LECTURE 34 AC CIRCUITS (RC & L & RL)

Instructor: Kazumi Tolich Lecture 34

2

¨ Reading chapter 23-11. ¤ Resistors and capacitors in AC circuits (RC circuit) ¤ Inductors in AC circuits ¤ Resistors and inductors in AC circuits (RL circuit) RC circuit

3

¨ Consider an RC circuit with a generator with a maximum voltage of Vmax, a resistor and a capacitor. ¨ The voltage across the resistor and the voltage across the capacitor are not in phase; they do not peak at the same time. � ≠ �, + �, ¨ Phasor diagram for an RC circuit:

¤ Current phasor with a magnitude Imax.

¤ Resistor-voltage phasor with a magnitude ImaxR. The resistor-voltage phasor is in phase with the current phasor.

¤ Capacitor-voltage phasor with a magnitude ImaxXC. The capacitor- voltage phasor lags current phasor by 90°. ¤ The total voltage phasor is the vector sum of the resistor-voltage and capacitor-voltage phasors. Impedance for an RC circuit

4

¨ The magnitude of the total voltage is

2 2 V = I R + I X = I R2 + X 2 max ( max ) ( max C ) max C ¨ This has the exact same form as Ohm’s law (V = IR) if we define the impedance, Z (in Ω) for an RC circuit:

2 " 1 % Z = R2 + X 2 = R2 + C #$ ωC &'

¨ The maximum current in an RC circuit is

Vmax Imax = Z Clicker question: 1

5 Example: 1

6

¨ An ac generator with a frequency of f = 105 Hz and an rms voltage of Vrms = 22.5 V is connected in series with a resistor with a resistance of R = 10.0 kΩ, and a capacitor with a capacitance of C = 0.250 µF. What is the rms current in this circuit? Phase angle

7

¨ A phase angle ϕ is the angle between the current phasor and the total voltage phasor. Power factor for an RC circuit

8

¨ The average power delivered to the circuit is ! V $ P = I 2 R = I rms R = I V cosφ av rms rms # Z & rms rms " % ¨ Therefore cos ϕ is called the power factor.

Purely resistive RC Purely capacitive Example: 2

9

a) Sketch the phasor diagram for an ac circuit with a resistor with a resistance of R = 105 Ω in series with a capacitor with a capacitance of C = 32.2 µF. T h e fre qu e n cy of the generator is f = 60.0 Hz.

b) If the rms voltage of the generator is Vrms = 120 V, what is the average power consumed by the circuit? Inductors in AC circuits

10

¨ Rms current and rms voltage, and max current and max voltage are related by V V I = rms I = max rms X max X L L where � is inductive reactance, defined by

X L ≡ ωL = 2π f L and has a unit of Ohms, Ω. I and V in an ac inductor circuit

11

¨ The voltage across an inductor leads the current by 90° or π/2.

¨ The current and voltage are I = I sin ωt V = V sin ωt + 90! max ( ) max ( ) ¨ The phase difference ϕ between the current and the voltage is -90°, giving the power factor of cos ϕ = 0. Power in an ac inductor circuit

12

¨ The instantaneous power for any circuit is P = IV.

¨ P > 0 in 0 < ωt < π/2: the inductor draws energy from the generator.

¨ P < 0 in π/2 < ωt < π: the inductor delivers energy to the generator.

¨ The average power as a function of time is zero. Clicker question: 2

13 Example: 3

14

¨ An inductor with an inductance of L = 0.22 µH is connected to an ac generator with an rms voltage of Vrms = 12 V. For what range of frequencies will the rms current in the circuit less than 1.0 mA? Phasor diagram for an RL circuit

15

¨ Consider an RL circuit with a generator oscillating at ω with a maximum voltage of Vmax, a resistor with a resistance R and an inductor with an inductance L.

¨ The voltage across the resistor and the voltage across the inductor are not in phase. � ≠ �, + �, . ¨ Phasor diagram for an RL circuit:

¤ Current phasor with a magnitude Imax.

¤ Resistor-voltage phasor with a magnitude ImaxR. The resistor-voltage phasor is in phase with the current phasor.

¤ Inductor-voltage phasor with a magnitude ImaxXL. The inductor-voltage phasor leads current phasor by 90°. ¤ The total voltage phasor is the vector sum of the resistor-voltage and inductor-voltage phasors. Impedance for an RL circuit

16

¨ The magnitude of the total voltage is

2 2 V = I R + I X = I R2 + X 2 max ( max ) ( max L ) max L ¨ This has the exact same form as Ohm’s law (V = IR) if we define the impedance, Z (in Ω) for an RL circuit:

2 Z = R2 + X 2 = R2 + ω L L ( )

¨ The maximum current in an RL circuit is

Vmax Imax = Z Power factor for an RL circuit

17

¨ The power factor for an RL circuit is: Irms comparisons

18

¨ Rms currents in a resistor-only, an RC, and an RL circuits as a function of angular frequency: Ferrite beads

19

¨ Ferrite beads are particularly common on data cables and on medical equipment.

¨ Electronic devices might be near other devices that radiate damaging high- frequency signals.

¨ A ferrite bead contains an inductor. Combined with its resistance, it acts like an RL circuit.

¨ High-frequency noise signals can be reduced by ferrite beads by dissipating the unwanted signals as heat.