# Harmonic Analysis of Switching Mode Transformer Using Voltage-Driven Source

In most cases, pulse width modulation and zero-current switched resonant converters can be considered as a voltage-source to the magnetic system which is always coupled to the external circuits, as shown in Figure 4.7. The current in the input circuits will be unknown, but saturation of the current waveform occurs because of the nonlinear characteristic of the magnetic core. This section discusses the HBFEM model and computed results, compared with experimental results.

## Numerical Model of Voltage Source to Magnetic System

When a transformer of power supply is excited by a voltage source, such as pulse-width modulation (PWM) converters and zero-current switched (ZCS) resonant converters,

Figure 4.2 Magnetic system with current-source excitation. (a) Magnetic configuration; (b) Hysteresis characteristic

the numerical analysis of the magnetic field should be carried out by taking account of the voltage source and the external circuits. If the excitation waveform is a square wave or a triangular wave, it can be considered as a linear combination of harmonics. Figure 4.8 shows a generalized model of voltage-source to the magnetic system, where the input current is unknown.

According to Faraday’s and Kirchhoff’s laws, the relationship between the voltage and the magnetic field for a single element can be obtained from:

where *V _{ink}* is the input voltage of circuit

*k, V*is the corresponding induced electromotive force,

_{k}*S*and

_{ck}*Z*are the cross-areas and impedance of winding k, respectively.

_{k}*C*and

_{k}*L*are the capacitance and inductance, respectively, in circuit

_{k }*k*.

Figure 4.3 The magnetic flux distribution for fundamental and third harmonic components. (a) Fundamental component; (b) Third harmonic component

Figure 4.4 The analysis model with an air gap and two slots at the central leg, and B-H curve

The induced electromotive force can be obtained based on Faraday’s law:

Figure 4.5 The experimental results compared with numerical computation results in the case of current source excitation, where ... indicates an experimental result and — indicates a numerical result. (a) Magnetic density Bi; (b) Magnetic density B_{2}; (c) Magnetic density B_{3}; (d) Excitation current density *J*

Therefore, *V _{k}* can be written as the following compact form:
where matrix [C

_{ck}] is the coefficients related to the voltage:

and *S _{ck}* and

*d*are the area of windings and the depth in the z-direction respectively. Then the input voltage can be defined:

_{0}Figure 4.6 The magnetic flux distribution for fundamental and third harmonic components. (a) Fundamental harmonic component; (b) Third harmonic component

Figure 4.7 Voltage-source to the magnetic system used for switch mode transformers

where *S _{ck}* is the area of windings. The input voltage {

*V*

_{ink}}, including all harmonic components which have a known value, is expressed as follows:

Figure 4.8 Generalized model of voltage-source to the magnetic system

where *V _{in},_{0k}* (= 0) is a DC component. The matrix [Z

_{k}] is the circuit impedance, including the resistance of windings and leakage inductance corresponding to the harmonics, and

In the series circuit connection, the impedance of external circuit or equivalent circuit of transformer at the *n*th harmonic can be expressed as:

using Galerkin’s method combined with harmonic balance techniques to discretize. The following harmonic balance matrix equation, considering only odd harmonics for a single element, can be obtained as

where the *D* and *N* matrix at no DC-biased case can be expressed as:
and harmonic matrix *N* is:

The system matrix equation related to current therefore can be rewritten as
where [G_{k}] is obtained from a single element, that is:

Combining Equations (4-11) and (4-18), the global system matrix equations for multiple input and output are obtained.

Figure 4.9 Magnetic core for a 2-D transformer structure and its B-H cure. (a) Magnetic core (b) B-H curve

where [*H]* (= [5] + [TV]) is FEM system matrix, [G] is current density-related coefficient matrix, [C_{ck}] is voltage-related coefficient matrix, and [Z] is impedance matrix. {A} and *{J _{k}}* are unknown, and can be calculated by solving the system matrix equation.

The harmonic balance FEM matrix equation with voltage driven source for a compact system matrix can be expressed as below: