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SIMPLIFIED RESONANT MODEL

In order to analyze this resonant circuit, a simple model is developed. The two switches provide a 50% square wave on the primary side of the transformer. This allows the inverter to be simplified as in Figure 3a. In this resonant circuit, not only do both leakage inductances from the primary and secondary sides join the resonance, but the magnetizing inductance does as well. So this resonant circuit can be simplified further as shown in Figure 3b. The equivalent resonant inductance can be calculated from (1).

(1)

(2)

This model makes it easy to analyze and design the forward resonant inverter. The following analysis and design procedures are based on this simple model.
In the practical implementation, the inductance required for the resonant circuit could be the stray inductance of the transformer winding. Cs is an external capacitor. Cp is the stray capacitance from the transformer and layout. R is the equivalent resistance of the CCFL.

(3a)

(3b)

Figure 3. Model for Forward Inverter.

CHARACTERISTICS OF SERIES-PARALLEL RESONANT CIRCUITS

There are two other variations of this inverter. In Figure 3b, if Cp is removed from the circuit, then it becomes a series resonant converter (SRC). If Cs is shorted, then it becomes a parallel resonant converter (PRC). If both Cs and Cp are used, this circuit is called a series-parallel resonant converter or LCC resonant converter.
An SRC can provide output current with low high-frequency harmonics content when adequately loaded. It does not have circulating currents between the tank and the input source because the only current path goes through the output load. However, when the output current is reduced, the filtering of the high-frequency harmonics decreases dramatically: at no load there is no attenuation. This characteristic makes the converter inadequate for CCFL application where the lamp impedance changes from several Megohms (before strike) to hundreds of ohms when lighted.
A PRC can be designed to provide good attenuation of high frequency harmonics from no load to full load. The fact that the output is in parallel with the resonant capacitor makes the converter behave as a voltage source. It can be designed to have very low load sensitivity. However, the high attenuation of the harmonics and the load insensitivity require a design with significant circulating current through the resonant capacitor. The switches have to conduct the load current plus the current through the resonant capacitor. This circuit can be used in the CCFL application too.
This LCC shows the intermediate characteristics of SRC and PRC. The advantages of LCC over PRC will be discussed later. The LCC circuit is described by the following parameters:

1. The ratio of two capacitances,
   
2. The equivalent capacitance C of C_s and C_p connected in series,
   
3. The corner frequency ω_o,
   
4. The relative angular frequency ω_n,
   
5. The characteristic impedance Z_O,
   
6. The loaded quality factor Q,
   
7. The output to input voltage gain M for LCC resonant converter can be calculated from (3).
   (3)

Figure 4 shows the output to input voltage gain with frequency and Q at A=0.5 and A=2 respectively. From Figure 4, the resonant frequencies decrease as the load increases, which prevents the circuit from operating below the resonant frequency where positive feedback would occur. This is a big advantage of the LCC resonant circuits over the PRC circuits, where the resonant frequency remains the same from no load to full load.
The normalized secondary current through L_eq can be described as (4).

(4)

Figure 5 shows the resonant inductor current with frequency variations and with loads of A=2 and A=0.5 respectively. This current reflects the conduction losses and is related to the system efficiency directly. This Figure shows that a smaller Q and a smaller A results in lower resonant inductor currents, which leads to higher efficiency. The resonant inductor current is even lower than for the PRC design for a proper selection of A. This is another advantage of LCC over PRC. However, A can't be further reduced, because in that case the converter behaves more as a SRC, resulting in insufficient attenuation for light loads. In the design example in section 4, A=0.5 is chosen.
So, in order to get higher efficiency, choose a smaller Q and a smaller A at full load and minimum input voltage, but also make sure that the switching frequency won't be lower than the resonant frequency, in which positive feedback will happen and the circuit will be out of control.

(a) A=2


(b) A=0.5
Figure 4. Output-input voltage gain vs. ω_n and Q.
(click thumbnail for larger image)

(a) A=2


(b) A=0.5
Figure 5. Resonant inductor current vs. ω_n and Q.
(click thumbnail for larger image)

LCC RESONANT TANK DESIGN

1. Transformer Design

The transformer is the most critical and bulky component in the whole circuit. All the resonant inductances come from the transformer. It needs bench measurements and modeling of the resonant circuit. Work with transformer manufacturers to get the right transformer, including leakage inductance, magnetizing inductance and turns ratio.
In order to let most of the resonant energy be transferred to the secondary side, make the leakage inductance as small as possible compared to magnetizing inductance. This factor is described by K, which is defined as (5)

(5)

The larger the K, the lower the needed turns ratio.
In CCFL application, there are a lot of parasitic capacitances from lamp to ground, which will reduce the efficiency at high operating frequency. 50~150KHz operation frequency range is preferred. So the equivalent resonant inductance L_eq from (1) shouldn't be too low.

The transformer turns ratio can be calculated by (6).

(6)

in which, V_lamp is the CCFL lamp voltage and V_in is the input voltage. M (A, ω_n , Q) is the normalized output-input gain and is determined by resonant parameters. It is described in (3) and Figure 4.

2. Choose parallel capacitance C_P and series capacitance C_S.

Parasitic capacitances from layout and transformer consist of the parallel resonant capacitance C_p. C_p is usually very small and about 10~20pF.
After C_p is decided, C_s can be chosen too. As discussed above, the ratio of C_p/C_s, which is A, affects the gain. A smaller A is preferred for higher efficiency. But A can't be further reduced, because in that case the converter behaves more as a SRC, resulting in insufficient attenuation for light loads. A=0.5 is chosen in the following design example.

Those LCC resonant tank parameters need bench measuring and tuning. The relationship between efficiency and resonant parameters is summarized in table 1.

Table 1 Relationship between efficiency and resonant parameters.

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