FACULTY OF SCIENCE, WINTER 2008

PHYS 126 LEC B3 : Fluids, Fields and Radiation (Instructor: Marc de Montigny)

Walker, Physics, Chapter 24: Alternating-Current Circuits

• Section 24-1: Alternating Voltages and Currents
  • The main results of this chapter are best summarized by this simulation
  • The current typically provided by a wall socket is not direct (DC); it is an alternating current (AC) and its frequency is 60 Hz.
  • The equations leading to the main results are:
    • V = RI for resistors (Ohm's Law), Eq. 21-2
    • V = Q/C for capacitors, Eq. 20-9 (Note that I = ΔQ/Δt)
    • V = LΔI/Δ for inductors, Eq. 23-12
  • Because of these relations, V and I will be in phase for resistors, but not for capacitors and inductors. In other words, we can write V_R(t) = (constant)× I_R(t), but V_C(t) ≠ (constant)× I_C(t) and V_L(t) ≠ (constant)× I_L(t).
  • Oscillating Functions
    • It might be useful to review Chapter 13 on Simple Harmonic Motion
    • Simulation showing the connection between rotation and oscillation. This is useful in order to understand the concept of phasor.
    • In this chapter, we consider oscillating voltages (P. 804, Eq. 24-1, V = V_max sin(ωt)) and oscillating currents (P. 804, Eq. 24-2, I = I_max sin(ωt))). Note that one may use cos instead of sin.
    • P. 805, Eq. 24-4, RMS (Root mean square) x_rms = x_max/2^1/2
    • P. 805, Figure 24-4 shows the square of a sinusoidally varying current. The average over a period is I²_av = I²_max/2.
    • The average power dissipated over a period is P_av=RI²_av= RI²_max/2 = R(I_rms)²
    • P. 832, Problem 2
  • Resistors and AC Current
    • P. 804, Figure 24-1 shows an AC resistor circuit.
    • Only case where V(t) = (constant)× I(t).
    • P. 804, Figure 24-2 shows that I(t) and V(t) are in phase. This is so because V(t) = R I(t).
    • Voltage is given by P. 804, Eq. 24-1: V = V_max sin(ωt)
    • Current is given by P. 804, Eq. 24-2: I = I_max sin(ωt)
    • Both I and V are described by a sinus function.
    • P. 805, Figure 24-3 displays the phasor diagram for an AC resistor circuit. For resistors, this is rather trivial.
    • According to the phasor representation, the phasor is a rotating vector, and the projection on the y-axis gives the current or voltage at time t.
• Section 24-2: Capacitors in AC Circuits
  • P. 809, Figure 24-6 shows an AC capacitor circuit.
  • From V = Q/C (Eq. 20-9 ) and the definition I = ΔQ/Δt, we find from calculus that if I(t) = I_max sin(ωt) then V(t) = -V_max cos(ωt) where V_max = X_C I_max.
  • P. 809, Eq. 24-8, Capacitive reactance X_C defined by V_rms = X_C I_rms
  • P. 809, Eq. 24-9: X_C = 1/(ωC). [SI Unit: ohm (Ω)]
  • P. 810, Figure 24-7 shows the behaviour of I(t) and V(t). Note that V(t) ≠ X_C I(t).
  • P. 811, Figure 24-8 displays the phasor diagram for an AC capacitor circuit.
  • Both Figure 24-7 and Figure 24-8 show that V lags I by 90 degrees.
• Section 24-3: RC Circuits [Omitted]
• Section 24-4: Inductors in AC Circuits
  • P. 817, Figure 24-13 shows an AC inductor circuit.
  • From V = LΔI/Δ (Eq. 23-12) we find using calculus that if I(t) = I_max sin(ωt) then V(t) = +V_max cos(ωt) where V_max = X_L I_max.
  • P. 817, Eq. 24-14, Inductive Reactance: X_L = ωL. [SI Unit: ohm (Ω)]
  • P. 818: V_rms = X_LI_rms
  • P. 817, Figure 24-15 shows the behaviour of I(t) and V(t). Note that V(t) ≠ X_L I(t).
  • P. 818, Figure 24-16 displays the phasor diagram for an AC inductor circuit.
  • Both Figure 24-15 and Figure 24-16 show that V leads I by 90 degrees.
  • P. 818, Eq. 24-15, Impedance in an RL Circuit: Z = (R² + X_L²)^1/2
• Section 24-5: RLC Circuits
  • P. 820, Figure 24-20 shows an alternating RLC circuit.
  • The phasor diagram shown in P. 820, Figure 24-21 provides the easiest way to describe the current and voltage.
  • P. 820: V_max = Z I_max
  • P. 820, Eq. 24-16, Impedance in an RLC Circuit: Z = [R² + (X_L-X_C)²]^1/2
  • SI Unit: ohm
  • Graphs of the current and the voltages across R, L and C.
  • Table for Z and φ for different combinations
  • The phasor diagram also show that V and I are not in phase. The phase angle φ between the total voltage and the current is given by P. 820, Eq. 24-17: tanφ = (X_L - X_C)/R
  • P. 834, Problem 42
  • P. 834, Problem 46
  • P. 834, Problem 36
  • P. 833, Problem 22
• Section 24-6: Resonance in Electrical Circuits [Omitted]

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