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bizuputyi
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Homework Statement
Given data of the given tuned R-LC circuit:
[itex] Q=1000 [/itex]
[itex] f_{resonance} = 1MHz [/itex]
[itex] I = 15 \mu A [/itex]
[itex] V_{s} = 2.5V [/itex]
[itex] R_{L} = 10kΩ[/itex]
[itex] C=2nF [/itex]
Calculate the bandwidth, half-power frequencies, the value of R and L.
Estimate the impedance offered to the supply at resonance and at the frequencies of [itex] \pm[/itex]2% from resonance.
Homework Equations
[itex] BW = \frac{f_{r}}{Q} [/itex]
[itex] f_{lower}=f_{r} \Big( \sqrt{\frac{1}{4Q^2}+1}-\frac{1}{2Q} \Big)[/itex]
[itex] f_{upper}=f_{r} \Big( \sqrt{\frac{1}{4Q^2}+1}+\frac{1}{2Q} \Big)[/itex]
[itex] f_{r}=\frac{1}{2\Pi} \sqrt{\frac{1}{LC}}[/itex]
[itex] Q=\frac{2\Pi f_{r}L}{R}[/itex]
[itex] R_{dynamic}=RQ^2[/itex]
[itex] \frac{Z}{R_{D}}=\frac{1}{1+j2Q\frac{δf}{f_{r}}}[/itex]
The Attempt at a Solution
[itex] BW=1000Hz[/itex]
[itex] f_{lower}=999.5kHz[/itex]
[itex] f_{upper}=1000.5kHz[/itex]
[itex] L=25.33 \mu A [/itex]
[itex] R=0.159Ω [/itex]
Do these calculations appear to be correct?
I'm struggling to find total impedance of the circuit, although I have some idea:
Is it simply [itex] Z=\frac{V_{s}}{I}=166.67kΩ[/itex]?
or [itex] Y=j2\Pi f_{r}C+\frac{1}{R+j2\Pi f_{r}L}[/itex] then [itex] Z=\frac{1}{Y}[/itex]
from which I get [itex] Z=159006+j4780 Ω [/itex] plus [itex] R_{L} [/itex] total impedance comes to [itex] Z_{t}=169006+j4780 Ω [/itex]
or from dynamic impedance equation [itex] Z=RQ^2=159kΩ [/itex] plus [itex] R_L [/itex] again [itex] Z_t=169kΩ [/itex]
or this is a bit complicated but I found it in the textbook:
[itex] Z= \frac{R \Big( 1+\frac{(2\Pi f_r)^2L^2}{R^2} \Big) }{1+j(2\Pi f_r)\frac{L}{R} \Big( \frac{C}{L}R^2+(2\Pi f_r)LC-1 \Big) } [/itex] from which I get [itex] Z=159138+j1700 Ω [/itex] plus [itex] R_L [/itex] again [itex] Z_t=169138+j1700Ω[/itex]
Which of the total impedance is right? Maybe all of them are acceptable? Or a fifth one?
And as of to find total impedance at the frequencies [itex] \pm [/itex] 2%:
From the relevant equation I've got [itex] Z=99-j3972Ω [/itex] plus [itex] R_L → Z_t=10099-3872Ω [/itex]
Your comments are greatly appreciated.