RLC Circuit (with variable frequency)

[meb09JW]

Homework Statement

Find the frequency at which the circuit becomes purely resistive and calculate the effective resistance.

Here is the circuit with the given solution-

The trouble is, whilst the solution is simple enough, there are many places to go wrong, and the whole process is time comsuming. I've managed to simplify a few other answers given in this answer booklet, but for this one, i can't find an easier method.

Any ideas? I can't find a similar example in Hughes.

Thanks,
Josh.

 

[ehild]

If Z is real so is 1/Z.

$$\frac{1}{Z}=\frac{R-j\omega L}{R^2+(\omega L)^2}+j\omega C=\frac{R}{R^2+(\omega L)^2}+j\omega( C-\frac{ L}{R^2+(\omega L)^2})$$

ehild

 

[kerimek]

I understand the frustration of complex solutions and the importance of finding easier methods. In this case, it may be helpful to first understand the concept of resonance in an RLC circuit. At resonance, the reactive components (inductance and capacitance) cancel each other out, leaving only the resistance in the circuit. This means that at resonance, the circuit becomes purely resistive.

To find the resonance frequency, you can use the formula f = 1/2π√(LC), where L is the inductance and C is the capacitance in the circuit. Once you have calculated the resonance frequency, you can calculate the effective resistance by using Ohm's Law (R=V/I) and measuring the voltage and current in the circuit at resonance.

It is also important to note that the effective resistance may vary depending on the quality factor (Q) of the circuit. A higher Q value means a sharper resonance peak and a more purely resistive circuit. Therefore, it may be helpful to also calculate the Q value of the circuit to get a better understanding of its behavior.

I recommend consulting your textbook or other resources for more examples and practice problems to improve your understanding of RLC circuits and resonance. With practice, you will become more familiar with the concepts and be able to find easier methods for solving these types of problems.