How Do You Calculate Total Impedance in a Tuned R-LC Circuit?

[bizuputyi]

Homework Statement

Given data of the given tuned R-LC circuit:

$Q=1000$
$f_{resonance} = 1MHz$
$I = 15 \mu A$
$V_{s} = 2.5V$
$R_{L} = 10kΩ$
$C=2nF$

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 $\pm$2% from resonance.

Homework Equations

$BW = \frac{f_{r}}{Q}$

$f_{lower}=f_{r} \Big( \sqrt{\frac{1}{4Q^2}+1}-\frac{1}{2Q} \Big)$

$f_{upper}=f_{r} \Big( \sqrt{\frac{1}{4Q^2}+1}+\frac{1}{2Q} \Big)$

$f_{r}=\frac{1}{2\Pi} \sqrt{\frac{1}{LC}}$

$Q=\frac{2\Pi f_{r}L}{R}$

$R_{dynamic}=RQ^2$

$\frac{Z}{R_{D}}=\frac{1}{1+j2Q\frac{δf}{f_{r}}}$

The Attempt at a Solution

$BW=1000Hz$

$f_{lower}=999.5kHz$

$f_{upper}=1000.5kHz$

$L=25.33 \mu A$

$R=0.159Ω$

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 $Z=\frac{V_{s}}{I}=166.67kΩ$?

or $Y=j2\Pi f_{r}C+\frac{1}{R+j2\Pi f_{r}L}$ then $Z=\frac{1}{Y}$

from which I get $Z=159006+j4780 Ω$ plus $R_{L}$ total impedance comes to $Z_{t}=169006+j4780 Ω$

or from dynamic impedance equation $Z=RQ^2=159kΩ$ plus $R_L$ again $Z_t=169kΩ$

or this is a bit complicated but I found it in the textbook:
$Z= \frac{R \Big( 1+\frac{(2\Pi f_r)^2L^2}{R^2} \Big) }{1+j(2\Pi f_r)\frac{L}{R} \Big( \frac{C}{L}R^2+(2\Pi f_r)LC-1 \Big) }$ from which I get $Z=159138+j1700 Ω$ plus $R_L$ again $Z_t=169138+j1700Ω$

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 $\pm$ 2%:

From the relevant equation I've got $Z=99-j3972Ω$ plus $R_L → Z_t=10099-3872Ω$

Your comments are greatly appreciated.

 

[rude man]

For high-Q circuits like this one you need to use approximations.

You should be able to derive the following:
|Z|/Zo ~ {1 + (2Qδ)²}^-1/2

where |Z| is magnitude of the RLC network impedance (as seen by R_L) at fractional freequency deviation δ, and Zo is the RLC network (real) impedance at resonance.

δ = (ω - ω_o)/ω_o
f = frequency, Hz
ω_o = resonant frequency
Q = quality factor at resonance.

You can then compute R and L for the RLC network and go on from there.

The sequence of computations might be ω_o → L → R → half-power δ → Z_o → |Z|_2%.

 

[bizuputyi]

Right, I think I get it. This is similar approach than mine and I get the same results. I was overthinking a bit on Z as such a high-Q circuit has a very sharp-edged graph.
That's one more tiny step towards my degree. Thank you!

 

[rude man]

Right, I think I get it. This is similar approach than mine and I get the same results. I was overthinking a bit on Z as such a high-Q circuit has a very sharp-edged graph.
That's one more tiny step towards my degree. Thank you!

Good luck to you!

 

[rezeew]

I would like to commend you on your efforts to solve this problem and find various solutions. It shows that you have a good understanding of the concepts involved in tuned R-LC circuits.

To answer your question about the total impedance, it is important to note that impedance is a complex quantity, meaning it has both real and imaginary components. Therefore, all of the solutions you have found are correct, as they represent different ways of calculating the total impedance at resonance. However, the most accurate solution would be the one that takes into account the dynamic resistance, as it considers the effect of the circuit's Q factor.

As for finding the impedance at frequencies \pm 2% from resonance, you have correctly used the formula provided in the homework. However, it is important to note that this formula assumes that the circuit is still operating in its resonant state. If there are any changes in the circuit parameters, such as the values of L and C, the impedance at these frequencies may not be accurately represented by this formula.

Overall, your approach to solving this problem is commendable, and your calculations appear to be correct. Keep up the good work and continue to explore different methods of solving problems to gain a deeper understanding of the concepts involved.