ADP1823ACPZ-R7 datasheet(20/32 Pages) AD

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ADP1823

Rev. A | Page 20 of 32

In Figure 26, the location of the ESR zero corner frequency

gives significantly different net phase at the crossover frequency.

GAIN

FREQUENCY

PHASE

LC FILTER BODE PLOT

PHASE CONTRIBUTION AT CROSSOVER

OF VARIOUS ESR ZERO CORNERS

fSW

fCO

fESR3

fESR2

fESR1

0dB

fLC

–40dB/dec

–20dB/dec

0°

–90°

–180°

Φ1

Φ2

Φ3

Figure 26. LC Filter Bode Plot

Using a linear approximation from Figure 26, the phase

contribution of the ESR zero at crossover can be estimated by

ESR

log

(25)

If φESR ≥ 70°, then Type II compensation is adequate.

If φESR < 70°, use Type III, as an additional zero is needed.

The total phase of the system at crossover is the sum of the

contributing elements, namely:

φT = φLC + φESR + φCOMP

(26)

where:

φLC = −180°

φESR is as calculated in Equation 25

φCOMP = −90° + φP + φZ

(27)

Note in the compensator phase expression shown in Equation 27,

the −90° term is the phase contributed by the initial integrator

pole. The φP is the additional phase contributed by the high

frequency compensation poles placed above crossover, and φZ is

the phase contributed by the compensation zeros placed below

crossover. For the system to be stable at crossover, phase boost

is required from the compensator.

For stability, the total phase at crossover is designed to be equal

to −120°:

φT = φLC + φESR + φCOMP

(28)

−120° = −180° + φESR + −90° + φP + φZ

(29)

Define phase boost, φB, to be the portion of the phase at cross-

over contributed by the compensator’s higher order poles and

zeros:

φB = φP + φZ

(30)

φB = 150° − φESR

(31)

Venable1 showed that an optimum compensation solution was

to place the zeros and poles symmetrically around the crossover

frequency. He derived a factor known as K with which the

frequencies of the compensation zeros and poles may be calculated.

K is calculated for the type of compensation selected in Figure 27.

Type II Compensator

(dB)

PHASE

–180°

–270°

VRAMP

CHF

RTOP

RBOT

FROM

VOUT

VREF

COMP

TO PWM

–1

SLO

–1

SLO

Figure 27. Type II Compensation

To calculate K for Type II compensation, use

⎟

⎠

⎞

⎜

⎝

⎛

tan

(32)

Values of K between 4 and 15 are practical for implementation;

if the selected type of compensation does not yield a reasonable

value of K, try the other type.

From K, the frequency of the added zeros, fZ, is below crossover by

Z =

for Type II

(33)

Similarly, the frequency of the added poles, fP, should be above

crossover:

P =

for Type II

(34)

1 D. Venable, “The K Factor: A New Mathematical Tool for Stability Analysis and Synthesis,” 1983.

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