Boost converter

The boost converter is a DC/DC converter in which the magnitude of the output voltage is always higher than that of the input voltage. The circuit diagram in figure shows the basic elements of the boost converter and their connection with a solar cell and a load resistor. The boost converter consists of a coil L at the input, a switch Q in the middle, a diode D and capacitsor C at the output.

Figure boost converter schematic

There are two states in the circuit:

• switch is closed (dTs)

• switch is open (1-d)Ts

In figure the switch is closed. A current flows from the solar cell through the coil L and the switch Q to set up the magnetic field in the coil. The diode D blocks the current flow from input to output in this state. The capacitor C discharges its charge to the load resistor. During this on-time dTs, the voltage across the coil is $V_{pv}$ and the current increases linearly to a peak. 

Figure boost on time

During the switch-off time (1-d)Ts, the coil tries to maintain the current flow and to release the energy that was stored in the form of the magnetic field. In the process, the polarity on the coil changes and the voltage increases. The voltage across the coil is now $(V_{pv}-V_{out})$. This causes the diode D to commutate because the voltage on the side of the coil exceeds the output voltage on the capacitor C, so the diode becomes conductive. The capacitor C at the output of the circuit now stores this additional voltage and the magnetic field of the coil is reduced. Figure shows the current flow during the off-time. In the next time window, the switch starts again in the closed state and the cycle begins again.

Figure boost off time

If the switch is switched with a fixed period Ts, a voltage can be measured at the output that is higher than the voltage at the input. To adjust the output voltage, the duty cycle d is varied. The duty cycle d describes the ratio between the time in the switched-on and switched-off state.

A change in the duty cycle also causes a change in the input resistance of the circuit. This point will be discussed in more detail in the following, as it is important for understanding the solar opdul optimiser.

Timing diagram

For this, the relationship between input and output voltage is first derived by considering the voltage across the coil over both states. With the timing diagram and the assumption that the magnitude of the voltage time surface of the coil over one period is zero, equation \eqref{InOut} can be established.

\[\label{InOut} V_{in} \cdot dT_{s} + (V_{in} - V_{out})(1-d)T_{s}=0\]

If the equation is transformed, the input-to-output voltage ratio we are looking for appears with the equation \eqref{InOutrel}.

\[\label{InOutrel} V_{out}=V_{in} \cdot \frac{1}{1-d}\]

For the ratio of the current across the input and output, consider the balance of the current from the capacitor across both states. During the on time $dTs$, the diode blocks and the capacitor is discharged only through the load resistor. During the off-time $(1-d)Ts$, the current from the coil flows through the conducting diode, charging the capacitor. In addition, part of the current goes to the load resistor. Over an entire period, current across the capacitor can be set to zero, as given in equation \eqref{InOutCurrent}.

\[\label{InOutCurrent} (-I_{out})dT_{s}+(I_{in}-I_{out})(1-d)T_{s}=0\]

The relationship between input and output current, after a transformation, shows the equation \eqref{InOutrelCurrent}

\[\label{InOutrelCurrent} I_{out}=I_{in} \cdot{1-d}\]

If the output voltage \eqref{InOutrel} is divided by the output current \eqref{InOutrelCurrent}, an equation \eqref{Rout} for the load resistance can be given. The factor of the first term can be replaced by the input resistance.

\[\label{Rout} R_{L}=\frac{V_{out}}{I_{out}}=\frac{V_{in}}{I_{in}} \cdot \frac{1}{(1-d)^{2}}=\frac{R_{in}}{(1-d)^{2}}\]

After a further transformation, the equation we are looking for \eqref{Rin} can be found for the input resistance.

\[\label{Rin} R_{in}=R_{L} \cdot (1-d)^{2}\]

The input resistance from this figure is thus dependent on the load connected to the converter and the duty cycle. This insight is the basis for the operation of the solar module optimiser. The next section shows how the solar module can be operated at the optimal operating point by modulating the duty cycle.

Figure Input Resistance