AC Sources and Phasors Circuits powered by a sinusoidal emf are called AC circuits, where AC stands for alternating current. Steady-current circuits are called DC circuits, for direct current. The instantaneous emf of an AC generator or oscillator can be written: Slide 35-21
AC Sources and Phasors An alternative way to represent the emf and other oscillatory quantities is with a phasor diagram, as shown. A phasor is a vector that rotates counterclockwise (ccw) around the origin at angular frequency. The quantity s value at time t is the projection of the phasor onto the horizontal axis. Slide 35-22
AC Sources and Phasors The figure below helps you visualize the phasor rotation by showing how the phasor corresponds to the more familiar graph at several specific points in the cycle. Slide 35-23
QuickCheck 35.1 This is a current phasor. The magnitude of the instantaneous value of the current is A. Increasing. B. Decreasing. C. Constant. D. Can t tell without knowing which way it is rotating. Slide 35-24
Example 1 What is the angular frequency, frequency, and instantaneous emf?
Example 2 A 60 Hz source of emf has a peak voltage of 170 V. Draw the emf phasor at t = 3.0 ms.
Resistor Circuits In Chapter 31 we used the symbols I and V for DC current and voltage. Now, because the current and voltage are oscillating, we will use lowercase i to represent the instantaneous current and v for the instantaneous voltage. Slide 35-26
Resistor Circuits The figure shows a resistor R connected across an AC generator of peak emf equal to V R. The current through the resistor is: where I R = V R /R is the peak current. Slide 35-27
Resistor Circuits The resistor s instantaneous current and voltage are in phase, both oscillating as cos t. Slide 35-28
Resistor Circuits Below is the phasor diagram for the resistor circuit. V R and I R point in the same direction, indicating that resistor voltage and current oscillate in phase. Slide 35-29
Capacitor Circuits The figure shows current i C charging a capacitor with capacitance C. Slide 35-30
Capacitor Circuits The figure shows a capacitor C connected across an AC generator of peak emf equal to V C. The charge sitting on the positive plate of the capacitor at a particular instant is: Slide 35-31
Capacitor Circuits The current is the rate at which charge flows through the wires, i C dq/dt, thus: We can most easily see the relationship between the capacitor voltage and current if we use the trigonometric identity: sin (x) = cos (x + /2) to write: Slide 35-32
Capacitor Circuits A capacitor s current and voltage are not in phase. The current peaks one-quarter of a period before the voltage peaks. Slide 35-33
Capacitor Circuits Below is the phasor diagram for the capacitor circuit. The AC current of a capacitor leads the capacitor voltage by /2 rad, or 90. Slide 35-34
QuickCheck 35.2 In the circuit represented by these phasors, the current the voltage A. leads B. lags C. is perpendicular to D. is out of phase with Slide 35-35
QuickCheck 35.3 In the circuit represented by these graphs, the current the voltage A. leads B. lags C. is less than D. is out of phase with Slide 35-37
Capacitive Reactance The peak current to and from a capacitor is I C CV C. We can find a relationship that looks similar to Ohm s Law if we define the capacitave reactance to be: Slide 35-39
QuickCheck 35.4 If the value of the capacitance is doubled, the capacitive reactance A. Is quartered. B. Is halved. C. Is doubled. D. Is quadrupled. E. Can t tell without knowing. Slide 35-40
QuickCheck 35.5 If the value of the capacitance is doubled, the peak current A. Is quartered. B. Is halved. C. Is doubled. D. Is quadrupled. E. Can t tell without knowing C. Slide 35-42
Example 3 A 20 nf capacitor is connected across an AC generator that produces a peak voltage of 5.0 V. At what frequency is the peak current 50 ma? What is the instantaneous value of the emf at the instant when i c = I c? Slide 35-44
RC Filter Circuits The figure shows a circuit in which a resistor R and capacitor C are in series with an emf oscillating at angular frequency. If the frequency is very low, the capacitive reactance will be very large, and thus the peak current I C will be very small. If the frequency is very high, the capacitive reactance approaches zero and the peak current, determined by the resistance alone, will be I R 0 /R. Slide 35-47
QuickCheck 35.6 Does V R + V C = 0? A. Yes. B. No. C. Can t tell without knowing. Slide 35-54
Using Phasors to Analyze an RC Circuit Step 1 of 4 Begin by drawing a current phasor of length I. This is the starting point because the series circuit elements have the same current i. The angle at which the phasor is drawn is not relevant. Slide 35-48
Using Phasors to Analyze an RC Circuit Step 2 of 4 The current and voltage of a resistor are in phase, so draw a resistor voltage phasor of length V R parallel to the current phasor I. The capacitor current leads the capacitor voltage by 90, so draw a capacitor voltage phasor of length V C that is 90 behind the current phasor. Slide 35-49
Using Phasors to Analyze an RC Circuit Step 3 of 4 The series resistor and capacitor are in parallel with the emf, so their instantaneous voltages satisfy v R v C. This is a vector addition of phasors. The emf is 0 cos t, hence the emf phasor is at angle t. Slide 35-50
Using Phasors to Analyze an RC Circuit Step 4 of 4 The length of the emf phasor, 0, is the hypotenuse of a right triangle formed by the resistor and capacitor phasors. Thus 0 2 V R2 + V C 2. Slide 35-51
RC Filter Circuits The relationship 02 V R2 V C2 is based on the peak values. The peak voltages are related to the peak current by V R IR and V C IX C, so: This can be solved for the peak current, which in turn gives us the two peak voltages: Slide 35-52
RC Filter Circuits The figure shows a graph of the resistor and capacitor peak voltages as functions of the emf angular frequency. The frequency at which V R V C is called the crossover frequency: Slide 35-53
RC Filter Circuits The figure below shows an RC circuit in which v C is the output voltage. This circuit is called a low-pass filter. Slide 35-56
RC Filter Circuits The figure below shows an RC circuit in which v R is the output voltage. This circuit is called a high-pass filter. Slide 35-57
Example 4 A capacitor is connected to a 15 khz oscillator. The peak current is 65 ma when the peak voltage is 6.0 V. What is the value of the capacitance. Slide 35-44