Electricity and Magnetism Unit Ac: Alternating Current Circuits

Prerequisite:E2, E4, W1

Overview:

In this unit we consider a circuit containing a resistor, inductor , and capacitor in series witn a sinusoidal applied voltage. The mathematical description of this system is equivalent to that of the forced damped harmonic oscillator discussed in unit W1 except for the names of the variables. In particular there can be resonance phenomena in such a circuit if the driving frequency matches the natural frequency of the circuit.

E. M. Purcell, Electricity and Magnetism, Berkeley Physics Course, Vol. 2, 2nd Ed. , Chapt. 4 -Electric Ciruits, Sec. 4.11; Chapt. 7 - Elecronmagnetic Induction, Sec. 7.9; Chapt. 8 - Alternating Current Circuits, Sec. 8.1-8.5.

Understand:

• How the voltages across a resistor, inductor, and capacitor is related to the current, rate of change of the current, and the charge, respectively.
• The use of complex impedance in the analysis of circuits.
• Free oscilaations (or transients) in LC circuits.
• Forced oscillations in LR, RC, and LRC circuits (see mFigs 8.5, 8.7, and 8.10 in Purcell) .
• Calculation of average power in circuit elements.
• Discussion:

Consider a resistor, inductor, and capacitor in series. The total voltage drop over all three is V_R + V_l + V_C = IR + L (dI/dt) + Q/C and is equal to the time-dependent apllied voltage V(t). Recalling that I = dQ/dt and dividing the equation by the inductance L we find the differential equation for Q as a fuction of the independent variable t: d^2Q/dt^2 +(R/L)dq/dt + (1/LC)Q = V(t)/L . This has exactly the form of the equation for a forced damped harmonic oscillator discussed in unit W1 with a re-labelling of variables. For example x becomes Q, the damping coefficient gamma becomes R/L and the square of the natural frequency omega_0^2 becomes 1/LC. This correspondence between the LCR circuit and the forced oscillaator, often called the mechanical analog, allows us to understand the behavior of the RLC circuit in terms of that of the forced oscillator previously studied.

Problems: Purcell 8.1,8.2(omit part (e)),8.3,8.4, 8.6, 8.10.