Find frequency such that two components have same average power

In summary, the two voltage dividers will behave identically if the components are placed in equivalent locations.
  • #1
Tobias Holm
3
1
Homework Statement
At what frequency is the average power in the two resistors the same? (Hint: Use a symmetry argument as calculations are tough). Use inductance ##L=2CR^2##.
Circuit: https://drive.google.com/file/d/17yHMDfH5zY7J42wewqI4P4nLTkpsXhQa/view?usp=sharing
Relevant Equations
Kirchhoff's loop law:
$$ \sum_j \tilde{V}_j = \sum_j \tilde{I}_j Z_{L, j} + \sum_j \tilde{I}_j Z_{C, j} + \sum_j \tilde{I}_j Z_{R, j} $$

Component impedance:
##Z_L = -i\omega L##, ##Z_C = i/(\omega C)## and ##Z_R = R##.

AC current:
$$ \tilde{I}(t) = |\tilde{I}_{max}| \cos (\omega t + \phi) $$

Average power:
$$P_{ave} = \frac{1}{2} |\tilde{I}_{max}|^2 R = \frac{V_0^2}{|Z|} \cos (\phi) $$
Where ##|\tilde{I}_{max}|## is the amplitude of the complex current and ##V_0## the amplitude of the AC voltage.
(This is my first time posting here, sorry in advance for any difficulties. )

All componenets of same type has same magnitude, so e.g. the two resistors both have $R$ resistance.
Given the difficulty of the previous exercises, I believe I'm over complicating the problem. However, here is what I've tried:

1) Use that the power in each resistor is half the total power.
I know the effective impedance is simply ##Z_{eff} = R## (calculated in earlier exercise). Thus the total average power is: $$P_{avg}^{tot} = \frac{1}{2} |\tilde{I}_{max}|^2 R$$ and each of the resistors must experience half of this power:
$$ P_{avg} = \frac{1}{2} |\tilde{I}_{max, R}|^2 R $$ $$ = \frac{1}{2}P_{avg}^{tot} = \frac{1}{4} |\tilde{I}_{max}|^2 R$$
I thought I might be able to find the angular frequency by using ##|\tilde{I}_{max, R}| =\frac{\tilde{I}(t)}{\cos (\omega t + \phi)}## but this seems rather complicated.

2) Set the two average power expressions equal to each other
This one I'm less certain about (I will leave out the tildes for brevity)
$$ P_{avg, 1} = P_{avg, 2} $$ $$ \frac{1}{2} I_{max, 1}^2 R_1 \cos(\alpha) = \frac{1}{2} I_{max, 2}^2 R_2 \cos (\beta)$$
I believe the trick is to say that ##\omega## is the resonance frequency, such that ##\alpha = \beta = 0##:
$$ I_{max, 1} = I_{max, 2} $$
But that does not give me an expression for ##\omega##, nor does it prove that the two currents actually are equal.

3) Use Kirchhoff's Loop- and Current Rule.
Since each resistor has same resistance, I should find an ##\omega## that makes the current flowing in the resistors equal to each other. If I write all the necessary equations for the loops and the condition:
$$ |\tilde{I}_{R1}| =|\tilde{I}_{R2}| $$
I get the result:
$$ \omega = \frac{1}{\sqrt{2} CR} = \frac{1}{\sqrt{LC}}$$
Which is the resonance frequency for a simple RLC-circuit. I believe this is the right answer, however it requires immense amount of algebra, and I have to do this exercise by hand, thus I'm inclined to believe this is not the correct method of obtaining the result.

Thanks in advance.

EDIT: Formatting
 
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  • #2
If you ignore the resistors for a moment, you'll see that you have a pair of voltage dividers consisting of series-connected L and C. What can you say about the reactances of L and C if both arrangements are to produce identical potential drops at their junctions?

1639323366636.png
 
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  • #3
If we ignore the resistors, we will see that for the two paths:
$$ V = I_1 (Z_1+Z_2) = I_2 (Z_2 +Z_1) $$
$$ I_1 =I_2 $$
So the current that runs through the pair is equal to each other.
However, I cannot see how we can just ignore the resistors? I get that we can conclude that their reactans is identical, but is that necessarily true for the original case, where the last one is in parallel with a resistor?
gneill said:
If you ignore the resistors for a moment, you'll see that you have a pair of voltage dividers consisting of series-connected L and C. What can you say about the reactances of L and C if both arrangements are to produce identical potential drops at their junctions?

View attachment 294034
 
  • #4
If the two voltage dividers behave identically at the given frequency, then adding equal resistors in equivalent locations will have identical results for each path.
 
  • #5
gneill said:
If the two voltage dividers behave identically at the given frequency, then adding equal resistors in equivalent locations will have identical results for each path.
I'm not sure exactly where you are getting at - Wouldn't adding the resistors at "equivalent locations" be on the same component, so e.g. we put the resistors on both capacitors not a capacitor and an inductor?
However, your comments made me realize the solution:
If I can make ##Z_L = Z_C## then the two voltage dividers is equal to each other, and thus the power in the resistors must be the same.

You have my thanks! The many hours I have spend on such a simple problem.
 
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Related to Find frequency such that two components have same average power

What is the meaning of "average power" in this context?

In this context, "average power" refers to the average amount of energy that is being transmitted by a signal over a certain period of time. It is a measure of the strength or intensity of the signal.

Why do we need to find the frequency at which two components have the same average power?

Finding the frequency at which two components have the same average power is important in order to achieve a balanced transmission of energy between the two components. This can help optimize the performance and efficiency of a system.

How is the frequency at which two components have the same average power calculated?

The frequency at which two components have the same average power can be calculated by using the formula P = VI, where P is the average power, V is the voltage, and I is the current. By setting the average power of both components to be equal, the frequency can be solved for.

What factors can affect the frequency at which two components have the same average power?

The frequency at which two components have the same average power can be affected by factors such as the type and properties of the components, the circuit design, and external interference. It is important to consider these factors in order to accurately determine the frequency.

What are some applications of finding the frequency at which two components have the same average power?

This concept is commonly used in the design and optimization of electrical circuits, such as in radio frequency systems, power distribution networks, and telecommunications. It can also be applied in various fields such as physics, engineering, and telecommunications research.

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