Buck Converters in DCM in Steady State

[dvscrobe]

Homework Statement

This problem comes out of a EET course called Power Electronics. See equation 3.48a. The text doesn’t show the derivation and I could not figure it out even after asking the instructor. This is for learning how DC-DC buck converters operate in a discontinuous mode. A buck converter uses transistors and passive components to step DC voltages down.

Relevant Equations

Power in = Power out

Let K=2Lfs and Pin=Pout,

Have,

(Vin-Vo)/k D^2 Vin =Vo ^2/R

(I am fine up to this part. I am equating input power to output power)

V^2=D^2 + D^2 R/2 Vo Vin - R/k D ^2 Vin =0

(This is where the instructor takes over. Can’t figure out how he got to V^2. Where did that come from? Thought I was just dealing with Vin and Vout.)

Let M=D^2 R/k

Vo=Vin/2 (sqrt(M^2+4M ) -M)

 

[anorlunda]

Is the problem a buck boost converter feeding a resistive load R?

I'm thinking of $P=\frac{V^2}{R}$

 

[dvscrobe]

anorlunda, After much headache, I realized the book is using a quadratic equation to solve for Vout. The instructor didn’t explain very well. If he would have said word quadratic, a light bulb would went off.