Question

A tank circuit uses a 0.09 uH inductor and a 0.4-uF capacitor. The resistance of the inductor is 0.3 ohms. Would the quality of the inductor be 158 and the bandwidth be 5.3?

Answer 1

Quality Factor of an inductor is defined as the ratio between the impedance of the inductor and its resistance:\[Q=\frac{ \omega_0 L }{ r }\] You need to know the angular frequency wo and this can be determined from the design of the tank circuit. As you know, its resonant frequency is given by:\[\omega_0=\frac{ 1 }{ \sqrt{LC} } \rightarrow Q=\frac{ \omega_0 L }{ r }=\frac{ L/\sqrt{LC} }{ r}=\frac{ 1 }{ r }\sqrt{\frac{ L }{ C }}\]In our case:\[Q=\frac{ 1 }{ 0.3 }\sqrt{\frac{ 0.09}{ 0.4}}=1.58\]Band Width is defined as:
\[BW=\frac{ f_0 }{ Q }=\frac{ f_0 }{ 2 \pi f_0L/r }=\frac{ r }{ 2 \pi L}=\frac{ 0.3 }{ 2 \pi· 0.09 ·10^{-6}}=530,516 Hz\]In order to get Q=158, the inductance would need to be 10,000 times bigger, say 0.9 mH. That would give Q=158 and BW=53

Answer 2

As far as increasing the inductive reactance by increasing the inductor's value, you will have to make a change in the value of C in order to keep the resonant frequency of the tank at its original value. The increase in L will lower the resonant frequency of the tank. You would then need to make C smaller to bring up the frequency.

Answer 3

It is the internal resistance of the inductor that is the major factor that reduces the Q of the tank.

Answer 4

@radar I agree with that required modification of C. I have eliminated frequency for Q and BW calculations for the sake of simplification and then it is not so obvious. In order to keep resonant frequency and get Q=158, the best thing would be to use 100xL (although 100xL will increase r too) and C/100 instead 10,000L as I had proposed. That would keep frequency but will make BW wider by 100 times, leading to BW=5300 Hz.

Answer 5

Yes, I noticed your solution did not involve finding the tank frequency. We both came up with a Q of 1.58, but my method was more work, I first found the tank freq, then the Inductive reactance at that freq' and then the Q. Your method was a more elegant method and I learned from it.

Answer 6

thank yoy, that is very kind!