Question

conversion ratio analysis of the SEPIC of Fig. 4 a) Suppose that the converter operates at the boundary between CCM and DCM DCM, with the following element and parameter values D 0.225 f-100 kHz V120 V C2 200 μF Sketch the diode current ID(t) waveform and the inductor current waveforms and I(t) and 12(t) Label the magnitudes of the ripples and de components of these waveforms. b) Suppose next that the converter operates in the discontinuous conduction mode, with a different choice of parameter and element values. Derive an analytical expression for the de conversion ratio M(D, K). e) Sketch the diode current waveform i(t), and the inductor current waveforms I() and 12() and for operation in the discontinuous conduction mode.

Answer 1

Given data for SEPIC Converter is,

D = 0.225

Vg = 120V

L1=50*10^-6H

C1=47*10^-6F

f=100KHZ

R=10 Ohmss

L2= 75*10^-6H

C2= 200*10^-6F

The Out put voltage is determined by equation given below.

V0=Vg (D/(1-D)) = 120*(0.225/(1-0.225)) = 34.838V

Average current in L1, IL1 =(V0^2)/(Vs*R) = (34.838*34.838)/(120*10) = 1.0114 Amps

The variation in iL1 when the switch is closed is found from

change in IL1 = (Vs*D)/L1*f = (120*0.225)/(50*10^-6* 100*10^3) = 5.4Amps

Maximum current in L1 , IL1 Max = IL1 + (change in IL1/2)= 1.0114+(5.4/2) = 3.7114Amps

Minimum current in L1, IL1 Min = IL1 - (chang ein IL1/2) = 1.0114 - (5.4/2) = -1.6886 Amps

the average current in L2 is IL2 = I0 = V0/R = 34.838/10 = 3.484 Amps

cahnge in IL2 = (Vs*D)/L2*f = (120*0.225)/(75*10^-6* 100*10^3) = 3.6 Amps

Maximum current in L2 , IL2 Max = IL2 + (change in IL2/2)= 3.484+(3.6/2) = 5.284Amps

Minimum current in L1, IL1 Min = IL1 - (chang ein IL1/2) = 3.484 -(3.6/2) = 1.684 Amps

Diode current is expressed as ID = zero, when switch is closed.

= IC1+IL2 , when switch is open

The graphical representation is as shown below.

c). discontinuous mode of operation graphs are shown below.

b).

, a discontinuous conduction mode (DCM) means that the third time interval of operation cycle is nonzero, not that either inductor current is discontinuous.

Three distinct time intervals appears there, namely D1Ts , D2Ts and D3Ts with D1 + D2 + D3 = 1

for a constant switching frequency. For DCM, besides of expressions of average inductor currents, average input and output currents, average voltage across the energy storage capacitor C1 and average currents transistor and diode, we have to determine the parameter D2 that fixes the decay interval of inductor currents and the dc voltage conversion ratio implicitly.

Concerning the waveforms of inductor currents, it can see that the shapes of the currents are similar to those CCM.

IL01= - IL02 = IL0

The currents through inductors can be expressed as

IL1=i1+IL0

IL2=i2-IL0

For the first two time intervals D1Ts and D2Ts , change in IL1 and change in IL2 keep their validity. For the third time interval ( D3Ts ), the voltages across inductors are zero:

vL1 = vL2 = 0V. Taking into account that the average voltages across the inductors over a switching period are zero, the following relationships result

VC1= V1 = (D2/D1)*V0

The above relationships yield the dc voltage conversion ratio

M = V0/Vi = D1/D2 and VC1= V1

The remark regarding the relationship VC1= V1 that is the same as change in IL2 = (Vs*D)/L2*f for the converter with CCM keeps its validity for the converter with DCM. The determination of the conversion ratio M needs the value of parameter D2 too. It is obvious that the conditions of ripple cancellation from the inductor current found for CCM remain unchanged for DCM. Also, the two effective inductances and two parameters of conduction through the inductors hold their expressions.

Similar way D2= sqrt( Kem);

Kem = 2 Lem*f/R

Lem = L1e//L2e