PHYS 221 General Physics II

Transcription

1 PHYS 22 General Physics II Resonance, Impedance, Transformers, Motors Spring 25 Assigned Reading: ecture 5

2 Review: Self-Induction: Changing current through a loop induces an opposing voltage in that same loop. i t Stored energy electric magnetic PEC 2 CV 2 PE ind 2 I 2 Reactance: X C C X Phys 22 Spring 24 ecture 5 2

3 AC circuit and Impedance R C We can use Kirchoff loop rule and solve differential eq. But we can also use phasors Phys 22 Spring 24 ecture 5 3

4 Relationship between I rms & V C,rms in RC circuit I rms V C, rms X C where X C C is the capacitive reactance VR, rms. XC is similar to R in Irms. R 2. Average power delivered to a capacitor in an ac circuit is zero. Phys 22 Spring 24 ecture 5 4

5 Relationship between I rms & V rms in R circuit I rms V, rms X where X is the inductive reactance. VR, rms. X is similar to R in Irms. R 2. Average power delivered to an inductor in an ac circuit is zero. Phys 22 Spring 24 ecture 5 5

6 Summary: V R RI R V max sint I R V max R sint R is resistance, in Ohms Impedances for,c,r V R I t V R I t V C Q C V max sint I C CV max cos t X C C is capacitive reactance, in Ohms I V C I t V C V I t V max sint I V max sin t 2 X = is inductive Reactance, in Ohms V I V I t 6

7 C Circuit The voltage and current in the circuit oscillate between positive and negative values The circuit behaves as a simple harmonic oscillator Charge: q = q max cos (2πƒt) Current: I = I max sin (2πƒt) Phys 22 Spring 24 ecture 5 7

8 C Circuits Consider the C and RC series circuits shown: Suppose that at t= the capacitor is charged to a value of Q C ---- R Is there is a qualitative difference in the time development of the currents produced in these two cases. Why?? Consider from point of view of energy! In the RC circuit, any current developed will cause energy to be dissipated in the resistor. In the (ideal) C circuit, there is NO mechanism for energy dissipation; energy can be stored in the capacitor and/or in the inductor! C Phys 22 Spring 24 ecture 5 8

9 C Circuit After t =, the charge moves from one capacitor plate to the other and current passes through the inductor. Eventually, the charge on each capacitor plate falls to zero. The inductor opposes change in the current, so the induced emf now acts to maintain the current at a non-zero value. This current continues to transport charge from one capacitor plate to the other, causing the capacitor s charge and voltage to reverse sign. The charge on the capacitor returns to its original value. Phys 22 Spring 24 ecture 5 9

10 C Oscillations Kirchoff s loop rule I Q Q VVC, where I t C t Q I C Differential equation 2 dq Q 2 dt C This is the solution: Q Q cos( t) Where: determined from equation Phase and initial charge Q from initial conditions C Phys 22 Spring 24 ecture 5

11 C Oscillations I Q Q C I V C I t V t Phys 22 Spring 24 ecture 5 t

12 C Oscillations: Energy Check Oscillation frequency is found from the C loop equation. The other unknowns are found from the initial conditions. Question: Does this solution conserve energy? Phys 22 Spring 24 ecture 5 2

13 Energy Check Energy in Capacitor 2 2 PEcap () t Qmax cos( t) 2C U E Energy in Inductor 2 2 PEind () t Imax sin ( t) 2 Conservation of energy PE PE cap,max 2 Q max I 2 C 2 Imax Q C ind,max 2 max max U B Phys 22 Spring 24 ecture 5 3 t t

14 U B versus U E Phys 22 Spring 24 ecture 5 4

15 i>clicker question At t = the capacitor has charge Q ; the resulting oscillations have frequency ω. The maximum current in the circuit during these oscillations has value I. What is the relation between ω and ω 2, the frequency of oscillations when the initial charge = 2Q? Q t = = Q C (A) ω 2 = /2 ω (B) ω 2 = ω (C) ω 2 = 2ω Phys 22 Spring 24 ecture 5 5

16 Phasors CR I max X C R I max X C I max R V max This picture corresponds to a snapshot at t=. The projections of the phasors on the vertical axis are the voltages at the given time. The current is continuous around the loop. The three VECTOR voltages add up vectorially to equal the V max I m and can now be solved graphically since the vector sum of the voltages V + V C + V R =V Phys 22 Spring 24 ecture 5 6

17 Phasors:Tips This phasor diagram was drawn at t = with the voltages being given as the projections along the y-axis. Sometimes it is easier to draw the diagram at a time when the current is along the x- axis (when I = ). I max X V max I max X I max X C V max I max R Full Phasor Diagram I max X C From this diagram, we can also create a triangle which allows us to calculate the impedance Z: Imax ImaxR Impedance Triangle Phys 22 Spring 24 ecture 5 7 Z I max R Imax X XC

18 CR circuits Results of the phasor analysis V V max sint I I max sint I max V max Z 2 2 Z R X XC tan X X C R X X C C Phys 22 Spring 24 ecture 5 8

19 Resonance For fixed R,C, the current I m will be a maximum at the resonant frequency which makes the impedance Z purely resistive. m m Im Z 2 R X X the frequency at which this condition is obtained is given from: o o C C o 2 reaches a maximum when: X XC Note that the resonant frequency is identical to the natural frequency of the C circuit by itself! At this frequency, the current and the driving voltage are in phase! X XC tan R Phys 22 Spring 24 ecture 5 9 C