ADP1821

Rev. B | Page 16 of 24

If the zero produced by the ESR of the output capacitor pro-

vides sufficient phase boost at crossover, Type II compensation

is adequate. If the phase boost produced by the ESR of the output

capacitor is not sufficient, another zero is added to the compen-

sation network, and thus Type III is used. A general rule to

determine the scheme whether the phase contribution of the

ESR zero is greater than 70 degrees at crossover.

In Figure 16, 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

0dB

–40dB/dec

–20dB/dec

0°

–90°

–180°

Figure 16. LC Filter Bode Plot

Using a linear approximation from Figure 16, the phase contri-

bution of the ESR zero at crossover can be estimated by

ESR

CO

ESR

f

f

×

×

=

10

log

45

ϕ

(25)

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

contributing elements, namely

(26)

where:

(27)

Note in the compensator phase expression shown in Equation 27,

the −90 degree term is the phase contributed by the initial inte-

high frequency compensation poles placed above crossover, and

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 degrees

(28)

(29)

crossover contributed by the compensator’s higher order

poles and zeros:

(30)

(31)

D. Venable, in his article “The K Factor: A New Mathematical

Tool for Stability Analysis and Synthesis,” 1983, 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 compen-

sation zeros and poles may be calculated. K is calculated for the

type of compensation selected Figure 17.

Type II Compensator

G

(dB)

PHASE

–180°

–270°

0V

VRAMP

FROM

VREF

EA

COMP

TO PWM

–1

SLO

PE

–1

SLO

PE

Figure 17. Type II Compensation

To calculate K for Type II compensation use

⎟

⎠

⎞

⎜

⎝

⎛

+

=

45

2

tan

B

φ

K

(32)

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

so if the selected type of compensation does not yield a reason-

able value of K, try the other type.