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DC distribution line voltage drop model

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mainakae:
There's not 2 without 3 (in spanish sounds better)

This time I'm not only asking, but also offering some equations and scripts that I've designed to solve a problem that I've faced during the design of the power distribution on my Dromo project.

The problem is to find the voltage drops along a dc power line, in which a number of devices are connected in series at different points in the line. All those devices are equipped with switching regulators, they can be feed at a quite range of voltages, and will consume different currents depending on the voltage at its input. Although it's not completely precise, I've modeled those devices as constant power devices, thus the current consumption will be a quotient between wats and volts. I've also considered a constant distance between devices, and the same power need for all of them. The line will be copper wire with a constant diameter all along.

The simplified equation to find the voltage at the input of a device would be something like this:

being X the device index, n the number of devices, Rl the resistance of the segment of line between this device and the previous one, and Id(j) the current consumption of the device indexed 'j'. As I told before, I've modelled the current consumption as a quotient of the constant wats the device consumes, and the voltage it sees at its input.

After simplifying the equations I find that I'm not that good at maths. For n=1 I got two solutions (quite interesting and curiously valid). For n=2 I got 4 solutions,

two of them are imaginary numbers (Even more interesting!) and so forth. I got tired of Dave's CAS, and tried an open CAS software: Maxima. I got solutions for systems of 15 equations in the blink of an eye (amazing stuff, really), but that was a complete mess. Then I found I could use the newton's method to iteratively solve the system given an initial solution, and here I give you what I came up with!:

--- Code: ---kill(all)\$
h(n):=[makelist(V[i]=V[i-1]-(17*(sum(1/V[j],j,i,n))*l*W)/(500*%pi*(dia/2)^2),i,1,n),makelist(V[i],i,1,n),makelist(V[0],n)]\$
consts:[n=8, l=10, dia=1, W=1.196, V[0]=24]\$
params:subst(consts,h(ev(n,consts)))\$
volts:mnewton(params[1],params[2],params[3])[1];
relvolts:makelist('dV[i-1] = rhs(volts[i-1])-rhs(volts[i]),i,2,length(volts))\$
watsseg:map(lambda([x], lhs(x)=float(ev(rhs(x),consts))),makelist('W[i-1]=(rhs(volts[i-1])-rhs(volts[i]))²/(l*0.068/(%pi*dia^2)),i,2,length(volts)))\$
waste:lsum(rhs(x),x,watsseg)\$
used:ev(W*n,consts)\$
print("total potential loss: ", dV:ev(V[0]-rhs(volts[n]),consts), "v, ", ev(dV*100/V[0],consts), "%")\$
print("total wasted wats:",waste)\$
print("used wats:",used)\$
print("efficiency:",used*100/(waste+used),"%")\$

--- End code ---

To run it, copy that into Maxima, change the values of n, l, dia, W, V[0] to your harts contempt, and press shift+return, et voila! (if it fails to come with a solution, it's because it's impossible to solve  :-DD )

If you have any fun reading this, or find that I'm a complete math disabled, please tell me :) Also, if you have a better solution (or in case this is just b*****t, a working one) please share!

IanB:

--- Quote from: mainakae on May 29, 2013, 12:00:18 pm ---The problem is to find the voltage drops along a dc power line, in which a number of devices are connected in series at different points in the line. All those devices are equipped with switching regulators, they can be feed at a quite range of voltages, and will consume different currents depending on the voltage at its input. Although it's not completely precise, I've modeled those devices as constant power devices, thus the current consumption will be a quotient between wats and volts. I've also considered a constant distance between devices, and the same power need for all of them. The line will be copper wire with a constant diameter all along.
--- End quote ---

A schematic diagram of the system would help to understand what you are trying to do here. I can't immediately make sense of it, and a diagram helps to clarify the thoughts both of the writer and of the reader before attempting to put down equations.

The most significant problem I have is that you describe "a number of devices connected in series", and yet you say "all these devices will consume different currents". This is, of course, impossible. If two or more devices are connected in series then all of them will be passing the same current, by definition. If you do not write equations that satisfy this constraint then your equations cannot be correct.

If each of the devices has a different power consumption, then each will have a voltage drop proportional to the power consumed, where

Vdrop = I x Preq

in which Vdrop is the voltage drop of the device, I is the current in the circuit and Preq is the power requirement of the device.

For this to work, each of the devices will need to be designed as constant current, variable voltage input devices. I think in general this is quite a rare and unusual design. Most all electronic devices are designed as fixed voltage, variable current loads. Certainly any standard switching regulator is designed this way (with an expectation of varying current draw at a given supply voltage).

So in summary, you cannot connect things in series like this. You need instead a two wire power feed (out and return), and you must connect your devices in parallel between the two wires. If you diagram this out, you will see the algebra for the voltage drop along the line becomes quite simple.

mainakae:
Sorry IanB, you are absolutely right, thanks for pointing that out. I wanted to say in parallel, but messed myself in the translation.

I've tried to represent the problem with this schematic:

IanB:
Ah, in which case it becomes a fascinating problem.

You can simplify the analysis by moving all the resistances to the V+ rail and considering V- as a common 0 V reference node. This means all your R values in the top rail will become 2R.

You then will have a problem you can solve by nodal analysis by doing a current balance around each V+ node and using Ohm's law to define the voltage drop across each resistance.

For the current in the loads you can do what you proposed and assume P = VI so that I = P/V.

If you do this you should get a set of linear simultaneous equations to solve. If all the R and P values are equal then the solution may indeed simplify down to a simpler expression. I have not done the working to compare with your result, but now you have me intrigued, so I dare say I will work it out at some point.

croberts:
I've always modeled my LED lights as constant power loads when designing the distribution of DC power from my battery bank. The LED drivers are designed to deliver a constant current of 1.1A over a DC input voltage range of 8V to 16V so for a given LED voltage drop of about 12V and a given driver efficiency the input power should be more or less constant. While the efficiency of the driver, temperature, etc. can all have an effect on the power demanded by the load, I think in cases like this the constant power load model is OK.

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