PHYSICS - CLUTCH CH 29: ALTERNATING CURRENT.

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2 CONCEPT: ALTERNATING VOLTAGES AND CURRENTS BEFORE, we only considered DIRECT CURRENTS, currents that only move in - NOW we consider ALTERNATING CURRENTS, currents that move in Alternating currents are produced by ALTERNATING VOLTAGES - ONLY alternating voltage we will consider is v(t) = V max cos (ωt) This alternating voltage produces an ALTERNATING CURRENT of - i(t) = (ω is the angular frequency of alternations) V I V max I max t t EXAMPLE: In North America, the frequency of AC voltage coming out of household outlets is 60 Hz. If the maximum voltage delivered by an outlet is 120 V, what is the voltage at 0.04 s? Page 2

3 PRACTICE: ALTERNATING CURRENT An AC source produces an alternating current in a circuit with the function i(t) = (1.5 A) cos[(250 s 1 )t]. What is the frequency of the source? What is the maximum current in the circuit? EXAMPLE: AC CIRCUIT GRAPHS Current and voltage in an AC circuit are graphed in the following figure. What are the functions that describe these values? I, V 11 V A t 0.05 s Page 3

4 PRACTICE: ANGULAR FREQUENCY OF ALTERNATING CURRENT The current in an AC circuit takes 0.02 s to change direction. What is the angular frequency of the AC source? Page 4

5 CONCEPT: RMS CURRENT AND VOLTAGE In alternating current circuits, what is the average of the voltage and the current? V I V max I max t t - The average of the voltage and the current is A better average value is the RMS VALUE, the To find the RMS value, you take the square, then the average, then the square root X X 2 (X 2 ) av (X 2 ) av The RMS CURRENT and VOLTAGE are defined by - I RMS = I max 2 - V RMS = V max 2 or or I max = 2I RMS V max = 2V RMS EXAMPLE: If the RMS voltage of an outlet in the US is 120 V, what is the maximum voltage of an outlet? If you complete a simple circuit with this AC source by connecting a 12 Ω resistor, what is the RMS and maximum current in this circuit? Page 5

6 PRACTICE: RMS CURRENT IN AN AC CIRCUIT An AC source operates with a 0.05 s period s after the current is at a maximum, the current is measured to be 1.4 A. What is the RMS current of this AC circuit? Page 6

7 CONCEPT: PHASORS A PHASOR is just a rotating vector, whose information lies in its X-COMPONENT. - Phasors make representing oscillating information, like voltage and current, easy: V t EXAMPLE 1: For the following voltage phasor, is the voltage positive or negative? ω Phasors obey all the same rules as vectors, such as addition, subtraction, etc. - To find the magnitude of a phasor, you can sum its components using the Pythagorean theorem, as with vectors. EXAMPLE 2: In the following phasor diagram, find the direction of the net phasor for the three phasors shown. Is the resulting quantity the phasor describes positive or negative? Page 7

8 PRACTICE: ANGULAR FREQUENCY OF A PHASOR The following phasor diagram shows an arbitrary phasor during its first rotation. Assuming that it begins with an angle of 0 o, if the phasor took s to get to its current position, what is the angular frequency of the phasor? 30 o EXAMPLE: CONVERTING BETWEEN A FUNCTION AND A PHASOR The current in an AC circuit is given by i(t) = (1.5 A) cos[(377 s 1 )t]. Draw the phasor that corresponds to this current at t = 15 ms, assuming the phasor begins at 0 o. Page 8

9 PRACTICE: DRAWING A VOLTAGE PHASOR An AC source oscillates with an angular frequency of 120 s -1. If the initial voltage phasor is shown in the following phasor diagram, draw the voltage phasor after 0.01 s. V PRACTICE: INSTANTANEOUS VALUE FROM A PHASOR A phasor of length 4 begins at 0 o. If it is rotating at ω = 250 s 1, what is the value of the phasor after s? Page 9

10 CONCEPT: RESISTORS IN AC CIRCUITS Remember! In an AC circuit, the current produced by the AC source is - i(t) = i MAX cos (ωt) Ohm s Law will give us the voltage across the resistor at any point in time: - v R (t) = i(t)r The VOLTAGE ACROSS THE RESISTOR is - v R (t) = EXAMPLE: A 10 Ω resistor is plugged into an outlet with an RMS voltage of 120 V. What is the maximum current in the circuit? What about the RMS current? For MULTIPLE resistors in an AC circuit, you would just combine them into a single, equivalent resistor, as before. Page 10

11 PRACTICE: OSCILLATING VOLTAGE ACROSS A RESISTOR The voltage across a resistor is found to be given by v R (t) = (10 V) cos[(120 s 1 )t]: a) At what frequency does the AC course operate? b) If the resistance is 12 Ω, what is the maximum current in this circuit? c) What is the RMS voltage of the AC source? EXAMPLE: RESISTORS IN PARALLEL IN AN AC CIRCUIT What is the current through the 10 Ω resistor in the following circuit? (5 V) cos[(200 s 1 )t] 5 Ω 3 Ω 10 Ω Page 11

12 CONCEPT: PHASORS FOR RESISTORS Remember! The voltage and current across a resistor at any time t is - i(t) = i MAX cos (ωt) - v R (t) = i MAX R cos (ωt) Because both cosines have the same angle (ωt), they are said to be IN PHASE. - This is reflected in their phasors: I V R I V R ωt ωt ωt Voltage across a resistor is IN PHASE with the current EXAMPLE: An AC source with an angular frequency of 20 s -1 is connected to a resistor with the circuit broken. 0.2 s after the circuit is completed, draw the voltage phasor and the current phasor. Page 12

13 PRACITCE: RESISTOR VOLTAGE AND CURRENT PHASORS A 12 Ω resistor is connected to an AC source. If the resistor s voltage phasor is initially at 0 o, and the figure below shows the phasor after 0.04 s, answer the following: a) What is the angular frequency of the source? Assume the phasor is on its first rotation. b) What does the current phasor diagram look like? c) What is the current in the circuit at this point (t = 0.04 s)? 5 V 42 o Page 13

14 CONCEPT: CAPACITORS IN AC CIRCUITS The current in an AC circuit at any time is - i(t) = Remember! The voltage across a capacitor is v C = - Using calculus, one can show q(t) = i MAX ω cos (ωt π 2 ) The VOLTAGE ACROSS A CAPACITOR in an AC circuit is - v C (t) = This means, if current and voltage across the capacitor are plotted, the voltage of a capacitor LAGS the current by 90 o : V I t The MAXIMUM voltage across the capacitor is V C = - This result looks A LOT like Ohm s Law, if we have some resistance-like quantity 1/ωC We define the CAPACITIVE REACTANCE as X C = 1/ωC EXAMPLE: An AC power source delivers a maximum voltage of 120 V at 60 Hz. What is the maximum current in a circuit with this power source connected to a 100 µf capacitor? Page 14

15 PRACTICE: MAXIMUM CHARGE IN A CAPACITOR AC CIRCUIT An AC source operates at a maximum voltage of 120 V and a frequency of 60 Hz. If it is connected to a 175 µf capacitor, what is the maximum charge stored on the capacitor? EXAMPLE: CURRENT IN A PARALLEL RC AC CIRCUIT An AC source operating at 160 s -1 and a maximum voltage of 15 V is connected in parallel to a 5 Ω resistor and in parallel to a 1.5 mf capacitor. What is the RMS current through the capacitor? Page 15

16 PRACTICE: OSCILLATION FREQUENCY OF A CAPCITOR CIRCUIT A 300 µf capacitor is connected to an AC source operating at an RMS voltage of 120 V. If the maximum current in the circuit is 1.5 A, what is the oscillation frequency of the AC source? Page 16

17 CONCEPT: PHASORS FOR CAPACITORS Remember! The voltage and current across a capacitor at any time t is - i(t) = i MAX cos (ωt) - v c (t) = i MAX X C cos (ωt π 2 ) Because both cosines have a DIFFERENT angle, they are said to be OUT OF PHASE The voltage LAGS the current - This is reflected in their phasors: I I ωt ωt π 2 V C V C Voltage across a capacitor LAGS the current EXAMPLE: An AC source is connected to a capacitor. At a particular instant in time, the voltage across the capacitor is positive and increasing in magnitude. Draw the phasors for voltage and current that correspond to this time. Page 17

18 PRACTICE: PHASORS IN A CAPACITOR CIRCUIT An AC source operates at a maximum voltage of 60 V and is connected to a 0.7 mf capacitor. If the current across the capacitor is i(t) = i MAX cos[(100 s 1 )t], a) What is i MAX? b) Draw the phasors for voltage across the capacitor and current in the circuit at t = 0.02 s. Assume that the current phasor begins at 0 o. Page 18

19 CONCEPT: INDUCTORS IN AC CIRCUITS Remember! The current in an AC circuit at any time is - i(t) = Remember! The voltage across an inductor is v L = - Using calculus, one can show Δi Δt (t) = i MAXω cos (ωt + π 2 ) The VOLTAGE ACROSS AN INDUCTOR in an AC circuit is - v L (t) = This means, if current and voltage across the capacitor are plotted, the voltage of a capacitor LEADS the current by 90 o : V I t The MAXIMUM voltage across the inductor is V L = - This result looks A LOT like Ohm s Law, if we have some resistance-like quantity ωl We define the INDUCTIVE REACTANCE as X L = ωl EXAMPLE: An AC power source delivers a maximum voltage of 120 V at 60 Hz. If an unknown inductor is connected to this source, and the maximum current in the circuit is found to be 5 A, what is the inductance of the inductor? Page 19

20 EXAMPLE: INDUCTORS AND GRAPHS The voltage across, and the current through, an inductor connected to an AC source are shown in the following graph. Given the information in the graph, answer the following questions: a) What is the peak voltage of the AC source? b) What is the frequency of the AC source? c) What is the inductive reactance of the circuit? 10 V -2.5 A t 0.1 s PRACTICE: CURRENT IN INDUCTOR AC CIRCUITS AT DIFFERENT FREQUENCIES Will a frequency f = 60 Hz or ω = 75 s 1 produce a larger max current in an inductor connected to an AC source? Page 20

21 CONCEPT: PHASORS FOR INDCUTORS Remember! The voltage and current across an inductor at any time t is - i(t) = i MAX cos (ωt) - v L (t) = i MAX X L cos (ωt + π 2 ) Because both cosines have a DIFFERENT angle, they are said to be OUT OF PHASE The current LAGS the voltage - This is reflected in their phasors: I V L V L I ωt ωt + π 2 Voltage across an inductor LEADS the current EXAMPLE: An AC source is connected to an inductor. At a particular instant in time, the current in the circuit is negative and increasing in magnitude. Draw the phasors for voltage and current that correspond to this instant in time. Page 21

22 PRACITCE: PHASORS IN A INDUCTOR CIRCUIT An AC source operates at a maximum voltage of 75 V and is connected to a 0.4 H inductor. If the current across the inductor is i(t) = i MAX cos[(450 s 1 )t], a) What is i MAX? b) Draw the phasors for voltage across the inductor and current in the circuit at t = 4.2 ms. Assume that the current phasor begins at 0 o. Page 22

23 CONCEPT: IMPEDANCE IN AC CIRCUITS We know how to find the current in any AC circuit with ONE element It s just the maximum voltage divided by the There are two types of circuits: series circuits and parallel circuits. - Whenever an AC circuit has multiple elements in series, the phasors line up - Whenever an AC circuit has multiple elements in parallel, the phasors line up Consider an AC source connected in series to a resistor and a capacitor. - In this case, the maximum voltage across the resistor and capacitor, V RC, will NOT be equal to V R + V C - These maximum voltages, V R and V C, occur at different times - Instead, the maximum voltage V RC will be the of the voltage phasors This leads us to V RC = I MAX R 2 + X C 2 = I MAX Z The IMPEDENCE in an AC circuit, Z, acts as the effective reactance in a circuit with multiple elements The MAXIMUM CURRENT output by the source is ALWAYS I MAX = EXAMPLE: What s the impedance of an AC circuit with a resistor and inductor in series? Page 23

24 EXAMPLE: IMPEDANCE OF A PARALLEL LR AC CIRCUIT What s the impedance of a parallel LR AC circuit? PRACTICE: IMPEDANCE OF A PARALLEL RC AC CIRCUIT What s the impedance of a parallel RC AC circuit? Page 24

25 PRACTICE: CURRENT IN A PARALLEL RC CIRCUIT An AC source operates at a maximum voltage of 120 V and an angular frequency of 377 s -1. If this source is connected in parallel to a 15 Ω resistor and in parallel to a 0.20 mf capacitor, answer the following questions: a) What is the maximum current produced by the source? b) What is the maximum current through the resistor? c) What is the maximum current through the capacitor? Page 25

26 CONCEPT: LRC CIRCUITS IN SERIES In a series LRC circuit, the through each element is the same In a DC circuit, we would simply say that V LRC = V L + V R + V C, since they are all in series - In an AC circuit, this isn t true, since the maximum voltages occur at different times In a series LRC circuit, the MAXIMUM voltage is - V LRC = The IMPEDANCE, Z, acts like the effective reactance of the circuit. - In a series LRC circuit, the impedance is Z = R 2 + (ωl 1 ωc ) 2 The maximum current produced by the source is given by i MAX = EXAMPLE: A circuit is formed by attaching an AC source in series to an 0.5 H inductor, a 10 Ω resistor and a 500 µf capacitor. If the source operates at a VRMS of 120 V and a frequency of 60 Hz, what is the maximum current in the circuit? Page 26

27 PRACTICE: VOLTAGE IN A SERIES LRC AC CIRCUIT An AC source operates at an RMS voltage of 70 V and a frequency of 85 Hz. If the source is connected in series to a 20 Ω resistor, a 0.15 H inductor and a 500 µf capacitor, answer the following questions: a) What is the maximum current produced by the source? b) What is the maximum voltage across the resistor? c) What is the maximum voltage across the inductor? d) What is the maximum voltage across the capacitor? Page 27

28 CONCEPT: RESONANCE IN SERIES LRC CIRCUITS The impedance of an LRC circuit depends on the frequency of the AC source: - The impedance is large at small ω and at large ω Recall that the impedance is Z = Z X L = ωl X C = 1 ωc R ω - The SMALLEST value of impedance, Z = R, occurs when X C = X L - When this occurs, the circuit is said to be in RESONANCE The RESONANT FREQUENCY of an LRC circuit is ω 0 = 1 LC Since resonance occurs when the impedance is SMALLEST, the current is LARGEST in resonance for series LRC EXAMPLE: An AC circuit is composed of a 10 Ω resistor, a 2 H inductor, and a 1.2 mf capacitor. If it is connected to a power source that operates at a maximum voltage of 120 V, what frequency should it operate at to produce the largest possible current in the circuit? What would the value of this current be? In a series LRC circuit, the current is the same through the inductor and the capacitor - In resonance, since X L = X C The voltage across the inductor and the capacitor is the same Page 28

29 PRACTICE: VOLTAGES IN A SERIES LRC CIRCUIT IN RESONANCE A series LRC circuit is formed with a power source operating at VRMS = 100 V, and is formed with a 15 Ω resistor, a 0.05 H inductor, and a 200 µf capacitor. What is the voltage across the inductor in resonance? The voltage across the capacitor? Page 29

30 CONCEPT: POWER IN AC CIRCUITS In AC circuits, the only element to have an average power not equal to zero is the - Whatever energy enters a(n) / equals the energy that leaves The MAXIMUM power of a resistor is P MAX = Since the power of a resistor is p(t) = i(t) 2 R, we have the following graphs of current and power through a resistor: P I t The AVERAGE POWER emitted by an AC circuit is P av = = EXAMPLE: An AC source operating at a maximum voltage of 120 V is connected to a 10 Ω resistor. What is the average 2 power emitted by this circuit? Is this equivalent to the RMS power, which would be i RMS R? Page 30