The input voltage is 5V for each circuit shown above. I need to find the gain of these two RLC circuits using the formula G(ω) = V_out / V_in. With this formula, I want to find the half-power frequencies of the two circuits by squaring |V_out|/|V_in| and then setting it equal to 0.5. I believe the resonance frequencies are are found by setting the reactance of of the inductor equal to that of the capacitor. This gives me f = sqrt(1/LC)/2π. Please tell me if there is a way to get it using only the gain formula.

I have made a plot of log10(f) vs. V_out/V_in in LTspice for each circuit to get an idea of what the resonance and half-power frequencies should look like.



The solid line is the magnitude of V_out/V_in while the dotted line is its phase. For the first circuit, the resonance frequency is f = 5033 Hz, and the half-power frequencies are around f = 1460 Hz and f = 17386 Hz based on this graph. For the second circuit, the resonance frequency is f = 15915 Hz, and the half-power frequency is around f = 23801 Hz.

I have tried taking the complex impedences of the inductor (jωL), the capacitor (1/jωC) and the resistor (R) then using voltage division to find the output voltage, but I am not getting the correct answers. When do this then solve for the half-power frequency, I always get 0.5=1/(2π*f^2*R^2*C^2), which does not give me the answers above. Perhaps someone can show me how to derive the gain correctly.

 

These might help -

https://www.google.com/url?sa=t&rct.../filters.pdf&usg=AOvVaw0x3JOHd_-6zfk4no27aQrl

https://www.google.com/url?sa=t&rct...ab11_s05.pdf&usg=AOvVaw2OANBHGp-e7Uz-3EvvvCjx

Regards, Dana.

 

Thanks, Dana.

One last thing. I now have G(ω) = jωRC/(1+jωRC-(ω^2)LC) and G(ω) = 1/(1+jωRC-(ω^2)LC) as the gain functions of both circuits. Now, I need to find the half-power frequencies. To do that, I need to set the gain functions equal to 0.5 and square the magnitudes of the gain expressions. This is tricky since I can't just multiply the denominators by their complex conjugates like I could for complex binomials.

How would I go about finding the magnitudes of each?

 

This should help -

https://electronics.stackexchange.c...ransfer-function-magnitude-how-to-deal-with-j

Half power point -

https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Half_power_point.html

So your source is 5, therefore Vout = Vin / root(2) = 5 x .707


Regards, Dana.