at the operating frequency between terminals In1 and In2 with Out1 and Out2 short-circuited. From these values, using the previous equations, we obtain the parameters of the equivalent circuit in Figure 1b. Listing 1 shows the PSpice subcircuit that represents the behavioral model of a nonideal transformer. You can use the subcircuit for both transient and ac analysis.

The input parameters of the subcircuit are the measured values of inductances L1, L2, and LS. You obtain the ideal transformer, T, by means of the voltage-controlled voltage source, E1, and the voltage-controlled current source, G1, connected back to back (Reference 1). The current source, G1, senses the current, I(Vsense), and provides the current I(Vsense)/n. The transformation ratio n is a function of inductances L1, L2, and LS. As an example, consider a transformer that provides an impedance transformation of 46 to 75W at 72 kHz. It uses an RM8 ferrite core with inductance factor AL=1600 nH. The measured inductances are L1=4.2 mH, L2=2.6 mH, and LS=20 µH. Figure 2 shows the simulated transfer function of the transformer. Click here to download Listing 1. (DI #2539).

REFERENCE

1. Coelho, J, "A Spice model for the ideal transformer," Electronic Design, June 28, 1999.