8.2.2.10 Compensating the Small Signal Control Loop

All continuous mode boost converters have a right half plane zero (ƒ_RHPZ) due to the inductor being removed from the output during charging. In a traditional voltage mode controlled boost converter, the inductor and output capacitor form a small signal double pole. For a negative feedback system to be stable, the fed back signal must have a gain less than 1 before having 180 degrees of phase shift. With its double pole and RHPZ all providing phase shift, voltage mode boost converters are a challenge to compensate. In a converter with current mode control, there are essentially two loops, an inner current feedback loop created by the inductor current information sensed across R_SENSE (40mΩ) and the output voltage feedback loop. The inner current loop allows the switch, inductor and modulator to be lumped together into a small signal variable current source controlled by the error amplifier, as shown in Figure 9.

Figure 9. Small Signal Model of a Current Mode Boost in CCM

The new power stage, including the slope compensation, small signal model becomes:

Equation 12.

Where

Equation 13.

Equation 14.

Equation 15.

And

Equation 16.

He(s) models the inductor current sampling effect as well as the slope compensation effect on the small signal response. Note that if Sn > Se, for example, when L is smaller than recommended, the converter operates as a voltage mode converter and the above model no longer holds.

The slope compensation in TPS61175 is shown as follows:

Equation 17.

Equation 18.

Where R4 is the frequency setting resistor

Figure 10 shows a Bode plot of a typical CCM boost converter power stage.

Figure 10. Bode Plot of Power Stage Gain and Phase

The TPS61175 COMP pin is the output of the internal trans-conductance amplifier. Equation 19 shows the equation for feedback resistor network and the error amplifier.

Equation 19.

where

• G_EA and R_EA are the amplifier’s trans-conductance and output resistance located in the Electrical Characteristics table.

Equation 20.

Equation 21.

C5 is optional and can be modeled as 10 pF stray capacitance.

and

Equation 22.

Figure 11 shows a typical bode plot for transfer function H(s).

Figure 11. Bode Plot of Feedback Resistors and Compensated Amplifier Gain and Phase

The next step is to choose the loop crossover frequency, f_C. The higher in frequency that the loop gain stays above zero before crossing over, the faster the loop response will be and therefore the lower the output voltage will droop during a step load. It is generally accepted that the loop gain cross over no higher than the lower of either 1/5 of the switching frequency, f_SW, or 1/3 of the RHPZ frequency, f_RHPZ. To approximate a single pole roll-off up to f_P2, select R3 so that the compensation gain, K_COMP, at f_C on Figure 11 is the reciprocal of the gain, K_PW, read at frequency f_C from the Figure 10 bode plot, or more simply:

K_COMP(f_C) = 20 × log(G_EA × R3 × R2/(R2+R1)) = 1/K_PW(f_C)

This makes the total loop gain, T(s) = G_PS(s) × H_EA(s), zero at the f_C. Then, select C4 so that f_Z ≅ f_C/10 and optional f_P2> f_C × 10. Following this method should lead to a loop with a phase margin near 45 degrees. Lowering R3 while keeping f_Z ≅ f_C/10 increases the phase margin and therefore increases the time it takes for the output voltage to settle following a step load.

In the TPS61175, if the FB pin voltage changes suddenly due to a load step on the output voltage, the error amplifier increases its transconductance for 8-ms in an effort to speed up the IC’s transient response and reduce output voltage droop due to the load step. For example, if the FB voltage decreases 10 mV due to load change, the error amplifier increases its source current through COMP by 5 times; if FB voltage increases 11 mV, the sink current through COMP is increased to 3.5 times normal value. This feature often results in saw tooth ringing on the output voltage, shown as Figure 13. Designing the loop for greater than 45 degrees of phase margin and greater than 10-db gain margin minimizes the amplitude of this ringing. This feature is disabled during soft start.