# LED Current Balance Using a Variable Voltage Regulator with Low Dropout \(v_{DS}\) Control
## 2 Aspect of Linear Current Regulator for LEDs
### LED String Module in LCR  
Bias Situation for a Single String LED  
An LED is essentially a family of diodes, whose current is temperature-dependent and varies, approximately in exponential form, in its forward bias [19]. The relation of the general LED current \(i_{D}\) and the voltage \(v_{D}\) can be simply expressed by the Shockley equation; for a single LED, this is presented as:  
\[i_{D}(v_{D})=I_{s}(e^{\frac{v_{D}}{\Psi T}}-1), \tag{2}\]  
and  
\[v(i_{D})=v(i_{DS})=\eta V_{T}\ln(\frac{i_{DS}+I_{s}}{I_{s}}), \tag{3}\]  
where \(I_{s}\) is the leakage current in reverse bias, \(V_{T}\) is the thermal voltage, \(\eta\) is the ideality factor, \(v_{D}\) is the forward voltage, and \(v_{DS}\) is the drain-to-source voltage of MOSFET. For a number of \(N\) LEDs connected in a series to form the string \(N\)-LED, the voltage \(v_{DN}\) of the forward bias can be given by:  
\[ \begin{array}{l}v_{DN}\equiv Nv_{D}=\eta NV_{T}\ln(\frac{i_{D}+I_{s}}{I_{s} })\\ =\eta NV_{T}\ln(\frac{i_{DS}+I_{s}}{I_{s}})\end{array} , \tag{4}\]  
where \(i_{DS}=i_{D}\) is due to the series connection of \(N\)-LED, and \(N\) is an integer. With the assumption of the diode ideality factor \(\eta=1\) in standard fabrication, and if \(i_{D}>>I_{s}\), we can simplify Equation (4) as:  
\[ \begin{array}{l}v_{DN}\approx NV_{T}\ln(\frac{i_{D}}{I_{s}})\\ \approx NV_{T}\ln(\frac{i_{DS}}{I_{s}})\end{array} , \tag{5}\]  
In Figure 2a, the loop equation for the LCR loaded with the string \(N\)-LED can be given by:  
\[V_{dc}=v_{DS}+Nv_{D}+i_{DS}R_{s}, \tag{6}\]  
Figure 2: (**a**) Typical linear current regulator (LCR) loaded with a string LED; (**b**) Small-signal model of the LCR. where \(V_{dc}\) is the dc supply voltage. The drain-to-source voltage of the MOSFET can be given by:  
\[v_{DS} =V_{dc}-Nv_{D}-i_{DS}R_{s} \tag{7}\] \[=V_{dc}-v\]  
where we let \(v\) be the voltage sum, i.e.,  
\[v=Nv_{D}+i_{DS}R_{s}, \tag{8}\]  
Form (6), the drain current of the MOSFET in the LCR can also be given by  
\[ \begin{array}{l}i_{DS}=f(v_{DS})\\ =\frac{1}{R_{s}}(V_{dc}-Nv_{D}-v_{DS})\end{array} , \tag{9}\]  
For the convenience of analysis, point \(M\) is set to measure the drain voltage of the MOSFET, \(v_{M}\), and Equation (6) can then be modified as:  
\[ \begin{array}{l}v_{M}=V_{dc}-Nv_{D}\\ =v_{DS}+i_{DS}R_{s}\end{array} , \tag{10}\]  
If a small variation of \(v_{D}\) is neglected, \(V_{M}\) instead of \(v_{M}\) is given by:  
\[ \begin{array}{l}V_{M}=V_{dc}-NV_{D}\\ =V_{DS}+l_{DS}R_{s}\end{array} , \tag{11}\]  
Equation (11) represents a LCR loaded with one string \(N\)-LEDs and supplied by a voltage \(V_{M}=V_{dc}-NV_{D}\). The combined characteristics of the one string \(N\)-LED and MOSFET of the LCR for bias and load-line descriptions, is shown in Figure 3a. Figure 3: Predicted bias situation of a LCR loaded with one string \(N\)-LEDs by a constant voltage \(V_{dc}\) supply: (**a**) Combined characteristics of the one string \(N\)-LED and MOSFET of a LCR for bias and load-line descriptions; (**b**) Bias situation of the LCR for \(T_{+}>T_{0}>T_{-}\). In this study, only the three kinds of temperature variations \(T_{+}>T_{0}>T_{-}\) for the string \(N\)-LED are considered. Because of temperature-dependence, the voltage change of the \(N\)-LED will lead to a change in \(V_{M}\), resulting in a move of the load line of MOSFET on the \(i_{DS}\)-\(v_{DS}\) plane, which is clearly explored in Figure 3a. Correspondingly, the characteristics of \(N\)-LED on the \(i_{D}\)-\(v_{D}\) plane for the mentioned temperature variation are also depicted. In addition, the bias situation in the LCR with a constant \(V_{dc}\) supply is predictively shown in Figure 3b under temperature variation \(T_{+}>T_{0}>T_{-}\), including the sense voltage \(v_{Rs}=i_{DS}R_{s}\), \(Nv_{D}\), \(v_{DS}\), and the voltage sum \(v\) etc., all are presented with respect to the current \(i_{DS}=i_{D}\). Ambient Temperature at \(T=T_{0}\)  
In this situation, \(T=T_{0}\), we presume that the MOSFET is initially biased at \(v_{GS2}\).