Capacitive and inductive circuits  (Page 4/4)

Lrc circuits, resonance and signal tuning

A circuit containing a resistor, a capacitor and an inductor all in series is called an LRC circuit. Because the components are in series, the current through each component at a particular time will be the same as the current through the others. The voltage across the resistor will be in phase with the current. The voltage across the inductor will be ${90}^{\circ }$ ahead of the current (the current always follows or lags the voltage in an inductor). The voltage across the capacitor will be ${90}^{\circ }$ behind the current (the current leads the voltage for a capacitor). The phases of the voltages for the inductor and capacitor relative to the common current are shown in [link] .

The reactance of the inductor is $2\pi fL$ , and the reactance of the capacitor is $1/2\pi fC$ but with the opposite phase. So the total reactance of the LRC circuit is $X=X_L-X_C=2\pi fL-\frac{1}{2\pi fC}$ The impedance of the circuit as a whole is given by $Z=\sqrt{X^2+R^2}=\sqrt{{\left(2,\pi ,f,L,-,\frac{1}{2\pi fC}\right)}^2+R^2}$ At different frequencies, the impedance will take different values. The impedance will have its smallest value when the positive inductive reactance cancels out the negative capacitive reactance. This occurs when $2\pi fL=\frac{1}{2\pi fC}$ so the frequency of minimum impedance must be $f=\frac{1}{2\pi \sqrt{LC}}$ This is called the resonant frequency of the circuit. This is the frequency at which you can get the largest current for a particular supply voltage. It is also called the natural frequency of the circuit. This means the frequency at which the circuit would oscillate by itself.

Resonance

Resonance occurs when a circuit is connected to an alternating voltage at its natural frequency. A very large current can build up in the circuit, even with minimal power input.

An LRC circuit is very useful when we have a signal containing many different frequencies, and we only want one of them. If a mixed signal like this is connected to an LRC circuit, then only the resonant frequency (and other frequencies close to it) will drive measureable currents. This means that an LRC circuit can select one frequency from a range. This process is called signal tuning .

Filters and signal tuning

1. Which component would you use if you wanted to block low frequencies?
2. Which component would you use if you wanted to block high frequencies?
3. Calculate the impedance of a series circuit containing a 50 $\Omega$ resistor, a 30 $\mu$ F capacitor and a 3 mH inductor for frequencies of (a) 50 Hz, (b) 500 Hz, and (c) 5 000 Hz.
4. Calculate the resonant frequency of the circuit in the previous question, part (c).
5. A radio station broadcasts at a frequency of 150 kHz. The tuning circuit in the radio contains a 0.3 mH inductor. What is the capacitance of the capacitor needed in the tuning circuit if you want to listen to this radio station?
6. State the relationship between the phase of the voltages across an inductor, a resistor and a capacitor in an LRC circuit.
7. Explain what is meant by resonance.
8. Explain how LRC circuits are used for signal tuning, for example in a radio.