# power supplied to a resistive load cannot exeed (V\2)^2/R^2

• October 26th 2010, 01:34 PM
bigwave
power supplied to a resistive load cannot exeed (V\2)^2/R^2
A battery has internal resistance $R_i$

and open circuit terminal voltage $V_b$

Show that the power supplied to a resistive load cannot exceed $\frac{({\frac{V_b}{2})^2}}{R_i}$

well I only changed this to

$\frac{{\frac{V_b}{2}^2}}{R_i}
\Rightarrow
({\frac{V_b}{2})^2}\times\frac{1}{R_i}
\Rightarrow
\frac{V_b^2}{4R_i}
$

but not sure how you determine what cannot be exceeded

any suggest...(Cool)

mod.... this should of been put in "other topics" its not an advanced topic.
• October 26th 2010, 02:00 PM
zzzoak
Internal resistance r.
External R.

$
\displaystyle { I=\frac{V}{r+R} }
$

$
\displaystyle { P=I^2 \; R \; = \; \frac{V^2}{(r+R)^2} \; R
}
$

Max power on R is

$
\displaystyle { \frac{dP}{dR} \; = \; 0
}
$

we get

$
R=r
$

and

$
\displaystyle { P(max)=\frac{V^2}{4r}
}
$
• October 26th 2010, 02:10 PM
Ackbeet
I've attached a circuit model of the situation. Would you agree with this model?

Attachment 19481

Let's say this model is accurate. Now what? Well, the one constant thing is the voltage of the battery. That doesn't change (at least, not theoretically). But as you change the load resistance, the relative voltage drops across each resistor changes. You want to maximize the power in the load. Now, Ohm's Law yields

$V_{\text{b}}=I(R_{\text{i}}+R_{\text{load}}).$

The power dissipated in the load resistor is $P_{\text{load}}=I^{2}R_{\text{load}}=\left(\dfrac{ V_{\text{b}}}{R_{\text{i}}+R_{\text{load}}}\right) ^{2}R_{\text{load}}.$

Now use the usual Calc I procedure of setting

$\dfrac{dP_{\text{load}}}{dR_{\text{load}}}=0,$ and solve for $R_{\text{load}}.$

What do you get?
• October 26th 2010, 02:29 PM
bigwave
verify
sure appreciate the help... now I see how this works...thanks...

there was a follow up question on this.... not sure if input it right

thus if $V_b=96V$ and $R_i = 50\Omega$.

Discrete loads of $150, 100, 50, 30,$ and $20m\Omega$ are connected, one at a time, across the battery. Plot the curve of power supplied versus the ohmic value of the load. I tried to use the Wolfram|Alpha to graph this but didn't get the expression right http://www.wolframalpha.com/input/?i=plot[P%3D%28%2896%29^2%29%2F%284\Omega%29%3B%28P%2C-50%2C200%29%2C%28\Omega%2C-50%2C200%29]

Hence, verify that the maximum power transfer occurs when $R_i = R_{load} = 50\Omega$
• October 26th 2010, 06:13 PM
Ackbeet
Well, here's a continuous plot. If you wanted to do a discrete plot, I'd use Excel.
• October 26th 2010, 08:59 PM
bigwave
appreciate the plot

well. I need a lot of help with this EE subject.

more ?? to post on the way
• October 27th 2010, 03:02 AM
Ackbeet
Did you mean you need more help with this problem, or just more help with problems in the same subject?
• October 27th 2010, 03:23 PM
bigwave
Im ok with this one

but I will be posting more new problems

its a new subject to me.
• October 27th 2010, 05:16 PM
Ackbeet
Ok. I'll be looking for more threads. Have a good one!