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U-97 APPLICATION NOTE

MODELLING, ANALYSIS AND

COMPENSATION OF THE

CURRENT-MODE CONVERTER

A b s t r a c t

As current-mode conversion increases in popularity, several peculiarities associated with fixed-frequency, peak-current

detecting schemes have surfaced These include instability above 50% duty cycle, a tendency towards subharmonic

oscillation, non-ideal loop response, and an increased sensitivity to noise. This paper will attempt to show that the

performance of any current-mode converter can be improved and at the same time all of the above problems reduced or

eliminated by adding a fixed amount of “slope compensation” to the sensed current waveform.

1.0 INTRODUCTION

The recent introduction of integrated control circuits designed specifically

for current mode control has led to a dramatic upswing in the

application of this technique to new designs. Although the advantages of

current-mode control over conventional voltage-mode control has been

amply demonstrated(l-5), there still exist several drawbacks to a fixed

frequency peak-sensing current mode converter. They are (1) open loop

instability above 50% duty cycle, (2) less than ideal loop response

caused by peak instead of average inductor current sensing, (3) tendency

towards subharmonic oscillation, and (4) noise sensitivity, particularly

when inductor ripple current is small. Although the benefits of current

mode control will, in most cases, far out-weight these drawbacks, a

simple solution does appear to be available. It has been shown by a

number of authors that adding slope compensation to the current

waveform (Figure 1) will stabilize a system above 50% duty cycle. If

one is to look further, it becomes apparent that this same compensation

technique can be used to minimize many of the drawbacks stated above.

In fact, it will be shown that any practical converter will nearly always

perform better with some slope compensation added to the current

waveform.

The simplicity of adding slope compensation - usually a single resistor -

adds to its attractiveness. However, this introduces a new problem - that

of analyzing and predicting converter performance. Small signal AC

models for both current and voltage-mode PWM’s have been

extensively developed in the literature. However, the slope compensated

or “dual control” converter possesses properties of both with an

equivalent circuit different from yet containing elements of each.

Although this has been addressed in part by several authors(l,2), there

still exists a need for a simple circuit model that can provide both

qualitative and quantitative results for the power supply designer.

FIGURE 1- A CURRENT-MODE CONTROLLED BUCK REGULATOR WITH SLOPE COMPENSATION.

3-43

APPLICATION NOTE U-97

The first objective of this paper is to familiarize the reader with the

peculiarities of a peak-current control converter and at the same time

demonstrate the ability of slope compensation to reduce or eliminate

many problem areas. This is done in section 2. Second, in section 3, a

circuit model for a slope compensated buck converter in continuous

conduction will be developed using the state-space averaging technique

outlined in (1). This will provide the analytical basis for section 4 where

the practical implementation of slope compensation is discussed.

2.1 OPEN LOOP INSTABILITY

An unconditional instability of the inner current loop exists for any fixed

frequency current-mode converter operating above 50% duty cycle -

regardless of the state of the voltage feedback loop. While some

topologies (most notably two transistor forward converters) cannot

operate above 50% duty cycle, many others would suffer serious input

limitations if greater duty cycle could not be achieved. By injecting a

small amount of slope compensation into the inner loop, stability will

result for all values of duty cycle. Following is a brief review of this

technique.

A.) DUTY CYCLE < 0.5

B.) DUTY CYCLE > 0.5

COMPENSATING

SLOPE

C.) DUTY CYCLE > 0.5 WITH SLOPE COMPENSATION

FIGURE 2- DEMONSTRATION OF OPEN LOOP INSTABILITY IN A

CURRENT-MODE CONVERTER.

Figure 2 depicts the inductor current waveform,

current

it may be seen graphically that

(3)

Therefore, to guarantee current loop stability, the slope of the

compensation ramp must be greater than one-half of the down slope of

the current waveform. For the buck regulator of Figure 1,

constant equal to

waveform should be chosen such that

vs nT for all n as in Figure 3, we observe a damped

sinusoidal response at one-half the switching frequency, similar to that of

an RLC circuit. This ring-out is undesirable in that it (a) produces a

ringing response of the inductor current to line and load transients, and

(b) peaks the control loop gain at ? the switching frequency, producing

a marked tendency towards instability.

FIGURE 3- ANALOGY OF THE INDUCTOR CURRENT RESPONSE TO

THAT OF AN RLC CIRCUIT.

It has been shown in (1), and is easily verified from equation 2, that by

choosing the slope compensation m to be equal to

2.3 SUBHARMONIC OSCILLATION

For steady state condition we can write

Gain peaking by the inner current loop can be one of the most

(8)

significant problems associated with current-mode controllers. This peaking occurs at one-half the switching frequency, and - because of

or loop to break into oscillation at one-half the switching frequency. This By using (9) to reduce (7), we obtain

instability, sometimes called subharmonic oscillation, is easily detected

as duty cycle asymmetry between consecutive drive pulses in the power (10)

in subharmonic oscillation (dotted waveforms with period 2T).

to an output current,

From figure 5, two equations may be

written

(4)(5)

Adding slope compensation as in figure 6 gives another equation

(6)

Using (5) to eliminate from (6) and solving for

yields

(7)FIGURE 6- ADDITION OF SLOPE COMPENSATION TO THE CONTROL

SIGNAL

write the small signal gain at f =

4TA

Loop gain =

(12)

to guarantee stability as

then for the case of m = 0 (no

compensation) we see the same instability previously discussed at 50%

duty cycle. As the compensation is increased to m =

system, the finite value of

we reach a point, m = where the

maximum. gain becomes independent of duty cycle. This is the point of critical damping as discussed earlier, and increasing m above this value will do little to improve stability for a regulator operating over the full duty cycle range.

2.4 PEAK CURRENT SENSING VERSUS

AVERAGE CURRENT SENSING

True current-mode conversion, by definition, should force the average inductor current to follow an error voltage - in effect replacing the inductor with a current source and reducing the order of the system by one. As shown in Figure 8, however, peak current detecting schemes are generally used which allow the average inductor current to vary with duty cycle while producing less than perfect input to output - or feedforward characteristics. If we choose to add slope compensation

equal to m = -?

APPLICATION NOTE

(A)

(B)

FIGURE 11

- BASIC BUCK CONVERTER (A) AND ITS SMALL SIGNAL EQUIVALENT CIRCUIT MODEL (B).

If we now perturb these equations - that in substitute

D +

duty cycle may be written from Figure 6 as

(18)

Perturbing this equation as before gives

(19)

By using 19 to eliminate from 16 and 17 we arrive at the state space equations

(20)

(21)

An equivalent circuit model for these equations is shown in Figure 11B and discussed in the next section.

3.2 A.C. MODEL DISCUSSION

The model of Figure 11B can be used to verify and expand upon our previous observations. Key to understanding this model is the interaction

U-97

between

then a double pole response will be formed by the LRC output filter similar to any voltage-mode converter. By appropriately adjusting m,

any condition between these two extremes can be generated.

Of particular interest is the case when Since the down

slope of the inductor current

At this point,

typical 12 volt buck regulator operating under the following conditions:

APPLICATION NOTE U-97

If a small ripple to D.C. current ratio is used. as is the case for

versus

frequency for the same example of Figure 12. The gains have all been

normalized to zero dB at low frequency to reflect the actual difference in

frequency response as slope compensation m is varied. At m =

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