AC Circuits II - Impedance and Resonance

3-2-98

Relevant sections in the book : 23.3 - 23.4

RLC Circuits

Consider what happens when resistors, capacitors, and inductors are combined in one circuit. If all three components are present, the circuit is known as an RLC circuit (or LRC). If only two components are present, it's either an RC circuit, an RL circuit, or an LC circuit.

The overall resistance to the flow of current in an RLC circuit is known as the impedance, symbolized by Z. The impedance is found by combining the resistance, the capacitive reactance, and the inductive reactance. Unlike a simple series circuit with resistors, however, where the resistances are directly added, in an RLC circuit the resistance and reactances are added as vectors.

This is because of the phase relationships. In a circuit with just a resistor, voltage and current are in phase. With only a capacitor, current is 90° ahead of the voltage, and with just an inductor the reverse is true, the voltage leads the current by 90°. When all three components are combined into one circuit, there has to be some compromise.

To figure out the overall effective resistance, as well as to determine the phase between the voltage and current, the impedance is calculated like this. The resistance R is drawn along the +x-axis of an x-y coordinate system. The inductive reactance is at 90° to this, and is drawn along the +y-axis. The capacitive reactance is also at 90° to the resistance, and is 180° different from the inductive reactance, so it's drawn along the -y-axis. The impedance, Z, is the sum of these vectors, and is given by:

The current and voltage in an RLC circuit are related by V = IZ. The phase relationship between the current and voltage can be found from the vector diagram: its the angle between the impedance, Z, and the resistance, R. The angle can be found from:

If the angle is positive, the voltage leads the current by that angle. If the angle is negative, the voltage lags the currents.

The power dissipated in an RLC circuit is given by:

Note that all of this power is lost in the resistor; the capacitor and inductor alternately store energy in electric and magnetic fields and then give that energy back to the circuit.

Resonance in an AC circuit

There is a special condition in an AC circuit when the capacitive impedance is equal to the inductive impedance. Consider what happens with the impedance equation:

When the inductive and capacitive reactances are equal, a number of things happen:

This condition is known as resonance.

Both the inductive and capacitive reactances depend on frequency, with the inductive reactance proportional to frequency and the capacitive reactance inversely proportional to frequency. This means there will be a particular frequency that corresponds to impedance. The resonance frequency can be calculated from:

Solving this for the resonance frequency gives:

AM/FM radios use a resonant circuit in which the current increases sharply at resonance (this requires a small resistance). When the radio is tuned to a particular radio station, which is broadcasting at a particular frequency, it means that the circuit has a resonance frequency which is the same as the station's broadcast frequency. This amplifies that frequency selectively, picking that radio station out from the many signals, at all sorts of frequencies, that are being picked up by the radio's antenna.To tune the radio to another station, you turn a knob that changes the value of the capacitor (and / or the inductor) in the circuit, changing the resonance frequency to the frequency of the other radio station, amplifying that signal selectively over signals of other frequencies.

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