Electronic – Finding value of Resistance for Maximum Power
powerresistance
I am badly stuck in finding value for R in this circuit for maximum power transfer in the network,
It is my assignment question and I am badly confused.
Please help ,
Thanks.
Best Answer
For maximum power transfer you are expected to not only know how to set up equivalent circuits, but also how to find maximums and minimums by setting a derivative to zero.
The first step, of course, is to develop the equivalent minimum circuit:
You should have been able to follow the above steps. If not, then you have some serious study to do and you won't yet be able to solve these problems until you can perform the above steps on your own.
Now that you have the final circuit, you can work out the following equation:
Now, in the above equation for \$P_3\$, \$V\$ and \$R\$ are known constants. \$R_3\$ is the variable whose value you want to find in order to maximize \$P_3\$. This is the point where you are supposed to know from calculus courses (or reading) that you take the derivative, set that new expression to zero as an equation (to find where the slope is zero), and then solve. This will give you either a minimum or a maximum.
They are writing the power as a function of Rl, taking the first derivative to find the maximum, and then using the second derivative test to ensure that it is a local maximum and not a minimum. Since Vth is squared, and normally resistance is positive (though incremental resistance can often be negative) it's not hard to meet the second derivative test.
Observe that in the original circuit the resistors R6 and R7 are short as they have a wire connecting their ends. You can just replace them with a short. Replacing it with a short you are left with R4 and R5 in parallel. It should be fairly easy to solve from here. It is
Best Answer
For maximum power transfer you are expected to not only know how to set up equivalent circuits, but also how to find maximums and minimums by setting a derivative to zero.
The first step, of course, is to develop the equivalent minimum circuit:
simulate this circuit – Schematic created using CircuitLab
You should have been able to follow the above steps. If not, then you have some serious study to do and you won't yet be able to solve these problems until you can perform the above steps on your own.
Now that you have the final circuit, you can work out the following equation:
$$\begin{align*} I &= \frac{V}{R+R_3} \\ \\ P_3 &= I^2 R_3 = \frac{V^2 R_3}{\left(R+R_3\right)^2} \end{align*}$$
Now, in the above equation for \$P_3\$, \$V\$ and \$R\$ are known constants. \$R_3\$ is the variable whose value you want to find in order to maximize \$P_3\$. This is the point where you are supposed to know from calculus courses (or reading) that you take the derivative, set that new expression to zero as an equation (to find where the slope is zero), and then solve. This will give you either a minimum or a maximum.
$$\begin{align*} \frac{\textrm{d}P_3}{\textrm{d} R_3}&=V^2\frac{R-R_3}{\left(R+R_3\right)^3} \\ \\ \textrm{so,} \\ \\ V^2\frac{R-R_3}{\left(R+R_3\right)^3}&= 0 \end{align*}$$
I'll leave the last step of solving that answer to you. But that's all that is left.