# Solar-array controller needs no multiplier to maximize power

-December 05, 2008

Solar-photovoltaic arrays are among the most efficient, cost-effective, and scalable “green” alternatives to fossil fuels, and researchers are almost daily announcing new advances in photovoltaic technology. But successful application of photovoltaics still depends on strict attention to power-conversion efficiency. Figure 1 shows one reason for this attention.

Figure 1 It is important to operate solar-photovoltaic arrays at their maximum power point.

A photovoltaic array’s delivery of useful power to the load is a sensitive function of load-line voltage, which in turn depends on insolation—that is, sunlight intensity—and array temperature. Operation anywhere on the current/voltage curve except at the optimal maximum-power-point voltage results in lowered efficiency and a waste of valuable energy. Consequently, methods for maximum-power-point tracking are common features in advanced solar-power-management systems because they can boost practical power-usage efficiency—often by 30% or more.

Because of its generality, a popular maximum-power-point-tracking-control algorithm is perturb and observe, which periodically modulates, or perturbs, the load voltage; calculates, or observes, the instantaneous transferred power response; and uses the phase relationship between load modulation and calculated power as feedback to “climb the hill” of the current/voltage curve to the maximum-power-point optimum. The perturb-and-observe algorithm is the basis of the maximum-power-point-tracking-control circuit (Figure 2, in yellow) but with a twist (in blue), which achieves a feedback function equivalent to a current-times-voltage power calculation but without the complexity of a conventional multiplier. The idea relies on the well-known logarithmic behavior of transistor junctions, VBE=(kT/q)log(IC/IS)=(kT/q)[log(IC)–log(IS)], where VBE is the base-to-emitter voltage. It also relies on the fact that adding logarithms is mathematically equivalent to multiplication. Here’s how.

Figure 2 This maximum-power-point-tracking controller relies on well-known logarithmic behavior of transistor junctions. (Click to enlarge)

Capacitor C2 couples a 100-Hz, approximately 1V-p-p-modulation or 1V-p-p-perturbation square wave from the S2/S3 CMOS oscillator onto the photovoltaic-input voltage, V. The current/voltage curve of the array causes the input current, I, to reflect the V modulation with a corresponding voltage-times-current input-power modulation. IC1A forces IQ1 to equal I×x1, where I is the solar-array current and x1 is a gain constant. IC1B forces IQ2 to equal V/499 kΩ, where V is the solar-array voltage. Thus, VQ1=(kT1/q)1[log(I)–log(IS1)+log(x1)], and VQ2=(kT2/q)[log(V) –log(IS2)–log(499 kΩ)]. VQ1 is the base-to-emitter voltage of Q1; k is the Boltzman constant; T1 is the temperature of Q1; q is the elementary charge of the electron; I is the current input from the solar panel’s negative terminal; IS1 is the saturation current of Q1; x1 is the arbitrary gain constant, which IC3 determines; V is the voltage input from the solar panel’s positive terminal; IS2 is the saturation current of Q2; K is degrees Kelvin; VPF is the power-feedback signal; and VIP is the calculated power-input signal. Because k, q, IS1, IS2, x1, and 499 kΩ are all constants and T1=T2=T, however, for the purposes of the perturb-and-observe algorithm, which is interested only in observing the variation of current and voltage with perturbation, effectively, VQ1=(kT/q)log(I), and VQ2=(kT/q)log(V).

The series connection of Q1 and Q2 yields VPF=VQ1+VQ2=(kT/q)[log(I)+log(V)]=(kT/q)log(VI), and, because of IC1B’s noninverting gain of three, VIP=3(kT/q)log(V I)≈765 µV/% of change in watts. The VIP log (power) signal couples through C1 to synchronous demodulator S1, and error integrator and control op amp IC1C integrates the rectified S1 output on C3. The IC1C integrated error signal closes the feedback loop around the IC3 regulator and results in the desired maximum-power-point-tracking behavior.

Using micropower parts and design techniques holds the total power consumption of the maximum-power-point-tracking circuit to approximately 1 mW, which avoids significantly eroding the efficiency advantage—the point of the circuit in the first place. Meanwhile, simplifying the interface between the maximum-power-point-tracking circuit and the regulator to only three connection nodes—I, V, and F—means that you can easily adapt the universal maximum-power-point-tracking circuit to most switching regulators and controllers. Therefore, this Design Idea offers the efficiency advantages of a maximum-power-point-tracking circuit to small solar-powered systems in which more complex, costly, and power-hungry implementations would be difficult to justify.