Document 6513313

 How to ensure DC-­‐DC converter stability Most power system designers are aware that the impedance of their power input source should be small compared to the input impedance of their switch-­‐mode DC-­‐DC converter to ensure its stable operation. Also, many designers connect a large electrolytic capacitor across their DC-­‐DC converter input to further increase stability. This article explains why these factors are important and offers designers advice on how to comply with them. The first point to remember is that a converter’s input impedance is not fixed – it varies with changes in both input voltage and output load, because the converter works to maintain its output voltage constant during any such changes. We can calculate the converter’s input impedance – or resistance, during its normal DC operation – in terms of its efficiency, load power and input voltage. Using Ohm’s law we can show that where the converter’s input resistance is , n is its efficiency, Vi is the input voltage and Po, the output power. Note also that an incremental increase in input current will cause an incremental decrease in input voltage and therefore input resistance. This incremental input resistance will therefore have a negative sign. The DC-­‐DC converter’s topology determines how it achieves these changes in input resistance. For basic pulse-­‐width modulator (PWM) forward converters, the input resistance is inversely proportional to the PWM duty cycle. Zero-­‐current switching (ZCS) converters use repetition frequency as the control variable; their input resistance is inversely proportional to the square of the repetition frequency. A DC-­‐DC converter with output voltage connected back to its ‘sense’ input is a closed-­‐loop control system. With its variable input resistance in mind, the converter can be tested and analysed for stability just as any other closed-­‐loop control system. Its response to a step change input, and therefore stability, can be predicted by considering its steady state open loop response in the frequency domain. Before applying an AC waveform for this test, therefore, the converter must be connected in an open-­‐loop configuration. On some converters this can be easily achieved simply by not connecting the output voltage feedback signal to the ‘sense’ input, or by disconnecting the feedback resistor. On other, sealed, units the procedure is a little more involved – a counteractive voltage must be injected into the output circuit using a high power, wide bandwidth transformer, to nullify the feedback voltage. Within a test set-­‐up or real application, the DC-­‐DC converter is connected to a power source having its own internal impedance, with inductive and capacitive impedances also present in the input circuit. We can call the open-­‐loop transfer function of the converter itself H(s). However our real interest is in the overall open-­‐loop transfer function of the entire DC-­‐DC supply system, or Vo/Vs, where Vo is the voltage across the load and Vs is the supply source’s internal voltage. We can write this down as 1 (1) where Zs is the input source impedance and Zp is the capacitive input impedance of the converter with its capacitive and resistive parallel components. We can see from this that, provided Zs is sufficiently small compared with Zp, it will not affect the overall transfer function. As a general practical rule, Zs should be kept at least ten times smaller than Zp. It is possible to apply a more detailed analysis to obtain more precise results. Within this introductory article we cover the key steps of this analysis without providing the full algebraic detail. The first step is to express the various impedances Z in terms of their resistive and reactive components. For example, the source impedance Zs can be expressed as Zs = Rs + sLs where Rs is the source resistance and Ls is the source inductance. The variable ‘s’ is frequency-­‐
dependent; in the frequency domain, the system’s stability is also frequency-­‐dependent. By substituting all the resistive and reactive components into the Z terms of equation (1), we can develop a more detailed equation describing the system’s transfer function and its susceptibility to instability. Equation (2) below is a generic representation of this equation: (2) The characteristic equation for the system contains all the terms for inductance, capacitance, resistance and frequency (s) that affect the system’s transfer function. For some systems, this can be a third order or higher polynomial, which is difficult to solve algebraically. In such cases the most popular approach is to use Nyquist’s stability criterion for a graphical solution. For our DC-­‐DC converter example, however, the characteristic equation is a second order or quadratic type of the form as2 + bs + c = 0, which can be solved algebraically. Solving the characteristic equation for our DC-­‐DC converter yields two conditions that must be met for stability: (3) And 2 (4) Where Rs = source resistance, Ls = source inductance and = incremental input resistance. We can draw two important conclusions from these conditions. The first is a repetition of our earlier observation that the source impedance must be lower than the converter’s input impedance. The second is that the source impedance cannot be arbitrarily small because if the condition of equation (3) is violated, the RLC input network will be underdamped and the system will show instability. As previously mentioned, designers often connect a large capacitor across the DC-­‐DC converter’s input terminals. This allows control of the converter’s stability, but it significantly affects the system’s characteristic equation and stability conditions. We can make some assumptions to characterise and simplify our analysis of the new input circuit. Firstly, we remove the source resistance Rs for simplicity. In this version of the circuit, Rs should be minimised as much as possible, as a resistive element in series with the converter creates power dissipation and reduces overall efficiency. Next, we allow for the parallel capacitor of capacitance Ce and equivalent series resistance (ESR) Re. The previous example’s internal capacitance Ci has now been included within Ce. This is an approximation, but it allows us to make a sufficiently accurate stability assessment without having to solve a third-­‐order polynomial equation. We can develop another characteristic equation using these new variables, solve it and find our new conditions for stability. These are revealed in equations (5) and (6) below: (5) And (6) Equation (5) states that Re should be smaller than . However, from (6) it is clear that if Re is made too small, for example, by using high-­‐quality input capacitors for Ce, the system could become under-­‐
damped and start to oscillate. For this reason, ceramic and film input capacitors are not recommended because their construction results in lower ESRs than electrolytic types. From (6) it is also possible to see that, the bigger the source inductance, the larger the input capacitance required for compensation. Also, for low-­‐voltage systems such as a 12 V DC-­‐DC converter that have lower input differential resistance , larger input capacitance is needed to ensure stable operation. Once we know the values of the input inductor Ls and input incremental resistance it is possible to replace the numbers in (5) and (6) to find the optimal combination for Ce with the proper ESR. By using these considerations for stability while allowing for ’s dependence on the converter’s input voltage and output power as described earlier, we can choose components for stability over the converter’s entire input voltage operating range. 3