PHYS208 5/8 Class

ac Circuits (contd.)

Reworking the formula for the current resulting from an ac emf applied to an RLC series circuit leads to the universal resonance formula.

Universal resonance with quality factor:

Quality factor for RLC series ac circuit:

Resonance in an RLC Series Circuit

Resonance occurs in a circuit when the maximum value of the current is achieved. Examining the numerous formulas associated with the RLC series circuit, we find that the following conditions are synonymous with resonance:

• I_m is a maximum.
• Z is at a minimum; Z = R; Z is purely resistive.
• X_L = X_C; inductive reactance cancels capacitive reactance; net reactance is zero.
• The phase angle is zero; the current is perfectly in phase with the applied emf; the tangent of the phase angle is zero.
• The driving frequency is identical to the natural frequency.
• The power factor is unity (see next class).

Note that at resonance, V_L = V_C, so that the net voltage across the reactive elements is zero (added as phasors). However, each reactive element itself may have a very large voltage. In a high-Q circuit, the voltage across the reactive elements may be many times larger than the applied emf.

Concept Check

An Interesting Application of Phasors

Quiz: LC Circuits - (Solution)

"http://www.physics.udel.edu/~watson/phys208/clas0508.html"
Last updated May 6, 1998.
Copyright George Watson, Univ. of Delaware, 1997.