sub harmonics in buck converter (req)

peak current control buck

hi, can u tell me , how the sub harmonics come into the picture of buck converter, how to eliminate this, please explain me both in theoritical and mathematical way, if u have any papers relating to this topic give me. thanx.

for current mode converters, ppls find that the converter's inductor current will undergo oscillation whenever Duty cycle(D) > 0.5
Consider peak current mode control with (continuous current mode),
Ipk = I1 + m1DT = I2 + m2D'T ;
where Ipk is the peak current control limit
I1 is the starting point of inductor current in the beginning of the period
I2 is the end point of inductor current in the end of the period
m1 slope for inductor current @ DT
m2 slope for inductor current @ D'T
T: period of the converter
D : duty cycle for the converter
D' = 1-D
I1 is perturbed to I1 + deltaI1 (due to input changes/ output changes for example)
you can find
Ipk = I1 + deltaI1 + m1(D+deltaD)T = I2 + deltaI2 + m2(D'-deltaD)T
and deltaI2 = -(D/D')deltaI1.
for the subharmonic oscillation takes place, |deltaI2| > |deltaI1|
this situation happens @ D>0.5

To avoid this subharmonic oscillation takes place,
normally a compensation ramp(mc>m2/2) is added in the inductor current.
For deadbeat control , mc (slope of compensation ramp) = m2.