In this section, the multi-string LED power *P**LED* and drive efficiency *η* were calculated to optimize the distribution of LEDs in sub-strings by adjusting *V**f[k]* *(k = 1, 2, …, n)*. And the product of *P**LED* and *η* was proposed as an optimization parameter to balance high *P**LED* and high *η*.

To illustrate the calculation process more clearly, this paper shows specific calculation examples with 380V line voltage. With linear LED driving schemes, one of the critical issues is the voltage fluctuation in power grids, which significantly impacts *P**LED* and *η*. In this paper, the input voltage *V**in* was considered within a tolerance of ± 10%. For 380V ± 10% line voltage, the corresponding *V**in* range is 420 ~ 593V, as shown in Fig. 4 (a).

*A. LED Power and Driving Efficiency*

For simplifying the calculation, suppose that the input constant current *I**in* *= I**[k]* *(k = 1, 2, …, n) = I**0* and *V**sat[k]* *(k = 1, 2, …, n)* = 0V (as *V**sat[k]* is much smaller than *V**f[k]*, which can be ignored). To achieve the maximum driving efficiency and the lowest flicker, the optimum Vf1 should be set at as same as the minimum *V**in*, so that the first LED sub-string operates continuously without being affected by *V**in* ripple, i.e., the input current is constant at *I**0*, as shown in Fig. 4 (b). Then the input power *P**in* can be calculated by Eq. (5)

$${P}_{in}= \frac{{\int }_{t=0}^{t=T}{I}_{0} {V}_{in}\left(t\right) dt}{T} \left(5\right)$$

Where *T* is the period of *V**in*, equal to 1/6 of the fundamental period of three-phase AC voltage.

Considering the symmetry of the *V**in* waveform in period, LED power *P**LED* is Eq. (6)

$${P}_{LED}=\frac{2\sum _{1}^{n}\left({\int }_{{t=t}_{n}}^{{t=t}_{n+1}}{I}_{0}{V}_{fn} dt\right)-{\int }_{t={t}_{n}}^{t={t}_{n+1}}{I}_{0} {V}_{fn} dt}{T} \left(6\right)$$

Where *t**[k]* *(k = 1, 2, …, n)* is the time when *V**in**(t) = V**f[k]* *+ V**sat[k]* *(k = 1, 2, …, n)* independently, which is also the time when *I**s[k]* *(k = 1, 2, …, n)* is turned on in sequence, and *t**n+1* is the time when *I**sn* is turned off.

Driving efficiency *η* is defined as Eq. (7)

$$\eta = \frac{{P}_{LED}}{{P}_{in}} \left(7\right)$$

*η* equals the ratio of the area under the stepped line representing *V**f[k]* *(k = 1, 2, …, n)* to the area under the arc line representing *V**in* in Fig. 4 (b). It is observed that the larger the number of LED sub-strings, the closer the ratio is to 1.

However, the number of LED sub-strings can’t be as large as possible in practice. On the one hand, the multi-string LED structure reduces the utilization ratio of LEDs, which increases the material cost of lamps; on the other hand, each MOSFET connected to the LED sub-string must be capable of withstanding high voltage, so each port needs to use a high-voltage MOS device. Multiple high-voltage MOS devices will increase the size and cost of the current regulator chip. Therefore, there is a trade-off between the total luminous flux of LEDs and the cost of multi-string LED schemes in lighting applications. This section only shows the optimization results and analysis of the double-string and triple-string LED schemes obtained by the formulas (5), (6), and (7) for reference.

*B. Optimization for Double-string LED*

In the double-string LED driving scheme, *V**f1* is set to 420V, i.e., the minimum value of *V**in*, to ensure that the first LED sub-string can be lit all the time. *V**f2* is set to 420 ~ 564V. *I**0* is set to 40mA to be consistent with the experiment.

Figure 5 (a) shows *P**LED* of the double-string LED with different *V**f2*. At *V**f2* *= V**f1* *=* 420V, it is equivalent to a single-string LED in the circuit, and there is almost no change in *P**LED* as the line voltage increases. At 420V < *V**f2* < 492V, *P**LED* increases with the increase of line voltage. At *V**f2* ≥ 492V, a breakpoint can be seen in *P**LED*, and it shifts to the right as *V**f2* increases. At this breakpoint, *V**f2* equals *Max(V**in**)*. That means that when *V**f1* *≤ Max(V**in**) < V**f2*, only the first LED sub-string operates and *P**LED* keeps constant; when *Max(V**in**) ≥ V**f2*, the second LED sub-string starts to operate and *P**LED* increases with the increase of *V**in*.

Figure 5 (b) shows *η* of the double-string LED with different *V**f2*. At 420V ≤ *V**f2* < 492V, *η* gradually decreases with the increase of line voltage. Similar to the trend of *P**LED*, the breakpoint also appears at *V**f2* ≥ 492V. Overall, *P**LED* and *η* are obviously impacted by line voltage fluctuations. *P**LED* remains stable with the decrease of *η* at small *V**f2*, while *P**LED* fluctuates significantly with the increase of *η* at high *V**f2*. Therefore, it is difficult to select *V**f2* intuitively.

Taking the balance of *P**LED* and *η* into account, the maximum average value of *P**LED* ⅹ *η* within ± 10% voltage fluctuations was selected as a candidate for optimizing *V**f2*. The optimization goal *G* in the double-string scheme can be expressed as Eq. (8)

$$G = \frac{{\int }_{ {(1-10\%)*U}_{0}}^{(1+10\%)* {U}_{0}}{P}_{LED} \left({V}_{f2} , U\right) \eta \left({V}_{f2} , U\right)dU }{{\int }_{ (1-10\%)* {U}_{0}}^{ (1+10\%)* {U}_{0}} dU} \left(8\right)$$

Where *U* is the input voltage *V**in*, *U**0* is the rated line voltage within ± 10% tolerance.

*G* rises firstly and then falls as the line voltage increases, exhibiting a parabolic-like trend in Fig. 5 (c). Within ± 10% voltage fluctuations, the maximum of *G* is achieved at optimal *V**f2* = 492V, and the corresponding *η* is 87% (@420V line voltage) ~ 95% (@365V line voltage).

*C. Optimization for Triple-string LED*

Similarly, in the triple-string LED driving scheme, *V**f1* is set to 420V and *I**0* is 40mA. *V**f3* is set to 420 ~ 564V. *V**f2* satisfies *V**f1* *< V**f2* *< V**f3*, and the optimal *V**f2* is calculated after its corresponding *V**f3* is determined.

The optimization goal G in the triple-string LED scheme is

$$G = \frac{{\int }_{ {(1-10\%)*U}_{0}}^{(1+10\%)* {U}_{0}}{P}_{LED} \left({V}_{f2} , {V}_{f3} , U\right) \eta \left({V}_{f2} , {V}_{f3} , U\right)dU }{{\int }_{ (1-10\%)* {U}_{0}}^{ (1+10\%)* {U}_{0}} dU} \left(9\right)$$

Figure 6 (a) shows the three-dimensional surface of *G* with different *V**f2* and *V**f3*. The maximum *G* is calculated at *V**f2* = 477V and *V**f3* = 522V. In the case of determining *V**f3*, the optimal *V**f2* corresponding to the maximum *G* is shown in the insets of Fig. 6 (b) and (c). The value of *V**f2* increases non-linearly with the increase of *V**f3*.

Figure 6 (b) and (c) show *P**LED* and *η* of the triple-string LED calculated with optimal *V**f2* and different *V**f3*, respectively. The trends of *P**LED* and *η* in the triple-string scheme are analogous to those in the double-string scheme. The difference is that two breakpoints appear in *P**LED* and *η* at *V**f3* ≥ 564V, which is relevant to the operating principle of the linear multi-string LED driving circuit mentioned in section Ⅱ. Under optimal conditions of *V**f2* = 477V and *V**f3* = 522V, and the corresponding *η* is 93% (@420V line voltage) ~ 96% (@387V line voltage).

Comparing the single-string, double-string, and triple-string LED driving schemes, it can be seen that *η* increases with the number of LED sub-strings.