Physics 2112 Unit 20. Outline: Driven AC Circuits Phase of V and I Conceputally Mathematically With phasors

Transcription

1 Physics 2112 Unit 20 Outline: Driven A ircuits Phase of V and I onceputally Mathematically With phasors Electricity & Magnetism ecture 20, Slide 1

2 Your omments it just got real this stuff is confusing the relationships between the capacitors, inductors, and resistors voltage This was the first somewhat challenging prelecture. I much prefer doing prelectures over reading a book. Please keep using this style next semester. Examples of the math on this, please! When the phi=0, what is happening- life is simple? Is the current across each element the same at a given instant? PHASOS!!!!!!!!!!!!!!! I don't understand reactance. what phasor diagrams do and how do we know if the current is out of phase with the capacitor? I don't understand why the voltage and current start where they do in the phaser diagrams. Would like to see examples in class per usual what is the relationship of the phase of current and voltage? Or between the voltage of the resistor and the inductor? Electricity & Magnetism ecture 20, Slide 2

3 How would we actually do this? Start with some initial V, I, Q, V Now take a tiny time step dt (1 ms) V d I dt dq Idt V Q V I epeat V V V V What would this look like? Electricity & Magnetism ecture 19, Slide 3

4 V V V I Unit 19, Slide 4

5 A Generator e V max sin(w d t) Driving frequency = natural frequency (w o ) Electricity & Magnetism ecture 20, Slide 5

6 Phase between I and V Simple ase - esistors I = V / Voltage goes up current goes up In phase Phase angle = 0 o I= V max / sin(w d t) Amplitude = V max / Electricity & Magnetism ecture 20, Slide 6

7 apacitors w d frequency of signal generator (not w o ) Q V V max sin(wt) I = V max w cos(wt) 90 o Amplitude V max /X where X 1/w is like the resistance of the capacitor X depends on w Unit 20, Slide 7

8 Inductors di V V max sin( wt) dt I V w max cos( wt) 90 o Amplitude V max /X where X w is like the resistance of the inductor X depends on w Electricity & Magnetism ecture 20, Slide 8

9 Phase Summary I V max sin( w t) d V and I in phase I V V max max w cos( w t) sin( w t d d 90 o ) I leads V I Vmax w Vmax cos( wt) sin( wt 90 o ) I lags V EI the IE man Electricity & Magnetism ecture 20, Slide 9

10 Question Does the current lead or lag the voltage in the above plots? A. the current leads the voltage B. the current lags the voltage. they are in phase These might be the current and voltage plots for a: A. capacitor B. inductor. resistor Electricity & Magnetism ecture 20, Slide 10

11 What does this look like together? Notice phase relationships Green inductor voltage Blue capacitor voltage ed resistor voltage Black current Electricity & Magnetism ecture 20, Slide 11

12 What does this look like together? apacitor and Inductor always 180 o out of phase apacitor/inductor and esistor always 90 o out of phase Green inductor voltage Blue capacitor voltage ed resistor voltage Black current esistor and the signal generator are some unknown phase angle out of phase Electricity & Magnetism ecture 20, Slide 12

13 What about current? urrent is always the same through all elements (in series) urrent and Voltage in phase across esistor urrent and voltage out of phase by unknown phase angle across signal generator Green inductor voltage Blue capacitor voltage ed resistor voltage Black current (We ll find this phase angle later.) Electricity & Magnetism ecture 20, Slide 13

14 Question The potential difference across a single inductor is determined by a sinusoidal signal with amplitude of 5V and a frequency of 60Hz. The maximum current through the inductor is I max. If I double the frequency of the potential signal to 120Hz, what with the value of the maximum current be? A. I max /4 B. I max /2. No change (I max ) D. 2*I max E. 4*I max Unit 20, Slide 14

15 Question The potential difference across a single resistor is determined by a sinusoidal signal with amplitude of 5V and a frequency of 60Hz. The maximum current through the resistor is I max. If I double the frequency of the potential signal to 120Hz, what with the value of the maximum current be? A. I max /4 B. I max /2. No change (I max ) D. 2*I max E. 4*I max Unit 20, Slide 15

16 eactance Summary Doesn t depend on w 1 X w w goes up, c goes down X w w goes up, goes up Electricity & Magnetism ecture 20, Slide 16

17 Example 20.1 (esistor esistance) A 60Hz signal with a V max = 5V is sent through a 100W resistor. What is the maximum current, I max, through the resistor? Electricity & Magnetism ecture 20, Slide 17

18 Example 20.2 (Inductor eactance) 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 ecture 20, Slide 18

19 Example 20.2 (Inductor eactance) A 60Hz signal with a V max = 5V is sent through a 50mH inductor. What is the maximum current, I max, through the inductor? A) 12mA B) 100mA ) 265mA D) 1660mA Electricity & Magnetism ecture 20, Slide 19

20 Phasors Think of same material graphically using phasors Phasor just thinks of sine wave as rotating vector Electricity & Magnetism ecture 20, Slide 20

21 ircuit using Phasors epresent voltage drops across elements as rotating vectors (phasors) V and V 180 o out of phase V and V 90 o out of phase emember V and I in phase I max X I max X I max Electricity & Magnetism ecture 20, Slide 21

22 The Voltages still Add Up At any instant: I max X V + V + V + e = 0 But now we are adding vectors: e max I max X I max X I max X I max e max I max I max I max I max X I max X I max X I max X e max Electricity & Magnetism ecture 20, Slide 22

23 Make this Simpler I max X e max I max X I max X I max X I max e max I max I max I max X I max X Electricity & Magnetism ecture 20, Slide 23

24 Make this Simpler I max X e max I max X I max X I max I max e max I max Z I max (X X ) I max I max X Electricity & Magnetism ecture 20, Slide 24

25 Make this Simpler I max X e max I max X I max e max I max Z I max (X X ) I max Electricity & Magnetism ecture 20, Slide 25

26 Make this Simpler I max X e max I max Z f I max (X X ) e max I max X I max I max Z 2 ( X X) 2 Impedance Triangle tan ( f) X X Electricity & Magnetism ecture 20, Slide 26

27 Summary I max X V max I max X V max I max X V max I max e max I max Z e max I max X I max I max e max / Z Z 2 ( X X ) 2 Z 2 tan ( f) 2 X X X X f Electricity & Magnetism ecture 20, Slide 27

28 Where we are: What are phasors? Just a graphical way of representing an alternating voltage. y value of the vector represents the instantenous value of the voltage at any given instant. Apply Kirchoff s oop ule numerically V + V + V + e = 0 Apply it graphically Electricity & Magnetism ecture 20, Slide 28

29 Summary I max X V max I max X V max I max X V max I max e max I max Z e max I max X I max I max e max / Z Z 2 ( X X ) 2 Z 2 tan ( f) 2 X X X X f Electricity & Magnetism ecture 20, Slide 29

30 Example: ircuit X c 0 I max X e max I max X I max e max I max Electricity & Magnetism ecture 20, Slide 30

31 heckpoint 1(A) A circuit is driven by an A generator as shown in the figure. Draw Voltage Phasors I max X The voltages across the resistor and generator are. A. always out of phase B. always in phase. sometimes in phase and sometimes out of phase e max I max Electricity & Magnetism ecture 20, Slide 31

32 heckpoint 1(B) A circuit is driven by an A generator as shown in the figure. Draw Voltage Phasors I max X e max The voltages across the resistor and inductor are. A. always out of phase B. always in phase. sometimes in phase and sometimes out of phase I max Electricity & Magnetism ecture 20, Slide 32

33 heckpoint 1() A circuit is driven by an A generator as shown in the figure. The UENT is THE UENT I max X The phase difference between the UENT through the resistor and inductor A. is always zero B. is always 90 o. depends on the value of and D. depends on, and the generator voltage f e max I max f is the phase between generator and current Electricity & Magnetism ecture 20, Slide 33

34 Example 20.2 () In the circuit to the right =500mH V max = 6V =47uF =100W V What is the maximum current and phase angle if w = 60rad/sec? What is the maximum current and phase angle if w = 400 rad/sec? What is the maximum current and phase angle if w = 206 rad/sec? Electricity & Magnetism ecture 20, Slide 34

35 What does this look like graphically? Electricity & Magnetism ecture 20, Slide 35

36 Phase urrent leads Voltage urrent lags Voltage In Sync 60rad/sec 400 rad/sec 206 rad/sec Electricity & Magnetism ecture 20, Slide 36

37 Max urrent urrent leads Voltage urrent lags Voltage In Sync 60rad/sec 400 rad/sec 206 rad/sec Electricity & Magnetism ecture 20, Slide 37

38 Point of confusion?? V + V + V + e = 0 V -max + V -max + V -max + e = 0 (Add like vectors) I max V max Z I V Z (I max and V max happen at different times.) Electricity & Magnetism ecture 20, Slide 38

39 heckpoint 2(A) A driven 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 = 0 B. V = V -max /2. V = V =max IX e I e I What does the voltage phasor diagram look like when the current is a maximum? IX IX c IX c Electricity & Magnetism ecture 20, Slide 39

40 heckpoint 2(B) I IX IX c e I e IX c A driven 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 = 0 B. V = V -max /2. V = V =max IX What does the voltage phasor diagram look like when the capacitor is fully charged? Electricity & Magnetism ecture 20, Slide 40

41 heckpoint 2() I IX IX c e I e IX c A driven circuit is represented by the above phasor diagram. When the voltage across the capacitor is at its positive maximum, V = +V -max, what is the magnitude of the voltage across the inductor? A. V = 0 B. V = V -max /2. V = V =max IX What does the voltage phasor diagram look like when the voltage across capacitor is at its positive maximum? Electricity & Magnetism ecture 20, Slide 41

42 Example 20.3 onsider the harmonically driven series circuit shown. V max 100 V I max 2 ma V max 113 V The current leads generator voltage by 45 o and are unknown. V ~ What is X, the reactance of the inductor, at this frequency? onceptual 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 Use V max and impedance triangle to determine X Electricity & Magnetism ecture 20, Slide 42

43 Example 20.3 (con t) onsider the harmonically driven series circuit shown. V max 100 V I max 2 ma V max 113 V The current leads generator voltage by 45 o and are unknown. V ~ What is X, the reactance of the inductor, at this frequency? ompare X and X at this frequency: A) X < X B) X X ) X > X D) Not enough information V This information is determined from the phase urrent leads voltage I V I max X V I max X V (phase of current) V 45 o V V leads Electricity & Magnetism ecture 20, Slide 43

44 Example 20.3 (con t) onsider the harmonically driven series circuit shown. V max 100 V I max 2 ma V max 113 V The current leads generator voltage by 45 o and are unknown. V ~ What is X, the reactance of the inductor, at this frequency? What is Z, the total impedance of the circuit? A) 70.7 kw B) ) 35.4 kw D) 50 kw 21.1 kw Vmax 100V Z 50kW I 2mA max Electricity & Magnetism ecture 20, Slide 44

45 Example 20.3 (con t) onsider the harmonically driven series circuit shown. V max 100 V I max 2 ma V max 113 V The current leads generator voltage by 45 o and are unknown. V ~ What is X, the reactance of the inductor, at this frequency? What is? A) 70.7 kw B) ) 35.4 kw D) 50 kw 21.1 kw Z 50kW sin(45).707 cos(45).707 Determined from impedance triangle 45 o Z cos(45 o ) (X X ) cos(45) Z 50 kw x kw Electricity & Magnetism ecture 20, Slide 45

46 Example 20.3 (con t) onsider the harmonically driven series circuit shown. V max 100 V I max 2 ma V max 113 V The current leads generator voltage by 45 o and are unknown. V ~ What is X, the reactance of the inductor, at this frequency? A) 91.9 kw B) ) 35.4 kw D) 50 kw 21.1 kw Z 50kW 35.4kW We start with the X X impedance triangle: tan( 45 ) 1 Z 45 o (X X ) X 56.5 kw 35.4 kw X X What is X? V max I max X 113 X 56. 5kW 2 Electricity & Magnetism ecture 20, Slide 46