LM3000: How Equation 29 is derived?

kingson zhang 

Hello, my name is Peter Miller, I am one of the other applications engineers supporting the LM3000 and assisting Britton Jones.

The Transient equations in the LM3000 datasheet are simplified estimations designed to provide an output capacitance that has a high-confidence of meeting the required transient response once the current mode control loop is compensated later in the design.  As such, they include some deliberate over-estimations to account for the non-ideal loop bandwidth response.  In addition, there appears to have been an error made in the second term's estimate of the output voltage ripple, which should not have Co in the numoerator.

The first term, as previously discussed, is based on the Charge flow through the inductor as it is fluxed up in response to a load step or defluxed in response to a load drop.  You are correct that this term is missing a 1/2 factor, as the average current through the period is ΔI / 2.  This is intentional and provides an over-estimation of the output capacitance to account for the control loop delay, which prevents the LM3000 from instantly adjusting it's duty cycle to 100% or 0% when the load changes.

The second term was suppose to be an additional term for the output ripple, to account for the fact that the transient may occur anywhere within the steady-state ripple on the output voltage.  It appears there was an error made in the derivation when they attempted to solve ΔVout from equation 25 by substituting ΔIL for for it's components Tsw x VL / L.  I would recommend replacing the second term with ΔVout from equation 25, to estimate Vp, or reduce Vp by ΔVout when calculating the minimum allowable capacitance for transient.

Again, this provides a small over-estimation of the required output capacitance so that when the loop compensation is done and the final loop bandwidth is established, the resulting transient voltage is lower than Vp used to obtain the output capacitance.  After you have selected Ren, Rcomp and Ccomp going through the compensation section, you can obtain a more accurate estimation of Vp by calculating the output capacitor impedance at the cross-over frequency using the equation:

Zout(f) = Rc + 1 / (2 × π × Fco × Co) 

If you are using multiple types of output capacitors, calculate each group of similar output capacitors separately, then add them in parallel using the formula Zeq = 1 / (1/Z1 + 1/Z2...1/Zn)