2 Understanding of Loop Stability Test

The purpose of a loop stability test is obtaining Bode plot of the open-loop transfer function in a closed-loop DC-DC converter. Consider a normally working converter circuit with a voltage feedback, shown in Figure 2-1(a).

In Figure 2-1(a) the feedback voltage directly comes from output voltage and controls the same output simultaneously, making the converter a closed loop. Now make the top-side feedback resistor not connected to output voltage but to an external voltage source V'_OUT as shown in Figure 2-1(b). Then output voltage will be automatically adjusted to V'_OUT. If V'_OUT is equal to V_OUT, the working status in Figure 2-1(a) and (b) are exactly the same.

Consider an extra AC voltage source with small amplitude V_INJ applied in series with the external source, shown in Figure 2-2, then there will be an AC component V_AC in feedback voltage and thus output voltage V_OUT will also have an AC component V_R. V_R and V_INJ normally have different amplitude and are out-of-phase but their frequencies are same which are much lower than switching frequency of the converter. So the open-loop transfer response in both Figure 2-1(b) and (a) can be expressed in Equation 3.

The circuit in Figure 2-2 can be considered as basis of any loop stability test methods for DC-DC converters, in another word, any loop stability test circuits should be equivalent to it. Otherwise the test will lead to a wrong Bode plot.

When the above circuit is directly applied in a practical loop stability test, several high-precision measuring equipments are needed. Thus some simplifications are applied and here consider the circuit in Figure 2-3.

In Figure 2-3 circuit, when the AC voltage V'_INJ is inserted in the closed-loop, there will be AC components with same frequency anywhere in the loop including output voltage, shown as V'_R. Here R_FBT will see an AC component as V'_R + V'_INJ which is exactly equivalent to the V_INJ in Figure 2-2 because V'_R and V'_INJ have a same frequency. The part of Figure 2-3 circuit circled with dash line is actually the same as Figure 2-2 circuit.

In a practical loop stability test, the connection outside dash line circle in Figure 2-3 is achieved by a frequency response analyzer, shown in Figure 2-4. The equivalent relations from Figure 2-2 to Figure 2-4 are described in the following equations.

The frequency response analyzer only collects and processes AC components of voltages at point A and B, so Equation 3 can be expressed in Equation 6:

When a frequency sweeping of injection signal is finished by analyzer, the gain and phase curves of G_OPEN(s) in Equation 6 will be displayed as the Bode plot of the converter loop.