GoBack Series RLC Circuit - Phasors
AC Circuits • Series RLC Circuit - Let’s analyze the following circuit. Phasors • • Phasor Analysis (A Single Moment in Time) • Finding Peak Current in RLC - Circuit • Impedance • Current and Driving Voltage
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 1 Series RLC Circuit - Phasors
AC Circuits • Series RLC Circuit - Let’s analyze the following circuit. Phasors • • Phasor Analysis (A Single Moment in Time) R • Finding Peak Current in RLC - Circuit • Impedance • Current and Driving Voltage VP sin(ωt) Root-Mean-Square L
Transforming Voltage C Amplitudes - AC - Circuits
Calculate the peak current through the circuit •
PHYS102 AC Circuits - Phasors – slide 1 Series RLC Circuit - Phasors
AC Circuits • Series RLC Circuit - Let’s analyze the following circuit. Phasors • • Phasor Analysis (A Single Moment in Time) R • Finding Peak Current in RLC - Circuit • Impedance • Current and Driving Voltage VP sin(ωt) Root-Mean-Square L
Transforming Voltage C Amplitudes - AC - Circuits
Calculate the peak current through the circuit • We need to keep track of the phase differences ◦
PHYS102 AC Circuits - Phasors – slide 1 Series RLC Circuit - Phasors
AC Circuits • Series RLC Circuit - Let’s analyze the following circuit. Phasors • • Phasor Analysis (A Single Moment in Time) R • Finding Peak Current in RLC - Circuit • Impedance • Current and Driving Voltage VP sin(ωt) Root-Mean-Square L
Transforming Voltage C Amplitudes - AC - Circuits
Calculate the peak current through the circuit • We need to keep track of the phase differences ◦ PHASORS!
PHYS102 AC Circuits - Phasors – slide 1 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
The voltage across the resistor is represented by the phasor • above since the driving voltage is sinusoidal.
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
The current is in phase with the voltage across the resistor. •
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit I • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
The current is in phase with the voltage across the resistor. •
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit I • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
The voltage across an inductor leads the current by π/2. •
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors VLP • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit I • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
The voltage across an inductor leads the current by π/2. •
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors VLP • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit I • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
The voltage across a capacitor lags behind the current by • π/2.
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors VLP • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit I • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - VCP Circuits
The voltage across a capacitor lags behind the current by • π/2.
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors VLP • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit I • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - VCP Circuits
Apply Kirchhoff’s Loop rule to find a relationship between all • the voltages.
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors VLP • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit I • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - VCP Circuits
Summing the phasors for the voltage across the capacitor and • inductor.
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit I • Impedance • Current and Driving VLP - VCP Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
Summing the phasors for the voltage across the capacitor and • inductor.
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit I • Impedance • Current and Driving VLP - VCP Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
Summing the phasors for the voltage across the resistor and • capacitor/inductor.
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors VP • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit ϕ • Impedance I • Current and Driving VLP - VCP Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
Summing the phasors for the voltage across the resistor and • capacitor/inductor.
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors VP • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit ϕ • Impedance I • Current and Driving VLP - VCP Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
The length of the resultant phasor represents the peak voltage • supplied by the AC Voltage source.
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors VP • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit ϕ • Impedance I • Current and Driving VLP - VCP Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
Finding relationships between the peak current in the circuit • and the peak voltages is now a trigonometry problem.
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors VP • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit ϕ • Impedance I • Current and Driving VLP - VCP Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
NOTE: The driving peak voltage is out of phase with the peak • current through the circuit.
PHYS102 AC Circuits - Phasors – slide 2 Phasor Analysis (A Single Moment in Time)
AC Circuits • Series RLC Circuit - Phasors VP • Phasor Analysis (A Single Moment in Time) VRP • Finding Peak Current in RLC - Circuit ϕ • Impedance I • Current and Driving VLP - VCP Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
− − tan ϕ = VLP VCP = χL χc • VRP R
2 2 VP = V +(VLP VCP ) q RP −
PHYS102 AC Circuits - Phasors – slide 2 Finding Peak Current in RLC - Circuit
AC Circuits • Series RLC Circuit - Phasors 2 • Phasor Analysis (A 2 Single Moment in Time) VP = VRP +(VLP VCP ) • Finding Peak Current q − in RLC - Circuit • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 3 Finding Peak Current in RLC - Circuit
AC Circuits • Series RLC Circuit - Phasors 2 • Phasor Analysis (A 2 Single Moment in Time) VP = VRP +(VLP VCP ) • Finding Peak Current q − in RLC - Circuit VRP = IP R • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 3 Finding Peak Current in RLC - Circuit
AC Circuits • Series RLC Circuit - Phasors 2 • Phasor Analysis (A 2 Single Moment in Time) VP = VRP +(VLP VCP ) • Finding Peak Current q − in RLC - Circuit VRP = IP R • Impedance • Current and Driving VLP = IP χL Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 3 Finding Peak Current in RLC - Circuit
AC Circuits • Series RLC Circuit - Phasors 2 • Phasor Analysis (A 2 Single Moment in Time) VP = VRP +(VLP VCP ) • Finding Peak Current q − in RLC - Circuit VRP = IP R • Impedance • Current and Driving VLP = IP χL Voltage
Root-Mean-Square VCP = IP χc
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 3 Finding Peak Current in RLC - Circuit
AC Circuits • Series RLC Circuit - Phasors 2 • Phasor Analysis (A 2 Single Moment in Time) VP = VRP +(VLP VCP ) • Finding Peak Current q − in RLC - Circuit VRP = IP R • Impedance • Current and Driving VLP = IP χL Voltage
Root-Mean-Square VCP = IP χc
Transforming Voltage 2 2 Amplitudes - AC - VP = IP R +(χL χC) Circuits q −
PHYS102 AC Circuits - Phasors – slide 3 Finding Peak Current in RLC - Circuit
AC Circuits • Series RLC Circuit - Phasors 2 • Phasor Analysis (A 2 Single Moment in Time) VP = VRP +(VLP VCP ) • Finding Peak Current q − in RLC - Circuit VRP = IP R • Impedance • Current and Driving VLP = IP χL Voltage
Root-Mean-Square VCP = IP χc
Transforming Voltage 2 2 Amplitudes - AC - VP = IP R +(χL χC) Circuits q − VP IP = ⇒ 2 2 R +(χL χC) q −
PHYS102 AC Circuits - Phasors – slide 3 Finding Peak Current in RLC - Circuit
AC Circuits • Series RLC Circuit - Phasors 2 • Phasor Analysis (A 2 Single Moment in Time) VP = VRP +(VLP VCP ) • Finding Peak Current q − in RLC - Circuit VRP = IP R • Impedance • Current and Driving VLP = IP χL Voltage
Root-Mean-Square VCP = IP χc
Transforming Voltage 2 2 Amplitudes - AC - VP = IP R +(χL χC) Circuits q − VP IP = (Resembles Ohm’s Law) ⇒ 2 2 R +(χL χC) q −
PHYS102 AC Circuits - Phasors – slide 3 Finding Peak Current in RLC - Circuit
AC Circuits • Series RLC Circuit - Phasors 2 • Phasor Analysis (A 2 Single Moment in Time) VP = VRP +(VLP VCP ) • Finding Peak Current q − in RLC - Circuit VRP = IP R • Impedance • Current and Driving VLP = IP χL Voltage
Root-Mean-Square VCP = IP χc
Transforming Voltage 2 2 Amplitudes - AC - VP = IP R +(χL χC) Circuits q − VP IP = (Resembles Ohm’s Law) ⇒ 2 2 R +(χL χC) q −
2 2 Z R +(χL χC) ≡ q −
PHYS102 AC Circuits - Phasors – slide 3 Finding Peak Current in RLC - Circuit
AC Circuits • Series RLC Circuit - Phasors 2 • Phasor Analysis (A 2 Single Moment in Time) VP = VRP +(VLP VCP ) • Finding Peak Current q − in RLC - Circuit VRP = IP R • Impedance • Current and Driving VLP = IP χL Voltage
Root-Mean-Square VCP = IP χc
Transforming Voltage 2 2 Amplitudes - AC - VP = IP R +(χL χC) Circuits q − VP IP = (Resembles Ohm’s Law) ⇒ 2 2 R +(χL χC) q −
2 2 VP Z R +(χL χC) IP = ≡ q − ⇒ Z
PHYS102 AC Circuits - Phasors – slide 3 Impedance
AC Circuits • Series RLC Circuit - The quantity Z is called the impedance of this series circuit. Phasors • • Phasor Analysis (A Single Moment in Time) • Finding Peak Current in RLC - Circuit • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 4 Impedance
AC Circuits • Series RLC Circuit - The quantity Z is called the impedance of this series circuit. Phasors • • Phasor Analysis (A Single Moment in Time) Impedance is a generalization of resistance to include the • Finding Peak Current • in RLC - Circuit frequency-dependent effects of capacitance and inductance. • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 4 Current and Driving Voltage
AC Circuits • Series RLC Circuit - In an AC circuit containing resistors, inductors, and capacitors, Phasors • • Phasor Analysis (A the current through the circuit will not be in phase with the Single Moment in Time) • Finding Peak Current driving voltage source. in RLC - Circuit • Impedance • Current and Driving Voltage
Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 5 Current and Driving Voltage
AC Circuits • Series RLC Circuit - In an AC circuit containing resistors, inductors, and capacitors, Phasors • • Phasor Analysis (A the current through the circuit will not be in phase with the Single Moment in Time) • Finding Peak Current driving voltage source. in RLC - Circuit • Impedance • Current and Driving χL χc Voltage tan ϕ = − R Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 5 Current and Driving Voltage
AC Circuits • Series RLC Circuit - In an AC circuit containing resistors, inductors, and capacitors, Phasors • • Phasor Analysis (A the current through the circuit will not be in phase with the Single Moment in Time) • Finding Peak Current driving voltage source. in RLC - Circuit • Impedance • Current and Driving χL χc Voltage tan ϕ = − R Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits A purely resistive circuit will have •
PHYS102 AC Circuits - Phasors – slide 5 Current and Driving Voltage
AC Circuits • Series RLC Circuit - In an AC circuit containing resistors, inductors, and capacitors, Phasors • • Phasor Analysis (A the current through the circuit will not be in phase with the Single Moment in Time) • Finding Peak Current driving voltage source. in RLC - Circuit • Impedance • Current and Driving χL χc Voltage tan ϕ = − R Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits A purely resistive circuit will have tan ϕ =0 •
PHYS102 AC Circuits - Phasors – slide 5 Current and Driving Voltage
AC Circuits • Series RLC Circuit - In an AC circuit containing resistors, inductors, and capacitors, Phasors • • Phasor Analysis (A the current through the circuit will not be in phase with the Single Moment in Time) • Finding Peak Current driving voltage source. in RLC - Circuit • Impedance • Current and Driving χL χc Voltage tan ϕ = − R Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits A purely resistive circuit will have tan ϕ =0 ϕ =0. • ⇒
PHYS102 AC Circuits - Phasors – slide 5 Current and Driving Voltage
AC Circuits • Series RLC Circuit - In an AC circuit containing resistors, inductors, and capacitors, Phasors • • Phasor Analysis (A the current through the circuit will not be in phase with the Single Moment in Time) • Finding Peak Current driving voltage source. in RLC - Circuit • Impedance • Current and Driving χL χc Voltage tan ϕ = − R Root-Mean-Square
Transforming Voltage Amplitudes - AC - Circuits A purely resistive circuit will have tan ϕ =0 ϕ =0. • ⇒ The current in a purely resistive circuit will be in phase with the • driving voltage.
PHYS102 AC Circuits - Phasors – slide 5 Power in AC Circuits
AC Circuits Can we talk about power in AC circuits? Root-Mean-Square • • Power in AC Circuits • Definition of Root-Mean-Square • Time-Averaged Power
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 6 Power in AC Circuits
AC Circuits Can we talk about power in AC circuits? Root-Mean-Square • • Power in AC Circuits • Definition of It is more difficult than DC Circuits because of the phase Root-Mean-Square ◦ • Time-Averaged Power shifts. Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 6 Power in AC Circuits
AC Circuits Can we talk about power in AC circuits? Root-Mean-Square • • Power in AC Circuits • Definition of It is more difficult than DC Circuits because of the phase Root-Mean-Square ◦ • Time-Averaged Power shifts. Transforming Voltage 2 Amplitudes - AC - Remember, without phases P = I R. Circuits ◦
PHYS102 AC Circuits - Phasors – slide 6 Power in AC Circuits
AC Circuits Can we talk about power in AC circuits? Root-Mean-Square • • Power in AC Circuits • Definition of It is more difficult than DC Circuits because of the phase Root-Mean-Square ◦ • Time-Averaged Power shifts. Transforming Voltage 2 Amplitudes - AC - Remember, without phases P = I R. Circuits ◦ There is a standard engineering technique that allows ◦ one to discuss the average power.
PHYS102 AC Circuits - Phasors – slide 6 Power in AC Circuits
AC Circuits Can we talk about power in AC circuits? Root-Mean-Square • • Power in AC Circuits • Definition of It is more difficult than DC Circuits because of the phase Root-Mean-Square ◦ • Time-Averaged Power shifts. Transforming Voltage 2 Amplitudes - AC - Remember, without phases P = I R. Circuits ◦ There is a standard engineering technique that allows ◦ one to discuss the average power. What is the average of a sinusoidally varying function ◦ over one period of oscillation?
PHYS102 AC Circuits - Phasors – slide 6 Power in AC Circuits
AC Circuits Can we talk about power in AC circuits? Root-Mean-Square • • Power in AC Circuits • Definition of It is more difficult than DC Circuits because of the phase Root-Mean-Square ◦ • Time-Averaged Power shifts. Transforming Voltage 2 Amplitudes - AC - Remember, without phases P = I R. Circuits ◦ There is a standard engineering technique that allows ◦ one to discuss the average power. What is the average of a sinusoidally varying function ◦ over one period of oscillation? ZERO. Does it make sense to talk about averages for ◦ sinusoidally varying functions?
PHYS102 AC Circuits - Phasors – slide 6 Power in AC Circuits
AC Circuits Can we talk about power in AC circuits? Root-Mean-Square • • Power in AC Circuits • Definition of It is more difficult than DC Circuits because of the phase Root-Mean-Square ◦ • Time-Averaged Power shifts. Transforming Voltage 2 Amplitudes - AC - Remember, without phases P = I R. Circuits ◦ There is a standard engineering technique that allows ◦ one to discuss the average power. What is the average of a sinusoidally varying function ◦ over one period of oscillation? ZERO. Does it make sense to talk about averages for ◦ sinusoidally varying functions? Yes, because the wall socket is a type of average.
PHYS102 AC Circuits - Phasors – slide 6 Definition of Root-Mean-Square
AC Circuits The average of a sine function (or cosine) is zero over one • Root-Mean-Square time period. • Power in AC Circuits • Definition of Root-Mean-Square • Time-Averaged Power
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 7 Definition of Root-Mean-Square
AC Circuits The average of a sine function (or cosine) is zero over one • Root-Mean-Square time period. • Power in AC Circuits • Definition of If we square a sine (or cosine) function, then its average is 1/2 Root-Mean-Square • • Time-Averaged Power over one time period. Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 7 Definition of Root-Mean-Square
AC Circuits The average of a sine function (or cosine) is zero over one • Root-Mean-Square time period. • Power in AC Circuits • Definition of If we square a sine (or cosine) function, then its average is 1/2 Root-Mean-Square • • Time-Averaged Power over one time period. Transforming Voltage Amplitudes - AC - Defining the root-mean-square (engineering practice) as: Circuits •
PHYS102 AC Circuits - Phasors – slide 7 Definition of Root-Mean-Square
AC Circuits The average of a sine function (or cosine) is zero over one • Root-Mean-Square time period. • Power in AC Circuits • Definition of If we square a sine (or cosine) function, then its average is 1/2 Root-Mean-Square • • Time-Averaged Power over one time period. Transforming Voltage Amplitudes - AC - Defining the root-mean-square (engineering practice) as: Circuits • 2 2 VRMS = VP sin ωt where denotes time-average q hi
PHYS102 AC Circuits - Phasors – slide 7 Definition of Root-Mean-Square
AC Circuits The average of a sine function (or cosine) is zero over one • Root-Mean-Square time period. • Power in AC Circuits • Definition of If we square a sine (or cosine) function, then its average is 1/2 Root-Mean-Square • • Time-Averaged Power over one time period. Transforming Voltage Amplitudes - AC - Defining the root-mean-square (engineering practice) as: Circuits •
V = VP sin ωt 2 2 VRMS = VP sin ωt where denotes time-average q hi
PHYS102 AC Circuits - Phasors – slide 7 Definition of Root-Mean-Square
AC Circuits The average of a sine function (or cosine) is zero over one • Root-Mean-Square time period. • Power in AC Circuits • Definition of If we square a sine (or cosine) function, then its average is 1/2 Root-Mean-Square • • Time-Averaged Power over one time period. Transforming Voltage Amplitudes - AC - Defining the root-mean-square (engineering practice) as: Circuits •
V = VP sin ωt 2 2 VRMS = V sin ωt where denotes time-average q P hi T 2 1 2 sin ωt = sin ωtdt where T is one period T Z 0
PHYS102 AC Circuits - Phasors – slide 7 Definition of Root-Mean-Square
AC Circuits The average of a sine function (or cosine) is zero over one • Root-Mean-Square time period. • Power in AC Circuits • Definition of If we square a sine (or cosine) function, then its average is 1/2 Root-Mean-Square • • Time-Averaged Power over one time period. Transforming Voltage Amplitudes - AC - Defining the root-mean-square (engineering practice) as: Circuits •
V = VP sin ωt 2 2 VRMS = V sin ωt where denotes time-average q P hi T 2 1 2 sin ωt = sin ωtdt where T is one period T Z 0 2 1 sin ωt = 2
PHYS102 AC Circuits - Phasors – slide 7 Definition of Root-Mean-Square
AC Circuits The average of a sine function (or cosine) is zero over one • Root-Mean-Square time period. • Power in AC Circuits • Definition of If we square a sine (or cosine) function, then its average is 1/2 Root-Mean-Square • • Time-Averaged Power over one time period. Transforming Voltage Amplitudes - AC - Defining the root-mean-square (engineering practice) as: Circuits •
V = VP sin ωt 2 2 VRMS = V sin ωt where denotes time-average q P hi T 2 1 2 sin ωt = sin ωtdt where T is one period T Z 0 2 1 sin ωt = 2 1 VRMS = VP √2
PHYS102 AC Circuits - Phasors – slide 7 Time-Averaged Power
AC Circuits The time-average product of voltage and current with an Root-Mean-Square • • Power in AC Circuits arbitrary phase difference ϕ is given by • Definition of Root-Mean-Square • Time-Averaged Power
Transforming Voltage Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 8 Time-Averaged Power
AC Circuits The time-average product of voltage and current with an Root-Mean-Square • • Power in AC Circuits arbitrary phase difference ϕ is given by • Definition of Root-Mean-Square • Time-Averaged Power P = IP sin(ωt + ϕ) VP sin ωt Transforming Voltage h i h i Amplitudes - AC - Circuits
PHYS102 AC Circuits - Phasors – slide 8 Time-Averaged Power
AC Circuits The time-average product of voltage and current with an Root-Mean-Square • • Power in AC Circuits arbitrary phase difference ϕ is given by • Definition of Root-Mean-Square • Time-Averaged Power P = IP sin(ωt + ϕ) VP sin ωt Transforming Voltage h i h 2 i Amplitudes - AC - = IP VP (sin ωt)(cos ϕ) + (sin ωt)(cos ωt)(sin ϕ) Circuits
PHYS102 AC Circuits - Phasors – slide 8 Time-Averaged Power
AC Circuits The time-average product of voltage and current with an Root-Mean-Square • • Power in AC Circuits arbitrary phase difference ϕ is given by • Definition of Root-Mean-Square • Time-Averaged Power P = IP sin(ωt + ϕ) VP sin ωt Transforming Voltage h i h 2 i Amplitudes - AC - = IP VP (sin ωt)(cos ϕ) + (sin ωt)(cos ωt)(sin ϕ) Circuits 1 P = IP VP cos ϕ h i 2
PHYS102 AC Circuits - Phasors – slide 8 Time-Averaged Power
AC Circuits The time-average product of voltage and current with an Root-Mean-Square • • Power in AC Circuits arbitrary phase difference ϕ is given by • Definition of Root-Mean-Square • Time-Averaged Power P = IP sin(ωt + ϕ) VP sin ωt Transforming Voltage h i h 2 i Amplitudes - AC - = IP VP (sin ωt)(cos ϕ) + (sin ωt)(cos ωt)(sin ϕ) Circuits 1 P = IP VP cos ϕ h i 2 VP = √2 VRMS
PHYS102 AC Circuits - Phasors – slide 8 Time-Averaged Power
AC Circuits The time-average product of voltage and current with an Root-Mean-Square • • Power in AC Circuits arbitrary phase difference ϕ is given by • Definition of Root-Mean-Square • Time-Averaged Power P = IP sin(ωt + ϕ) VP sin ωt Transforming Voltage h i h 2 i Amplitudes - AC - = IP VP (sin ωt)(cos ϕ) + (sin ωt)(cos ωt)(sin ϕ) Circuits 1 P = IP VP cos ϕ h i 2 VP = √2 VRMS and IP = √2 IRMS
PHYS102 AC Circuits - Phasors – slide 8 Time-Averaged Power
AC Circuits The time-average product of voltage and current with an Root-Mean-Square • • Power in AC Circuits arbitrary phase difference ϕ is given by • Definition of Root-Mean-Square • Time-Averaged Power P = IP sin(ωt + ϕ) VP sin ωt Transforming Voltage h i h 2 i Amplitudes - AC - = IP VP (sin ωt)(cos ϕ) + (sin ωt)(cos ωt)(sin ϕ) Circuits 1 P = IP VP cos ϕ h i 2 VP = √2 VRMS and IP = √2 IRMS
P = IRMS VRMS cos ϕ h i
PHYS102 AC Circuits - Phasors – slide 8 Time-Averaged Power
AC Circuits The time-average product of voltage and current with an Root-Mean-Square • • Power in AC Circuits arbitrary phase difference ϕ is given by • Definition of Root-Mean-Square • Time-Averaged Power P = IP sin(ωt + ϕ) VP sin ωt Transforming Voltage h i h 2 i Amplitudes - AC - = IP VP (sin ωt)(cos ϕ) + (sin ωt)(cos ωt)(sin ϕ) Circuits 1 P = IP VP cos ϕ h i 2 VP = √2 VRMS and IP = √2 IRMS
P = IRMS VRMS cos ϕ h i
cos ϕ (is called the power factor.)
PHYS102 AC Circuits - Phasors – slide 8 Transformers
AC Circuits
Root-Mean-Square Transforming Voltage Now that we have power dissipated through an RLC series Amplitudes - AC - • Circuits circuit, let’s address an important issue. • Transformers • Transformers - Picture • Transformers - Voltage • Transformers - Power
PHYS102 AC Circuits - Phasors – slide 9 Transformers
AC Circuits
Root-Mean-Square Transforming Voltage Now that we have power dissipated through an RLC series Amplitudes - AC - • Circuits circuit, let’s address an important issue. • Transformers • Transformers - Picture Not all devices require 120-V AC. • Transformers - • Voltage • Transformers - Power
PHYS102 AC Circuits - Phasors – slide 9 Transformers
AC Circuits
Root-Mean-Square Transforming Voltage Now that we have power dissipated through an RLC series Amplitudes - AC - • Circuits circuit, let’s address an important issue. • Transformers • Transformers - Picture Not all devices require 120-V AC. Some devices require only • Transformers - • Voltage 12-V AC. • Transformers - Power
PHYS102 AC Circuits - Phasors – slide 9 Transformers
AC Circuits
Root-Mean-Square Transforming Voltage Now that we have power dissipated through an RLC series Amplitudes - AC - • Circuits circuit, let’s address an important issue. • Transformers • Transformers - Picture Not all devices require 120-V AC. Some devices require only • Transformers - • Voltage 12-V AC. • Transformers - Power How do we “transform” the amplitude of the voltage ◦ provided by the power company to another amplitude?
PHYS102 AC Circuits - Phasors – slide 9 Transformers
AC Circuits
Root-Mean-Square Transforming Voltage Now that we have power dissipated through an RLC series Amplitudes - AC - • Circuits circuit, let’s address an important issue. • Transformers • Transformers - Picture Not all devices require 120-V AC. Some devices require only • Transformers - • Voltage 12-V AC. • Transformers - Power How do we “transform” the amplitude of the voltage ◦ provided by the power company to another amplitude? We go back to Faraday’s Law of Induction. ◦
PHYS102 AC Circuits - Phasors – slide 9 Transformers
AC Circuits
Root-Mean-Square Transforming Voltage Now that we have power dissipated through an RLC series Amplitudes - AC - • Circuits circuit, let’s address an important issue. • Transformers • Transformers - Picture Not all devices require 120-V AC. Some devices require only • Transformers - • Voltage 12-V AC. • Transformers - Power How do we “transform” the amplitude of the voltage ◦ provided by the power company to another amplitude? We go back to Faraday’s Law of Induction. ◦ If we strategically place two different solenoids near each • other in an AC circuit, then the EMF through the solenoids will have different values.
PHYS102 AC Circuits - Phasors – slide 9 Transformers
AC Circuits
Root-Mean-Square Transforming Voltage Now that we have power dissipated through an RLC series Amplitudes - AC - • Circuits circuit, let’s address an important issue. • Transformers • Transformers - Picture Not all devices require 120-V AC. Some devices require only • Transformers - • Voltage 12-V AC. • Transformers - Power How do we “transform” the amplitude of the voltage ◦ provided by the power company to another amplitude? We go back to Faraday’s Law of Induction. ◦ If we strategically place two different solenoids near each • other in an AC circuit, then the EMF through the solenoids will have different values. A device which uses an arrangement of coils to vary the • amplitude of the primary voltage source is called a transformer and one of its circuit symbol is shown above in the title.
PHYS102 AC Circuits - Phasors – slide 9 Transformers
AC Circuits
Root-Mean-Square Transforming Voltage Now that we have power dissipated through an RLC series Amplitudes - AC - • Circuits circuit, let’s address an important issue. • Transformers • Transformers - Picture Not all devices require 120-V AC. Some devices require only • Transformers - • Voltage 12-V AC. • Transformers - Power How do we “transform” the amplitude of the voltage ◦ provided by the power company to another amplitude? We go back to Faraday’s Law of Induction. ◦ If we strategically place two different solenoids near each • other in an AC circuit, then the EMF through the solenoids will have different values. A device which uses an arrangement of coils to vary the • amplitude of the primary voltage source is called a transformer and one of its circuit symbol is shown above in the title.
PHYS102 AC Circuits - Phasors – slide 9 Transformers - Picture
AC Circuits The artist rendition below is that of a typical transformer. Root-Mean-Square •
Transforming Voltage Amplitudes - AC - Circuits • Transformers • Transformers - Picture • Transformers - Voltage • Transformers - Power
PHYS102 AC Circuits - Phasors – slide 10 Transformers - Picture
AC Circuits The artist rendition below is that of a typical transformer. Root-Mean-Square •
Transforming Voltage Amplitudes - AC - Circuits • Transformers • Transformers - Picture • Transformers - Voltage • Transformers - Power
PHYS102 AC Circuits - Phasors – slide 10 Transformers - Picture
AC Circuits The artist rendition below is that of a typical transformer. Root-Mean-Square •
Transforming Voltage Amplitudes - AC - Circuits • Transformers • Transformers - Picture • Transformers - Voltage • Transformers - Power
Iron core used to concentrate magnetic flux which ensures •
PHYS102 AC Circuits - Phasors – slide 10 Transformers - Picture
AC Circuits The artist rendition below is that of a typical transformer. Root-Mean-Square •
Transforming Voltage Amplitudes - AC - Circuits • Transformers • Transformers - Picture • Transformers - Voltage • Transformers - Power
Iron core used to concentrate magnetic flux which ensures • the magnetic flux through primary and secondary coils is ◦ the same.
PHYS102 AC Circuits - Phasors – slide 10 Transformers - Picture
AC Circuits The artist rendition below is that of a typical transformer. Root-Mean-Square •
Transforming Voltage Amplitudes - AC - Circuits • Transformers • Transformers - Picture • Transformers - Voltage • Transformers - Power
Iron core used to concentrate magnetic flux which ensures • the magnetic flux through primary and secondary coils is ◦ the same.
PHYS102 AC Circuits - Phasors – slide 10 Transformers - Voltage
AC Circuits Since the magnetic flux is the same through both coils, the rate • Root-Mean-Square of change of magnetic flux is the same through the two coils. Transforming Voltage Amplitudes - AC - Circuits • Transformers • Transformers - Picture • Transformers - Voltage • Transformers - Power
PHYS102 AC Circuits - Phasors – slide 11 Transformers - Voltage
AC Circuits Since the magnetic flux is the same through both coils, the rate • Root-Mean-Square of change of magnetic flux is the same through the two coils. Transforming Voltage Amplitudes - AC - Circuits dΦB • Transformers VP = NP • Transformers - Picture dt • Transformers - Voltage • Transformers - Power
PHYS102 AC Circuits - Phasors – slide 11 Transformers - Voltage
AC Circuits Since the magnetic flux is the same through both coils, the rate • Root-Mean-Square of change of magnetic flux is the same through the two coils. Transforming Voltage Amplitudes - AC - Circuits dΦB • Transformers VP = NP • Transformers - Picture dt • Transformers - Voltage • Transformers - Power dΦ V = N B S S dt
PHYS102 AC Circuits - Phasors – slide 11 Transformers - Voltage
AC Circuits Since the magnetic flux is the same through both coils, the rate • Root-Mean-Square of change of magnetic flux is the same through the two coils. Transforming Voltage Amplitudes - AC - Circuits dΦB • Transformers VP = NP • Transformers - Picture dt • Transformers - Voltage • Transformers - Power dΦ V = N B S S dt
V V P = S ⇒ NP NS
PHYS102 AC Circuits - Phasors – slide 11 Transformers - Voltage
AC Circuits Since the magnetic flux is the same through both coils, the rate • Root-Mean-Square of change of magnetic flux is the same through the two coils. Transforming Voltage Amplitudes - AC - Circuits dΦB • Transformers VP = NP • Transformers - Picture dt • Transformers - Voltage • Transformers - Power dΦ V = N B S S dt
V V P = S ⇒ NP NS
NS VS = VP ⇒ NP
PHYS102 AC Circuits - Phasors – slide 11 Transformers - Power
AC Circuits It seems that the secondary voltage can be arbitrarily large. Root-Mean-Square •
Transforming Voltage Amplitudes - AC - Circuits • Transformers • Transformers - Picture • Transformers - Voltage • Transformers - Power
PHYS102 AC Circuits - Phasors – slide 12 Transformers - Power
AC Circuits It seems that the secondary voltage can be arbitrarily large. • Root-Mean-Square Does this violate conservation of energy? Transforming Voltage • Amplitudes - AC - Circuits • Transformers • Transformers - Picture • Transformers - Voltage • Transformers - Power
PHYS102 AC Circuits - Phasors – slide 12 Transformers - Power
AC Circuits It seems that the secondary voltage can be arbitrarily large. • Root-Mean-Square Does this violate conservation of energy? Transforming Voltage • Amplitudes - AC - Circuits No. • Transformers ◦ • Transformers - Picture • Transformers - Voltage • Transformers - Power
PHYS102 AC Circuits - Phasors – slide 12 Transformers - Power
AC Circuits It seems that the secondary voltage can be arbitrarily large. • Root-Mean-Square Does this violate conservation of energy? Transforming Voltage • Amplitudes - AC - No. Circuits ◦ • Transformers A transformer can not increase power. • Transformers - Picture • Transformers - ◦ Voltage • Transformers - Power
PHYS102 AC Circuits - Phasors – slide 12 Transformers - Power
AC Circuits It seems that the secondary voltage can be arbitrarily large. • Root-Mean-Square Does this violate conservation of energy? Transforming Voltage • Amplitudes - AC - No. Circuits ◦ • Transformers A transformer can not increase power. • Transformers - Picture • Transformers - ◦ Voltage Ideal transformers transfer all the power supplied by the • Transformers - Power • primary source to the secondary.
PHYS102 AC Circuits - Phasors – slide 12 Transformers - Power
AC Circuits It seems that the secondary voltage can be arbitrarily large. • Root-Mean-Square Does this violate conservation of energy? Transforming Voltage • Amplitudes - AC - No. Circuits ◦ • Transformers A transformer can not increase power. • Transformers - Picture • Transformers - ◦ Voltage Ideal transformers transfer all the power supplied by the • Transformers - Power • primary source to the secondary.
IP VP = IS VS (Statement of Conservation of Energy)
PHYS102 AC Circuits - Phasors – slide 12