ADP1823 Datasheet, PDF (20/32 Pages) Analog Devices – Dual, Interleaved, Step-Down DC-to-DC Controller with Tracking

ADP1823
In Figure 26, the location of the ESR zero corner frequency
gives significantly different net phase at the crossover frequency.
LC FILTER BODE PLOT
PHASE CONTRIBUTION AT CROSSOVER
OF VARIOUS ESR ZERO CORNERS
GAIN
0dB
f_LC f_ESR1 f_ESR2 f_ESR3 f_CO
f_SW
–40dB/dec
FREQUENCY
–20dB/dec
PHASE
0°
Φ₁
–90°
Φ₂
–180°
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
φESR = 45° × log ^{10 × f CO}
(25)
f _ESR
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
G
(dB)
–1 _SLOPE
PHASE
f_Z
–1 _SLOPE
f_P
–180°
–270°
C_HF
R_Z
C_I
FROM
V_OUT
R_TOP
R_BOT
EA
VREF
COMP
TO PWM
VRAMP
0V
Figure 27. Type II Compensation
To calculate K for Type II compensation, use
K = tan⎜⎛ ^φB + 45°⎟⎞
(32)
⎝2
⎠
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
f_Z
=
f_CO
K
for Type II
(33)
Similarly, the frequency of the added poles, fP, should be above
crossover:
f _P = f_CO K for Type II
(34)
1 _{D. Venable, “The K Factor: A New Mathematical Tool for Stability Analysis and Synthesis,” 1983.}
Rev. A | Page 20 of 32