# [Help] Power Loss in power cable

1. Dec 6, 2015

### Jamie Bristow

A small factory has decided to go Carbon neutral and has installed its own wind turbine. The wind turbine is installed on land belonging to the factory 1 km away. The factory owner needs 2km of copper wire of diameter 3mm to connect the turbine to the factory. The resistance of the cable is 1.2 Ω.

a) What is the resistivity of the wire?

The turbine generates 12V dc electricity and needs to supply the factory with 1kW of power. The factory owner has the choice inverting the supply to 230V or transmitting it at 12V. By calculating:

b) The current that passes along the cables if they are to deliver 1kW of power to the factory

c) The power that would be lost in the connecting wires

d) The total power that must be generated

e) The efficiency of the transmission

For both voltages decide which is a more efficient mode of transmission?

For: Resistivity
p= RA/L
I got 4.242x10^-9 ohmmeters to 4 s.f.

not sure how to implement the Power loss Formulas:
P = IV = I^2R = V^2/R

Would I use V=IR to find current and Voltage drop and the add to the power loss formulas or use I=p/v for current and use the initial voltages?

How do I go about finding the total power generated to deliver the 1KW of power to the destination? would it be a case of simple addition or is there a formula for this?

2. Dec 6, 2015

### Staff: Mentor

In order to deliver 1 kW of power to the factory, the factory end of the cable must have a combination of voltage and current such that $I V = 1kW$.

The thing is, for any given current there will be a voltage drop along the cable due to its resistance. In the case of the 12 V source the factory won't see that 12 V, it'll see 12 V less whatever drop occurred on the cables. See if you can set up an equation that reflects this. Write an expression that will give you the voltage $V_{end}$ at the factory end given that some current $I$ is being drawn. Find the current that will make $I V_{end} = 1 kW$. (Hint: it may not be possible!)

3. Dec 6, 2015

### Jamie Bristow

I was thinking this because this poor factory owner has built a windmill that generates 12 volts.... surely this Imaginary businessman's not that stupid.
I tried V=IR and P=VI but any combination just results in silly numbers.
Thank you for the response by the way.

4. Dec 6, 2015

### Jamie Bristow

For 12 volts I got

I=100012=83.3
V=2503(1.2)=100

So lost 88 more volts than started with. no use in continuing with 12v?

For 230 Volts:
I=1000/230=4.347826087
V=4.35(1.2)=5.217391304
Power loss = 22.68431002W

dont know if i got this right?

also not sure about the formula for finding the corrected power input to get 1KW output.

Last edited: Dec 6, 2015
5. Dec 7, 2015

### CWatters

You haven't calculated the current correctly yet. You assumed 1000W was sent rather than 1000W was received. You will need to write two equations before you can solve them.

One equation is for the delivered Power at the factory end in terms of the voltage at the factory Vf and the current eg..

Pf = Vf * I

The other one you need is an equation for the voltage at the factory end Vf in terms of the source voltage Vs (=230V) and the voltage loss in the cable. I'll let you write that one.

Solve for I and Vf

6. Dec 7, 2015

### Jamie Bristow

Power generated=? Power Received = 1KW
Voltage generated 12 or 230V Voltage recieved = ?
Current = ?
Resistance of wire= 1.2 ohms

So if I did V=IR and found the current along the line by using
I=12v/1.2ohms = 10
or
I=230v/1.2ohms = 191.6

then used the current for P=IV at the factory then
V=1000w/10=100 volts
or
V = 1000w/191.6 = 5.219 volts

so the voltage drop was
12 - 100 = -88
and
230 - 5.219 = 224.781

so the new p.d. at factory is either zero or 224v.

which makes some sense but if I add this back to P=IV or P=V2/R then I get silly numbers.

I cant see any other way of doing it

7. Dec 7, 2015

### Staff: Mentor

Here's a diagram we can work to:

Write KVL for the loop using $V_g$ and $V_f$ for the potentials at each end of the cable.

Consider how you might change or what you might do to the equation so that there will be a term for the factory power in it.

8. Dec 7, 2015

### Jamie Bristow

So I'm not that great with KVL since I only just read about it.
Are you Implying that Vg(Pw)-Vf(Pf)=0?

also is that +- on the resistor correct?
I thought this was going to be simple.

Last edited: Dec 7, 2015
9. Dec 7, 2015

### Staff: Mentor

No, I'm not implying that.

You should be able to write KVL for a loop if you've been given this problem as an assignment. I would think that Kirchhoff's laws would be one of the first things covered after Ohm's law.

KVL states that the sum of all potential changes around a closed path is zero (that is, if you set off from one spot in a circuit and "walk" around a path that ends up back where you started, then all the changes in electric potential that you crossed along the way will sum to zero and you'll be back at the same potential).

See if you can write that for this loop.
Yes, it represents the change in potential across the resistor due to the current flowing through it. Think Ohm's law.

10. Dec 7, 2015

### Jamie Bristow

The course im studying is mostly home study, we get one lesson a week on physics and one month to study for an exam. I have kind of neglected KVL since I know that most of the Exam is going to be using maxwells laws and drift velocities of elctrons and so on.
I can understand this is a simple task but I just wrote 27 pages on everything else to do with electricity and my brain is in overload right now.

Ok so KVL makes sence because you can't get something from nothing right? except our universe but thats off topic.

so if current starts at windmill with 0v and gain voltage, gets pushed anticlockwise and looses voltage toward factory then It must lose voltage as as it returns since the resistivity is constant through the wire?
Im just not getting something and its bugging me since I usually get this stuff really easily.

11. Dec 7, 2015

### Staff: Mentor

Going around the loop clockwise starting from the bottom left corner of the circuit:

Gain of Vg due to windmill generator
Drop across 1.2 Ohms
Drop of Vf across factory

Write it as an expression.

12. Dec 7, 2015

### Jamie Bristow

for 230V -

The voltage drop in the wire is 5.22V am i correct?

so starting from the bottom-left

gain of 230V
minus 5.22V across the first wire
minus 219.56V across the factory
and minus 5.22V across the second wire

Last edited: Dec 7, 2015
13. Dec 7, 2015

### CWatters

No that assumes you know what the current it. Don't substitute values yet. The voltage drop in the resistor is I*R.

I is unknown but don't worry about that for the moment just write the equation.

14. Dec 7, 2015

### Jamie Bristow

Ok so V0+Vg-V1-Vp=V0?

or I could do V2/R to find powerfrom windmill?

Last edited: Dec 7, 2015
15. Dec 7, 2015

### Jamie Bristow

I don't get it? I'll just Find another way of doing it.
I'm sorry you guys are probably frustrated with me, I have a learning disability and I get that a lot.

cant you just give me the order of the formulas I need to use?

because that is what is confusing me.

16. Dec 7, 2015

### Staff: Mentor

You need to be able to write a KVL equation using the given voltages, component values (resistors) and current variables. That's a basic necessity.

There's only one current (unknown, variable: $I$)
There's one resistor (known value: R = 1.2 Ω)
One voltage source (known value(s): $V_g$ = 12 V or 230 V)
One load voltage (unknown, variable: $V_f$)

There is one constraint given: Load power is to be 1kW.

I don't recognize some of those variables from the diagram. You'll have to define what they are for the equation to make sense. Also, V0, whatever that is, appears on both sides of the equation, so should vanish (cancel).

I'm making a guess that your V1 is meant to represent the voltage drop across the cable resistance? If so, replace it with the voltage drop due to current through the resistor (Ohm's law).
No, that would be true if the factory end of the cable was shorted out (no load, the wire just loops back to the generator). In that case all the generator would be accomplishing is heating the cable!

Note that the power lost on the cable $I^2 R$ is not the power that the factory sees. The factory sees $P_f = I V_f$.

Can you take another stab at writing the equation for the loop? I think you're getting very close.

17. Dec 7, 2015

### Jamie Bristow

Ok so ill try this
Vg-Vd-Vp=0?
Just got a nice book out on KVL and KCL ill give this a read for an hour or so.

18. Dec 7, 2015

### Staff: Mentor

I don't know what $V_d$ or $V_p$ are: They haven't been defined. You'll want to use the current and resistor values to express potential drops when possible.
Okay.

19. Dec 7, 2015

### Jamie Bristow

Sorry, I meant f, not p

So: Vg - 1.2Ω - Vf = 0?

20. Dec 7, 2015

### Staff: Mentor

Multiply the resistance by $I$ and you've got it. Ohm's law: $V = I R$. $I$ is the (as yet unknown) current passing through the resistor, so the (as yet unknown) potential drop across the resistor is $I~1.2 Ω$. All the terms of an equation must have the same units. In this case, volts. So the lone resistance value kind of stuck out.

Now we don't yet know what the factory voltage $V_f$ is, nor do we know the current $I$ that it draws. However! We do know that the factory wants to use 1 kW of power. Power is given by $I V$. So how might you modify your KVL equation to incorporate that constraint? Here's the equation in its current form:

$V_g - I R - V_f = 0$

Note the $V_f$ variable sitting there. It's an unknown...now if only it were the load power instead....