#  Driven AC Circuits  Phase of V and I  Conceputally  Mathematically  With phasors Physics 2112 Unit 20 Outline: Electricity & Magnetism Lecture 20,

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• Slide 1
• Driven AC Circuits Phase of V and I Conceputally Mathematically With phasors Physics 2112 Unit 20 Outline: Electricity & Magnetism Lecture 20, Slide 1
• Slide 2
• AC Generator V max sin( d t) Electricity & Magnetism Lecture 20, Slide 2 Driving frequency = natural frequency ( o )
• Slide 3
• Phase between I and V R I R = V R /R I= V max /R sin( d t) Amplitude = V max /R Electricity & Magnetism Lecture 20, Slide 3 Voltage goes up current goes up Simple Case - Resistors In phase Phase angle = 0 o
• Slide 4
• Capacitors C I = V max C cos( t) where X C 1/ C is like the resistance of the capacitor X C depends on Amplitude V max /X C Q CV CV max sin( t) 90 o Unit 20, Slide 4
• Slide 5
• Inductors L where X L L is like the resistance of the inductor X L depends on 90 o Electricity & Magnetism Lecture 20, Slide 5 Amplitude V max /X L
• Slide 6
• Phase Summary L Electricity & Magnetism Lecture 20, Slide 6 R C I leads V V and I in phase I lags V ELI the ICE man
• Slide 7
• What does this look like together? Electricity & Magnetism Lecture 20, Slide 7 Notice phase relationships
• Slide 8
• What does this look like together? Electricity & Magnetism Lecture 20, Slide 8 Capacitor and Inductor always 180 o out of phase Capacitor/Inductor and Resistor always 90 o out of phase Resistor is some unknown phase angle out of phase is signal generator
• Slide 9
• What about current? Electricity & Magnetism Lecture 20, Slide 9 Current is always the same through all elements (in series) Current and Voltage in phase across Resistor Current and voltage out of phase by unknown phase angle across signal generator (Well find this phase angle later.)
• Slide 10
• Reactance Summary L Electricity & Magnetism Lecture 20, Slide 10 R C goes up, c goes down Doesnt depend on goes up, L goes up
• Slide 11
• Example 20.1 (Inductor Reactance) L A 60Hz signal with a V max = 5V is sent through a 50mH inductor. What is the maximum current, I max, through the inductor? Electricity & Magnetism Lecture 20, Slide 11
• Slide 12
• Think of same material graphically using phasors Electricity & Magnetism Lecture 20, Slide 12 Phasors Phasor just thinks of sine wave as rotating vector
• Slide 13
• I max X L I max X C I max R V L and V C 180 o out of phase Electricity & Magnetism Lecture 20, Slide 13 Circuit using Phasors Represent voltage drops across elements as rotating vectors (phasors) V L and V R 90 o out of phase Remember V R and I in phase
• Slide 14
• I max R max I max Z I max (X L X C ) R (XL XC)(XL XC) Impedance Triangle Make this Simpler Electricity & Magnetism Lecture 20, Slide 14 I max X L I max X C I max R L R C max
• Slide 15
• R (XL XC)(XL XC) V Cmax I max X C V Lmax I max X L V Rmax I max R Summary I max max / Z max I max Z Electricity & Magnetism Lecture 20, Slide 15 I max X L I max X C I max R L R C max
• Slide 16
• CheckPoint 1(A) Electricity & Magnetism Lecture 20, Slide 16 A RL circuit is driven by an AC generator as shown in the figure. The voltages across the resistor and generator are. A.always out of phase B.always in phase C.sometimes in phase and sometimes out of phase
• Slide 17
• CheckPoint 1(B) Electricity & Magnetism Lecture 20, Slide 17 A RL circuit is driven by an AC generator as shown in the figure. The voltages across the resistor and inductor are. A.always out of phase B.always in phase C.sometimes in phase and sometimes out of phase
• Slide 18
• CheckPoint 1(C) Electricity & Magnetism Lecture 20, Slide 18 A RL circuit is driven by an AC generator as shown in the figure. The phase difference between the CURRENT through the resistor and inductor A.is always zero B.is always 90 o C.depends on the value of L and R D.depends on L, R and the generator voltage
• Slide 19
• Example 20.2 (LCR) Electricity & Magnetism Lecture 20, Slide 19 C R V In the circuit to the right L=500mH V max = 6V C=47uF R=100W L What is the maximum current and phase angle if = 60rad/sec? What is the maximum current and phase angle if = 206 rad/sec? What is the maximum current and phase angle if = 400 rad/sec?
• Slide 20
• What does this look like graphically? Electricity & Magnetism Lecture 20, Slide 20
• Slide 21
• Point of confusion?? Electricity & Magnetism Lecture 20, Slide 21 V L + V C + V R + = 0 V L-max + V C-max + V R-max + = 0 (Add like vectors) (I max and V max happen at different times.)
• Slide 22
• CheckPoint 2(A) A driven RLC circuit is represented by the phasor diagram to the right. The vertical axis of the phasor diagram represents voltage. When the current through the circuit is maximum, what is the potential difference across the inductor? A.V L = 0 B.V L = V L-max /2 C.V L = V L=max
• Slide 23
• CheckPoint 2(B) Electricity & Magnetism Lecture 20, Slide 23 A driven RLC circuit is represented by the above phasor diagram. When the capacitor is fully charged, what is the magnitude of the voltage across the inductor? A.V L = 0 B.V L = V L-max /2 C.V L = V L=max
• Slide 24
• CheckPoint 2(C) Electricity & Magnetism Lecture 20, Slide 24 A driven RLC circuit is represented by the above phasor diagram. When the voltage across the capacitor is at its positive maximum, V C = +V C-max, what is the magnitude of the voltage across the inductor? A.V L = 0 B.V L = V L-max /2 C.V L = V L=max
• Slide 25
• Conceptual Analysis The maximum voltage for each component is related to its reactance and to the maximum current. The impedance triangle determines the relationship between the maximum voltages for the components Strategic Analysis Use V max and I max to determine Z Use impedance triangle to determine R Use V Cmax and impedance triangle to determine X L ~ C R L V Example 20.3 Consider the harmonically driven series LCR circuit shown. V max 100 V I max 2 mA V Cmax 113 V The current leads generator voltage by 45 o L and R are unknown. What is X L, the reactance of the inductor, at this frequency? Electricity & Magnetism Lecture 20, Slide 25 Get your calculators out