# Voltage multiplier circuit - analytical expression

Discussion in 'General Electronics Chat' started by nicrip, Dec 2, 2008.

1. ### nicrip Thread Starter New Member

Dec 2, 2008
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I have been assigned a task that I have found to be very difficult - to try and develop an alaytical (mathematical) expression for the output voltage with respect to time of the cockroft-walton voltage doubler shown below (so that I can plot the transient response), when the input voltage is a sinusoidal source. I was wondering if anybody would know how this could be done, or if such a expression has already been developed. The problem is that the diodes continually switch and charge the caps, but I cannot assume that the caps (C1) would charge to the peak input voltage in one cycle.

Similarly with the rectifier circuit below (assume the resistor is replaced by a capacitor), how would I develop a mathematical expression for the output voltage if you cannot assume that the cap voltage reaches the peak input voltage in one cycle? Would I somehow have to find the points at which the diode switches, and continually calculate the voltage on the cap as it gains voltage over several cycles of the input voltage? I appreciate any help.

Jan 28, 2005
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3. ### The Electrician AAC Fanatic!

Oct 9, 2007
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Is this classwork or job work? This will indeed be difficult.

Why must you have an analytical expression? There will not be a simple expression. Can't you just simulate the transient response with SPICE for the particular transient conditions you care about?

4. ### nicrip Thread Starter New Member

Dec 2, 2008
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The work is part of a job project. I already expressed my belief that it was extremely difficult to solve analytically, but I was not believed (although the circuit appears simple, to solve it would be very difficult, and i do not think he knows too much about circuit analysis). I guess the best way to solve the problem would be numerically with spice.

5. ### steveb Senior Member

Jul 3, 2008
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I wanted to respond to this question before, but was not sure enough. However, I'll give my opinion now.

I think developing an analytical expression will prove very difficult, if not impossible. This is because rectifier circuits are time dependent nonlinear systems which do not lend themselves to easy simplification.

However, there are cases where simplifications may be made. For example, if you look at references for 3-phase (also called 6 pulse) bridge rectifiers used on the output of 3-phase generators, you can find analytical expressions (though they are complex) using what is called an average model. However, a key assumption of this technique is that the output current changes slowly compared to the generator frequency which is usually only true for an inductive load. With non-inductive loads, it is well-known that numerical solutions are needed.

I mention these facts so that you can decide if pursuing an analytical approach is worth the time investment and risk of failure. Also, should you decide to pursue this, the 6-pulse bridge rectifier analysis is a good place to start.

see the following reference paper and text book for details:

S.D. Sudhoff, H.J. Hegner and K.A. Corzine, “Transient and dynamic average-value modeling of synchronous machine fed load-commutated converters”, IEEE Trans. on Energy Conversion, Vol. 11, No. 3, September 1996.

P.C. Krause, O. Wasynczuk and S.D. Sudhoff, “Analysis of Electric Machinery and Drive Systems”, 2nd edition, IEEE Series on Power Engineering, Mohamed E. El-Hawary, Series Editor, Wiley Inter-science, ISBN 0-471-14326-X, 2002.

The other avenue is to check books and references on switching power converters. Perhaps this problem has been studied before, and approximate techniques may already exist.

Last edited: Dec 3, 2008
6. ### The Electrician AAC Fanatic!

Oct 9, 2007
2,301
339
A paper was published in 1979 giving a good solution to the problem of a single phase rectifier with capacitor filter, but only for the steady state result, not for the initial transient response. It is quite complicated. To solve this problem, you must know the leakage inductance of the transformer, the ohmic resistance of the windings, the ESR of the capacitors, the I-V characteristic of the diodes and the load, etc.

If you would like a copy to show your boss, post an email address, and I'll email it to you. You should doctor the email address so spammers can't harvest it, but so humans can figure it out.