In last month's article, it was found that the right-half-plane zero (RHPZ) presence forces the designer to limit the maximum duty-cycle slew rate by rolling off the crossover frequency. This is because the average inductor current cannot instantaneously change and is also slew-rate limited by the available transient average voltage across the inductor. By clamping the duty-cycle speed of change, the designer offers a way to let the inductor current build up at a pace where it can follow the output current increase demand. Failure to do this exposes the converter to instability. The second part of this series deals with the analytical description of the RHPZ in both voltage-mode and current-mode converters.
Eq. 1 (view equations for this articles) was introduced in the first article and represented the nonlinear large signal expression of the diode current in the CCM boost converter we have studied:
To deal with a small-signal ac equation in which poles and zeros could appear, we need to apply a linearization process around an operating point. There are two ways to do this:
Perturb all dc terms with a small ac modulation. That is to say, replace all terms susceptible to change by a static value plus an ac modulation:
Collect and sort dc terms and ac terms to form two different equations. Get rid of the ac cross products, as they are of negligible contribution (small by small leads to a smaller result):
We now have a dc equation that gives us the bias point of our boost converter. The ac equation is the small-signal response of the output current to a perturbation in the inductor current and the duty cycle. This is the equation for which we are looking.
The second method deals with partial derivative. In some cases, the individual variable perturbations can lead to complicated expressions where the final sort of dc and ac equations represents a tedious exercise. When the bias point is already known, it is faster to use partial derivatives. A partial derivative actually evaluates the sensitivity of a function to its individual variables. The result is then the ac equation we are looking for, without dc terms and without neglecting ac cross products.
Applying the method to Eq. 1, we have:
In Eqs. 5 or 6, the ac inductor current îL appears. What is the expression of an ac inductor current? It is simply the ac inductor voltage divided by the inductor impedance. Let's find the expression of the ac inductor voltage by first deriving its average large signal expression, already found in the first article:
On average, when the converter is at the equilibrium, this equation gives zero. However, under an ac excitation, the average inductor voltage is also ac modulated across zero. By using the partial derivative option, we can see that the ac inductor voltage, in this case, is expressed by:
From Eq. 7, we can see that the input term VIN has disappeared because the input voltage is considered constant during the ac analysis. Furthermore, if we consider a large output capacitor, its impedance at the ac excitation can be considered close to zero, helping to further simplify the expression to:
Having the ac inductor voltage, it is easy to obtain the ac inductor current we are looking for:
Substituting Eq. 10 into Eq. 6 gives the final ac output current expression:
The average inductor current IL is the source current IIN. Considering a 100% efficiency power conversion, we can write:
From which we have:
Substituting Eq. 13 into Eq.11, we obtain:
Now, factoring the first term and rearranging, we have:
From the above expression, we can see a pole at the origin given by the inductor L and a zero featuring a positive root: This is the RHPZ for which we are looking. Please note that both depend on the duty cycle and are moving in relationship to the input/output conditions.
Applying the boost converter numerical values from our previous example, we have the following positions:
In the low frequency domain, for s<<ωz2, the ac output current is dominated by the inductor pole and the phase lags to -90°. The gain drops with a -1 slope until it crosses the 0-dB axis at ω0. It then continues to further drop until the RHPZ kicks in. With a LHP zero, the slope would brake from -1 to zero, as it does, but the phase would return to -90° when the frequency further increases. Given the negative sign in Eq. 15, the phase will further lag by -90°, reaching a total of -180° in higher frequencies. We can easily calculate the asymptotic phase limits, using Eq. 15:
The average model introduced in Reference 1 lends itself very well to plotting Eq. 15. To further check the resulting curves, we have entered this equation into Mathcad® and superimposed both results. As Fig. 1 confirms, they are equivalent, showing the phase lag to -180° at high frequency.
The compensation of a system featuring such RHPZ is almost impossible given the phase stress as the crossover frequency approaches the RHPZ position. The only solution is to reduce the bandwidth to 20% to 30% of the worst-case RHPZ position, where the total phase stress remains manageable. By reducing the crossover frequency, the resulting duty-cycle slew rate stays within acceptable boundaries where the inductor current can always keep up with the demand.
In current-mode control, the controller does not directly drive the duty cycle but rather the inductor peak current. However, as the overall structure of the converter does not change, Eq. 6 remains the same. Because the duty cycle is now a consequence of the inductor peak current set point imposed by the control voltage VC, let's rework Eq. 10 to extract the duty cycle as a function of the inductor current:
If we now substitute the above equation in Eq. 11, we obtain:
The ac inductor peak current is imposed by the control voltage across the sense resistor and follows:
If we substitute Eq. 25 in Eq. 24 and rearrange the result, we have:
Unlike the voltage-mode expression (Eq. 15), Eq. 26 teaches us the presence of a static gain G0 independent from the frequency below the RHPZ location. This is the consequence of the current-mode technique whose inner current loop removes the inductor pole present in voltage-mode control. One immediate comment concerns the RHPZ, which is still present in current mode and occupies a same location as with the voltage-mode case.
Fig. 2 depicts the same boost converter as before but now using a peak current-mode controller, also described in Ref. 1. This new average mode is capable of modelling subharmonic instabilities and can toggle between DCM and CCM modes. The control voltage is adjusted to deliver 24 V, and it corresponds to a similar duty cycle as before: 58.3%. Applying the boost converter numerical values, the static gain G 0 reaches -7.6 dB. Again, we have entered Eq. 26 in Mathcad and the resulting calculations are plotted in Fig. 3, together with the SPICE-simulated waveforms. The agreement is fairly good until the subharmonic poles kick in at half the switching frequency and further degrade the phase response.
Reference 2 offers another interesting way to evaluate the RHPZ position in a boost converter. Using the high-frequency small-signal response of the converter, the author calculates the temporal response of the transfer function to a duty-cycle transient step. In a separate paragraph, he graphically calculates the output voltage variation related to the average output current change engendered by a similar abrupt duty-cycle change. As both voltages should be equal, the RHPZ is further unveiled in a unique way.
Reference 3 also documents the RHPZ aspects and is worthwhile to consult.
Converters implementing an indirect energy transfer-type of conversion suffer from the presence of a RHPZ when operated in CCM. These converters must first store the energy in the inductor during a certain time before dumping it into the output capacitor during the rest of the cycle. If the duty cycle quickly changes in response to a perturbation, then the inductor naturally limits the current slew rate and the output voltage drops. A way to limit the vicious effects of the RHPZ is to limit the available loop bandwidth to 20% to 30% of the worst-case RHPZ position. That way, the duty-cycle slew rate is limited and remains always slower than the minimum inductor slew rate. The calculations show that the RHPZ exists in CCM fixed-frequency voltage-mode and current-mode techniques, occupying a similar position. In Part 3, we will show how to compensate a converter featuring a RHPZ with the help of SPICE models.
Basso, C. “Switch Mode Power Supplies: SPICE Simulations and Practical Designs,” McGraw-Hill, 2008.
Vorpérian, V. “Fast Analytical techniques for Electrical and Electronic Circuits,” Cambridge, 0-521-62442-8.
Dixon, L. “The Right-Half-Plane Zero — A Simplified Explanation,” Unitrode Seminars SEM-500.