- #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
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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