Understanding the Right-Half-Plane Zero Part 1

The small signal analysis of power converters reveals the presence of poles and zeros in the transfer functions of interest (e.g., the control to the output variable). The zeros occur in the numerator of the expression, whereas the poles are located in the denominator. The stability analysis of the transfer function consists in looking at the position these poles and zeros occupy in the s-plane. For a stable converter, one condition is that both the zeros and the poles reside in the left-half of the plane: We're talking about negative roots. For a pole, a position in the left plane implies an exponentially decaying temporal response, hence asymptotically stable. Conversely, when placed on the right side in the s-plane, a step response will lead to a diverging response as the associated exponential term exhibits a positive exponent. This is a positive root. For some converter architectures, a zero may be the positive root to the numerator of the control-to-output transfer function. How this can happen and the consequences of such a positive zero — also called a right-half-plane zero (RHPZ) — are the subject of this 4-part series.

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Fig. 1 represents a classical boost converter where two switches appear: a power switch SW (usually a MOSFET) and a diode D (sometimes called the catch diode). In the continuous conduction mode (CCM) of operation, the inductor current I _L flows in the power switch SW during the on time or DTsw. During the off time, or (1-D)Tsw, the power switch is open and the inductor current goes to the output diode, further feeding an output network made of the capacitor and the load. Regardless of whether it is voltage- or current-mode control, this configuration assumes that energy is first stored in the inductor during the on time and then transferred to the output during the off time.

THE DIODE CURRENT

Fig. 2 is an equivalent representation of the boost converter where the switch/diode network has been replaced by a single-pole double-throw switch that alternatively routes the inductor current in the two different branches: the power switch or the output diode. If we were to observe the currents circulating in the output diode, we would see Fig. 3's typical waveforms. Our boost converter is designed to deliver power to a given load. Thus, the variable of interest, in our case, is the available output current I _OUT. This current is made of a dc portion on which is superimposed a switching ripple. In theory, the ripple goes into the capacitor, and the dc current circulates in the load. The dc current delivered by the boost converter is nothing more than the diode average current, I_D. Mathematically, this current can be expressed by:

Where I_D is the average diode current, also equal to the dc output current I_OUT, and D is the duty cycle.

On the left side of Fig. 3, note that the current in the diode jumps to the peak inductor current as soon as the switch opens. Then, the current decays with a slope imposed by the voltage across the inductor during the off time. The diode average current in the left picture is I _d0 and obeys Eq. 1. On the right side of Fig. 3, the duty cycle has slightly increased. The inductor current peaks a bit higher, but given the reduction of the term (1-D) in Eq. 1 due to the increase of D, the average current I_d1 is lower than before.

As seen from Eq. 1, if D suddenly increases to correct a perturbation, then, to let I_OUT follow up, there must be an immediate increase in the inductor current I_L as well. The problem relates to the average inductor current, which is limited in slew rate. If the inductor average current change is slower than the duty-cycle change , then the output current I_OUT goes down immediately until the inductor current builds up and eventually catches up with the set point imposed by the loop. However, if I_OUT goes down, then so does V_OUT, which is immediately sensed by the feedback loop. The controller increases the duty cycle and sees a decrease in the voltage, the reverse of what the loop polarity is supposed to be. This is the physical effect of the RHPZ located in the control-to-output transfer function.

What is the pace at which the average inductor current can change? Lenz's law says that the instantaneous current change rate in an inductor obeys the following formula:

On average, it simply follows:

Where I_L and V_L, respectively, represent the average inductor current and voltage values. The exercise now consists in calculating the average value across our inductor. By considering the weighted period of time during which V_IN or V_OUT - V_IN are applied across L, we have:

Assume the following boost operating parameters:

V_IN = 10 V

D₀ = 0.583

R_load = 240 Ω

L = 1 mH

With a 58.3% duty cycle, the converter delivers 24 V. Now, suppose the duty cycle jumps to D₁ = 59% or a difference of 0.7%. What is the inductor average current slope in this case? Considering a large output capacitor, the output voltage stays constant during the duty cycle change. Applying Eq. 4 gives a transient average inductor voltage of:

Referring to Eq. 3, the maximum average current slope is therefore:

a rather low value.

When the duty cycle changes from 58.3% to 59%, it implies an output voltage change of:

With a constant 240 Ω load, the output current increases to:

Brought back to the inductor change, the output current variation given by Eq. 8 must be accompanied by an average inductor current variation of:

AVERAGE MODEL TO VISUALIZE EFFECTS

Given an average inductor slope 160 µA/µs, this current variation will only be possible within a time frame of:

If the duty cycle is swept from 58.3% to 59% in much less time than 42.8 µs, the inductor current will not build up at a sufficient pace to make the output current rise at the same speed. As an immediate result, the output current drops rather than increases. Conversely, if the duty cycle sweep is slow enough, the current can increase in the inductor at sufficient speed to compensate the reduction in (1-D). Thus, the output voltage goes up. This is the reason why a reduction in the available loop bandwidth naturally limits the duty-cycle slew rate and gives time for the inductor current to build up.

Fig. 4 depicts a voltage-mode boost converter modelled using the newly derived auto-toggling model based on the PWM switch model. ¹ In this figure, we will sweep the duty cycle from 58.3% to 59% at different speeds and then observe the pertinent waveforms:

The results appear in Fig. 5 and Fig. 6. In Fig. 5, the duty cycle is slowly swept in 200 µs, and we can see that the output voltage rises up without any negative portion. The inductor current can keep up with the duty cycle change and the converter responds, in time, to the step. The situation differs in Fig. 6, where the sweep time is reduced to 10 µs. In this case, the inductor average current cannot positively answer the required change, and the output current drops. The same occurs in the output voltage and, if a voltage loop would be involved, an oscillation would occur.

Based on the above observations, we can state that:

REFERENCES

In CCM, the average inductor current is limited in slew rate by the available voltage during the duty-cycle change. A large inductor worsens the situation while a small inductor improves it.
If the duty-cycle change imposed by the feedback loop tries to set an output current variation beyond the inductor slew rate capabilities, then the output voltage drops and oscillations occurs. On the contrary, slower duty-cycle changes will correctly propagate to the output without endangering the loop stability.
As a preliminary conclusion, if we limit the duty-cycle slew rate or simply truncate the available loop bandwidth, we have a means to fight the control-to-output RHPZ inherent to the CCM boost converter.

Part 1 shows how the limited current slew rate in the inductor represents the basis for the instability in the CCM boost converter. A way to fight this RHPZ presence is to roll-off the crossover frequency at a position that naturally limits the duty-cycle variation rate. By doing so, the inductor current has enough time to build up, and the converter can supply the required current without issues. Part 2 will show how to unveil the RHPZ from a small signal study.