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# Creating a Linear Transformer Model for Circuit Simulations Transformer for 3-phase power distribution

I think every engineer hopes their systems act just as was taught in their electronics classes, but this is not always the case. Real transformers are a perfect example of this. As much as we would like to ignore parasitics and nonideal effects in circuit models, accounting for these gives you a more accurate view of your system’s behavior and helps you make the right design choices.

## Ideal vs. Nonideal Linear Transformer Model

Before getting into building a linear transformer model for a circuit simulation, we need to consider what happens in a real (i.e., nonideal) transformer, such as a flyback transformer. In an ideal transformer, the voltage/current induced in the secondary coil only depends on the ratio of turns in the primary and secondary coils. In this model, There is no magnetizing inductance or resistance, and magnetic flux is perfectly confined between the two coils. In other words, the flux produced by the primary coil is equal to flux seen at the secondary coil. As a result, the impedance of the primary coil is equal to the impedance of the secondary coil, multiplied by the turns ratio.

Unfortunately, the magnetic field produced by the primary coil is not perfectly confined in the core, thus the magnetic flux that is seen at the secondary coil is slightly less than the magnetic flux produced by the primary coil. In addition, the conductors used to form each coil have some resistance, so the voltage dropped across the primary coil is slightly reduced from the supply voltage due to the resistance in the coil. A linear transformer model accounts for this flux reduction in the derivation for the voltage/current induced in the secondary coil. As a result, the effective impedance of the secondary coil is slightly larger than the ideal case.

This is included in a circuit model by defining a coupling coefficient k, which ranges from 0 to 1. A value of k = 1 indicates perfect coupling (i.e., no flux loss), while k = 0 indicates perfect isolation (i.e., none of the flux reaches the secondary coil). Calculating the coupling coefficient directly from the geometry of the transformer can be difficult, but it is nominally defined in terms of the mutual inductance (M) between the two coils, the inductance of each coil: Coupling coefficient in a nonideal linear transformer model

As long as you can properly calculate the coupling coefficient based on the geometry of the coils in your transformer, you can get an accurate view of the behavior of a linear transformer model in your circuit. This might require using a 3D field solver, or you can determine the coupling coefficient using measurements from a real transformer.

## Linear Transformer Model and Equivalent Circuit

A simple linear transformer model in PSpice is shown in the figure below. The load on the output side is R1, and R2 and R3 define the effective series resistance of the primary and secondary coils, respectively. Each coil has self inductance L1 and L2. Note that SPICE simulations generally do not allow the user to define k = 1, although you can run simulations for 0 < k < 1 to examine how the transformer steps up/steps down the voltage in this linear transformer model. Linear transformer model in PSpice

The equivalent circuit for the linear transformer model is shown below. This model accounts for the effective series resistance in the coils and the leakage flux from the coil by including the coupling coefficient in the primary coil. The transformer on the right side of the circuit diagram is an ideal linear transformer, and adding the remaining circuit elements causes this linear transformer model to exhibit non-ideal behavior. Circuit diagram for a nonideal linear transformer model

## What a Linear Transformer Model Can’t Tell You

The fact that the linear transformer model requires a user-defined coupling coefficient should tell you something important. This model cannot account for the relative arrangement of the coils in the transformer; it requires the user to determine coupling coefficient manually before building a linear transformer model. This illustrates the advantage of using a post-layout simulation tool to account for the geometry of your transformer and its placement in your PCB.

Any linear transformer can act like a nonlinear transformer when the driving voltage is large enough. In the nonlinear regime, the behavior of a transformer deviates from the linear model in that the induced voltage and current on the secondary coil are nonlinear functions of the input voltage/current on the primary coil. As an example, the magnetic core in a transformer exhibits magnetization saturation, which depends on the frequency of the input signal. Additional losses between the primary and secondary coil also arise due to eddy currents in the transformer core and the skin effect (which converts some of the input and output power to heat).

Once the magnetic core saturates, it exhibits hysteresis as the input voltage and current oscillate. Magnetic saturation and hysteresis both distort the voltage/current waveform seen at the output from the transformer. This distortion results from the additional harmonics that are generated in the transformer core. Accounting for this behavior requires defining a transfer curve (not a transfer function, they are not the same thing) based from the magnetization curve for the core.

Any circuit simulation can quickly become complicated, but you can account for many of the nonideal aspects of many components, including a nonlinear or linear transformer model, when you use the right PCB design and analysis software package. The simulation tools in OrCAD PSpice Simulator and the full suite of analysis tools from Cadence allow you to simulate the behavior of circuits with transformers and any other components you can imagine.