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PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

• 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) = 𝑽𝒎𝒂𝒙𝐜𝐨𝐬(𝝎𝒕)

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?

● This alternating voltage produces an ALTERNATING CURRENT of

- 𝒊(𝒕) = ___________________ (𝝎 is the angular frequency of alternations)

V

t

Vmax

I

t

Imax

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 2

• PRACTICE: ALTERNATING CURRENT

An AC source produces an alternating current in a circuit with the function 𝑖(𝑡) = (1.5𝐴) cos[(250𝑠−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

t

11 V

- 2.5 A

0.05 s

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 3

• 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?

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 4

• CONCEPT: RMS CURRENT AND VOLTAGE ● In alternating current circuits, what is the average of the voltage and the current?

- 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 → 𝑿𝟐 → (𝑿𝟐)𝒂𝒗 → √(𝑿𝟐)𝒂𝒗

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?

V

t

Vmax

I

t

Imax

● The RMS CURRENT and VOLTAGE are defined by

- 𝑰𝑹𝑴𝑺 =𝑰𝒎𝒂𝒙

√𝟐 or 𝑰𝒎𝒂𝒙 = √𝟐𝑰𝑹𝑴𝑺

- 𝑽𝑹𝑴𝑺 =𝑽𝒎𝒂𝒙

√𝟐 or 𝑽𝒎𝒂𝒙 = √𝟐𝑽𝑹𝑴𝑺

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 5

• PRACTICE: RMS CURRENT IN AN AC CIRCUIT An AC source operates with a 0.05 s period. 0.025 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?

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 6

• 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:

EXAMPLE 1: For the following voltage phasor, is the voltage positive or negative?

● 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?

V

t

𝜔

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 7

• 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 0o, if the phasor took 0.027 s to get to its current position, what is the angular frequency of the phasor?

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 0o.

30o

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 8

• 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.

PRACTICE: INSTANTANEOUS VALUE FROM A PHASOR

A phasor of length 4 begins at 0o. If it is rotating at 𝜔 = 250 𝑠−1, what is the value of the phasor after 0.007 s?

V

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 9

• CONCEPT: RESISTORS IN AC CIRCUITS ● Remember! In an AC circuit, the current produced by the AC source is

- 𝑖(𝑡) = 𝑖𝑀𝐴𝑋cos(𝜔𝑡) ● Ohm’s Law will give us the voltage across the resistor at any point in time:

- 𝑣𝑅(𝑡) = 𝑖(𝑡)𝑅

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.

● The VOLTAGE ACROSS THE RESISTOR is

- 𝒗𝑹(𝒕) = ___________________

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 10

• PRACTICE: OSCILLATING VOLTAGE ACROSS A RESISTOR

The voltage across a resistor is found to be given by 𝑣𝑅(𝑡) = (10𝑉) cos[(120𝑠−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

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 11

• CONCEPT: PHASORS FOR RESISTORS ● Remember! The voltage and current across a resistor at any time t is

- 𝑖(𝑡) = 𝑖𝑀𝐴𝑋cos(𝜔𝑡)

- 𝑣𝑅(𝑡) = 𝑖𝑀𝐴𝑋𝑅cos(𝜔𝑡)

● Because both cosines have the same angle (𝜔𝑡), they are said to be IN PHASE.

- This is reflected in their phasors:

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.

𝜔𝑡

𝐼

𝜔𝑡

𝑉𝑅

𝜔𝑡

𝑉𝑅 𝐼

● Voltage across a resistor is IN PHASE with the current

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 12

• 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 0o, 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 = 0.04𝑠)?

42o

5 V

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 13

• CONCEPT: CAPACITORS IN AC CIRCUITS ● The current in an AC circuit at any time is

- 𝑖(𝑡) = ____________________ ● Remember! The voltage across a capacitor is 𝑣𝐶 = ___________

- Using calculus, one can show 𝑞(𝑡) =𝑖𝑀𝐴𝑋

𝜔cos (𝜔𝑡 −

𝜋

2)

● This means, if current and voltage across the capacitor are plotted, the voltage of a capacitor LAGS the current by 90o:

● The MAXIMUM voltage across the capacitor is 𝑉𝐶 = ________________

- This result looks A LOT like Ohm’s Law, if we have some resistance-like quantity 1/𝜔𝐶

We define the CAPACITIVE REACTANCE as 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?

● The VOLTAGE ACROSS A CAPACITOR in an AC circuit is

- 𝑣𝐶(𝑡) = _______________________

I

t

V

𝑿𝑪 = 𝟏/𝝎𝑪

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 14

• 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?

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

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• 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?

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 16

• CONCEPT: PHASORS FOR CAPACITORS ● Remember! The voltage and current across a capacitor at any time t is

- 𝑖(𝑡) = 𝑖𝑀𝐴𝑋 cos (𝜔𝑡)

- 𝑣𝑐(𝑡) = 𝑖𝑀𝐴𝑋XC 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:

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.

𝜔𝑡

𝐼

𝜔𝑡 −𝜋

2

𝑉𝐶

𝐼

𝑉𝐶

● Voltage across a capacitor LAGS the current

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 17

• 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) = 𝑖𝑀𝐴𝑋 cos[(100 𝑠−1)𝑡],

a) What is 𝑖𝑀𝐴𝑋? 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 0o.

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 18

• 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 𝑣𝐿 = ___________

- Using calculus, one can show Δ𝑖

Δ𝑡(𝑡) = 𝑖𝑀𝐴𝑋𝜔 cos (𝜔𝑡 +

𝜋

2)

● This means, if current and voltage across the capacitor are plotted, the voltage of a capacitor LEADS the current by 90o:

● The MAXIMUM voltage across the inductor is 𝑉𝐿 = ________________

- This result looks A LOT like Ohm’s Law, if we have some resistance-like quantity 𝜔𝐿

We define the INDUCTIVE REACTANCE as 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?

● The VOLTAGE ACROSS AN INDUCTOR in an AC circuit is

- 𝑣𝐿(𝑡) = _______________________

I

t

V

𝑿𝑳 = 𝝎𝑳

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 19

• 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?

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?

t

10 V

-2.5 A

0.1 s

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 20

• CONCEPT: PHASORS FOR INDCUTORS ● Remember! The voltage and current across an inductor at any time t is

- 𝑖(𝑡) = 𝑖𝑀𝐴𝑋 cos (𝜔𝑡)

- 𝑣𝐿(𝑡) = 𝑖𝑀𝐴𝑋XL 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:

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.

𝜔𝑡

𝐼

𝜔𝑡 +𝜋

2

𝑉𝐿 𝐼

𝑉𝐿

● Voltage across an inductor LEADS the current

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 21

• 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) = 𝑖𝑀𝐴𝑋 cos[(450 𝑠−1)𝑡],

a) What is 𝑖𝑀𝐴𝑋? 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 0o.

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 22

• 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, 𝑉𝑅𝐶 , will NOT be equal to 𝑉𝑅 + 𝑉𝐶

- These maximum voltages, 𝑉𝑅 and 𝑉𝐶, occur at different times

- Instead, the maximum voltage 𝑉𝑅𝐶 will be the ______________________ of the voltage phasors

This leads us to 𝑉𝑅𝐶 = 𝐼𝑀𝐴𝑋√𝑅2 + 𝑋𝐶2 = 𝐼𝑀𝐴𝑋𝑍

EXAMPLE: What’s the impedance of an AC circuit with a resistor and inductor in series?

● 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 𝐼𝑀𝐴𝑋 = __________________

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 23

• 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?

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 24

• 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?

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 25

• 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 𝑉𝐿𝑅𝐶 = 𝑉𝐿 + 𝑉𝑅 + 𝑉𝐶 , since they are all in series

- In an AC circuit, this isn’t true, since the maximum voltages occur at different times

● The IMPEDANCE, 𝒁, acts like the effective reactance of the circuit.

- In a series LRC circuit, the impedance is

The maximum current produced by the source is given by 𝑖𝑀𝐴𝑋 = ________________ 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?

● In a series LRC circuit, the MAXIMUM voltage is

- 𝑉𝐿𝑅𝐶 = _____________________________

𝒁 = √𝑹𝟐 + (𝝎𝑳 −𝟏

𝝎𝑪)𝟐

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 26

• 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?

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 27

• 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 = _________________________________

- The SMALLEST value of impedance, 𝑍 = 𝑅, occurs when 𝑋𝐶 = 𝑋𝐿

- When this occurs, the circuit is said to be in RESONANCE

● 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 𝑋𝐿 = 𝑋𝐶 The voltage across the inductor and the capacitor is the same

● The RESONANT FREQUENCY of an LRC circuit is

𝜔0 =1

√𝐿𝐶

𝜔

𝑋𝐶 =1

𝜔𝐶

𝑋𝐿 = 𝜔𝐿

𝑅

𝑍

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 28

• 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?

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 29

• 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

● Since the power of a resistor is 𝑝(𝑡) = 𝑖(𝑡)2𝑅, we have the following graphs of current and power through a resistor:

EXAMPLE: An AC source operating at a maximum voltage of 120 V is connected to a 10 Ω resistor. What is the average

power emitted by this circuit? Is this equivalent to the RMS power, which would be 𝑖𝑅𝑀𝑆2 𝑅?

I t

P

● The AVERAGE POWER emitted by an AC circuit is

𝑷𝒂𝒗 = _________________ = __________________

𝑷𝑴𝑨𝑿 = _________________

PHYSICS - CLUTCH

CH 29: ALTERNATING CURRENT

Page 30