# A High Power Factor LED Driver with Intrinsic Current Balancing Capability
## 3 Mathematical Equations for Parameters Design
### Class-\(D\) Series Resonant Converter  
The equivalent circuit of the series resonant converter is shown in Figure 5a. According to the analysis of operation modes, the input voltage of the resonant converter is expressed as  
\[v_{AB}(t)=\left\{\begin{array}{ll}V_{dc},&\text{for }0<\omega t\leq\pi\\ 0,&\text{for }\pi<\omega t\leq 2\pi\end{array}\right.. \tag{23}\]  
This voltage can be expanded into Fourier series,  
\[v_{AB}(t)=\frac{V_{dc}}{2}+\sum_{n=1,\ 3,\ 5,\...}\frac{2V_{dc}}{n\pi}\sin(n \omega t). \tag{24}\]  
The root-mean-square value of the fundamental voltage of \(v_{AB}\) is given by  
\[V_{1,\ rms}=\frac{\sqrt{2}V_{dc}}{\pi}. \tag{25}\]  
Figure 4: Conceptual waveform of \(T_{1}\) primary current and input current.  
It can be known from Equations (9) and (14) that \(v_{CB}\) is also a square wave expressed as  
\[v_{CB}(t)=\left\{\begin{array}{ll}0.5(V_{o1}+V_{o3}),\;\text{for}\;\theta<\omega t \leq\theta+\pi\\ -0.5(V_{o2}+V_{o4}),\;\text{for}\;\theta+\pi<\omega t\leq\theta+2\pi\end{array} \right.. \tag{26}\]  
In the practical circuit, the number of LEDs in each string is equal, and LEDs of the same type have almost the same voltage, so  
\[V_{o1}\approx V_{o2}\approx V_{o3}\approx V_{o4}\approx V_{LED}, \tag{27}\]  
where \(V_{LED}\) represents the total voltage of each LED sting. Combining Equations (26) and (27), the Fourier expansion for \(v_{CB}\) is  
\[v_{CB}(t)\ =\ \sum\limits_{n=1,\ 3,\ 5,\...}\frac{4V_{LED}}{n\pi}\sin(2n\pi f _{s}t)\ \ n\ =\ 1,\ 3,\ 5\ldots \tag{28}\]  
The root-mean-square value of the fundamental voltage of \(v_{CB}\) is given by  
\[V_{o1,\ rms}=\frac{2\sqrt{2}V_{LED}}{\pi}. \tag{29}\]  
When a series resonant circuit has a high loaded quality factor, the current in the resonant tank would be approximately sinusoidal, and the series resonant converter can be approximately analyzed by using the fundamental wave method (FWM). The equivalent circuit for FWM analysis is shown in Figure 5b, where \(R_{o,equ}\) is the equivalent load resistance [21; 22].  
\[R_{o,equ}\ =\ \frac{V_{o1,rms}}{I_{r,rms}}. \tag{30}\]  
The load quality factor of a series resonant circuit is defined as  
\[Q_{L}\ =\ \frac{\sqrt{L_{r}/C_{r}}}{R_{o,equ}}. \tag{31}\]  
The LED current is supplied from the resonant current. At steady-state operation, the following equation must be satisfied.  
\[\frac{2}{T_{s}}\int\limits_{0}^{\frac{T_{s}}{2}}\sqrt{2}I_{r,rms}\sin(2\pi f_ {s}t)dt=4I_{LED}. \tag{32}\]  
Figure 5: Equivalent circuits (a) the series resonant converter, (b) fundamental approximated analysis.  
The following equation is obtained by solving Equation (32),  
\[I_{r,rms}=\sqrt{2}\pi I_{LED}, \tag{33}\]  
where \(I_{LED}\) represents the current in each LED string. The phasor relationship between \(V_{o1}\) and \(V_{1}\) can be expressed as follows:  
\[\vec{V_{1}}=V_{o1}\angle 0^{\circ}+jX_{s}I_{r}\angle 0^{\circ}, \tag{34}\]  
where the impedance of the resonant tank is  
\[X_{s} = 2\pi f_{s}L_{r}-\frac{1}{2\pi f_{s}C_{r}}. \tag{35}\]  
Using Equation (34), the following equation is derived:  
\[\left(V_{1,rms}\right)^{2}=\left(I_{r,rms}X_{s}\right)^{2}+\left(V_{o1,rms} \right)^{2}. \tag{36}\]  
Referring to Figure 2a,b, the following equations can be obtained.  
\[v_{CB}(t)=\left\{\begin{array}{l}V_{pri}+0.7+V_{o1}=-V_{sec}+0.7+V_{o3}\ \text{when}\ D_{o1}\text{and}\ D_{o3}\ \text{are on.}\\ V_{pri}-0.7-V_{o2}=-V_{sec}-0.7-V_{o4}\ \text{when}\ D_{o2}\ \text{and}\ D_{o4}\ \text{are on.}\end{array}\right. \tag{37}\]  
From Equation (37), the primary voltage of \(T_{2}\) can be expressed as  
\[V_{pri}(t)=V_{sec}(t)=\left\{\begin{array}{l}0.5(V_{o3}-V_{o1})\ \text{when}\ D_{o1}\text{and}\ D_{o3}\ \text{are on.}\\ 0.5(V_{o2}-V_{o4})\ \text{when}\ D_{o2}\ \text{and}\ D_{o4}\ \text{are on.}\end{array}\right. \tag{38}\]