Announcements: Today: more AC circuits

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Ruby Taylor
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1 Announcements: Today: more AC circuits

2 I 0 I rms Current through a light bulb I 0 I rms I t = I 0 cos ωt I 0 Current through a LED I t = I 0 cos ωt Θ(cos ωt ) Theta function (is zero for a negative argument)

3 I 0 I rms Current through a light bulb I t = I 0 cos ωt I 0 I rms Resistance is a linear element I 0 Current through a LED I t = I 0 cos ωt Θ(cos ωt ) Theta function (is zero for a negative argument) LED (Diode) is a non-linear element

4 AC circuits for linear elements

5 AC current: I t = I 0 cos ωt Amplitude Angular frequency Can be represented by a phasor diagram:

6 AC current: I t = I 0 cos ωt Amplitude Angular frequency Can be represented by a phasor diagram: Im This is mathematically equivalent to : Re I t = I 0 cos ωt = I 0 Re{e iωt } Real part

7 AC current: I t = I 0 cos ωt Amplitude Angular frequency Can be represented by a phasor diagram: Im This is mathematically equivalent to : I t = I 0 cos ωt = I 0 Re{e iωt } = Re{I 0 e iωt } = Re{ሚI t } Re To indicate that it is complex real (true) current Real part complex current

8 AC circuit examples:

9 AC circuit 1 (AC source and resistor):

10 circuit-construction-kit-ac_en

11 AC circuit 1 (AC source and resistor): Time dependences of voltage and current are the same (only amplitudes are different)

12 AC circuit 1 (AC source and resistor): Time dependences of voltage and current are the same (only amplitudes are different) The Current and Voltage phasors are aligned

13 AC circuit 1 (AC source and resistor): V t = R ሚI(t) ሚI t = I 0 e iωt V t = RI 0 e iωt Time dependences of voltage and current are the same (only amplitudes are different) The Current and Voltage phasors are aligned

14 AC circuit 2 (AC source and inductor):

15 circuit-construction-kit-ac_en

16 AC circuit 2 (AC source and inductor): Time dependences of voltage and current are out of phase (and amplitudes are different)

17 AC circuit 2 (AC source and inductor): Time dependences of voltage and current are out of phase (and amplitudes are different) (For an inductor we have voltage=ldi/dt, so v>0 if di/dt>0)

18 AC circuit 2 (AC source and inductor): Time dependences of voltage and current are out of phase (and amplitudes are different) The Current and Voltage phasors are perpendicular (For an inductor we have voltage=ldi/dt, so v>0 if di/dt>0)

19 AC circuit 2 (AC source and inductor): Time dependences of voltage and current are out of phase (and amplitudes are different) (For an inductor we have voltage=ldi/dt, so v>0 if di/dt>0) The Current and Voltage phasors are perpendicular V t RI (t)

20 AC circuit 2 (AC source and inductor): V t = iωl ሚI(t) Time dependences of voltage and current are out of phase (and amplitudes are different) (For an inductor we have voltage=ldi/dt, so v>0 if di/dt>0) The Current and Voltage phasors are perpendicular V t RI (t)

21 AC circuit 2 (AC source and inductor): V t = iωl ሚI(t) ሚI t = I 0 e iωt V t = L d ሚI dt = iωli 0e iωt = ωli 0 e i(ωt+π 2 ) = ωl ሚI t e iπ 2 Time dependences of voltage and current are out of phase (and amplitudes are different) The Current and Voltage phasors are perpendicular (For an inductor we have voltage=ldi/dt, so v>0 if di/dt>0) V t RI (t)

22 AC circuit 3 (AC source and capacitor):

23 circuit-construction-kit-ac_en

24 AC circuit 3 (AC source and capacitor): Time dependences of voltage and current are out of phase (and amplitudes are different)

25 AC circuit 3 (AC source and capacitor): Time dependences of voltage and current are out of phase (and amplitudes are different) (For a capacitor we have voltage=q/c=>dv/dt=i/c, so dv/dt>0 if I>0)

26 AC circuit 3 (AC source and capacitor): Time dependences of voltage and current are out of phase (and amplitudes are different) The Current and Voltage phasors are perpendicular (For a capacitor we have voltage=q/c=>dv/dt=i/c, so dv/dt>0 if I>0)

27 AC circuit 3 (AC source and capacitor): V t = i 1 ωc ሚ I(t) Time dependences of voltage and current are out of phase (and amplitudes are different) The Current and Voltage phasors are perpendicular (For a capacitor we have voltage=q/c=>dv/dt=i/c, so dv/dt>0 if I>0)

28 AC circuit 3 (AC source and capacitor): V t = i 1 ωc ሚ I(t) ሚI t = I 0 e iωt V t = 1 C න ሚI t = i 1 ωc I 0e iωt Time dependences of voltage and current are out of phase (and amplitudes are different) = 1 I ωc 0e i(ωt π 2 ) = 1 I ሚ t e iπ 2 ωc (For a capacitor we have voltage=q/c=>dv/dt=i/c, so dv/dt>0 if I>0) The Current and Voltage phasors are perpendicular

29 Definitions: R=resistive reactance ωl X L inductive reactance 1 ωc X C capacitive reactance

30 What are the units of reactance? A. Volts B. Amperes C. Ohms D. Joule E. Watt

31 Phasors and complex numbers: Basic operations phasor representation: Addition: z = z 1 + z 2 = (x 1 +iy 1 ) + (x 2 +iy 2 ) = x 1 + x 2 + i(y 1 + iy 2 ) Imaginary axis y = y 1 + y 2 z 2 z 1 z = re iθ = r cos θ + i r sin θ x = x 1 + x 2 Real axis x iy

32 AC circuit 2 (AC source and inductor): The Current and Voltage phasors are perpendicular

33 AC circuit 4 (AC source L+R+C): RLC circuit

34 circuit-construction-kit-ac_en

35 AC circuit 4 (AC source L+R+C): RLC circuit

36 AC circuit 4 (AC source L+R+C): phasor representation: Imaginary axis ωl R RLC circuit 1 ωc Real axis

37 AC circuit 4 (AC source L+R+C): phasor representation: Imaginary axis ωl R RLC circuit 1 ωc Real axis

38 AC circuit 4 (AC source L+R+C): phasor representation: Imaginary axis ωl R φ RLC circuit tan φ = ωl 1 ωc R 1 Real axis ωc

39 AC circuit 4 (AC source L+R+C): Z = R 2 + (ωl 1 ωc )2 tan φ = ωl 1 ωc R φ phase difference between current and voltage Z = R + iωl i 1 ωc Impedance = Voltage ሚCurrent RLC circuit

40 AC circuit 4 (AC source L+R+C): Z = R 2 + (ωl 1 ωc )2 tan φ = ωl 1 ωc R φ phase difference between current and voltage Z = R + iωl i 1 ωc Impedance = Voltage ሚCurrent RLC circuit Impedance is complex

41 AC circuit 4 (AC source L+R+C): Z = R 2 + (ωl 1 ωc )2 tan φ = ωl 1 ωc R φ phase difference between current and voltage Z = R + iωl i 1 ωc Impedance = Voltage ሚCurrent RLC circuit resistance is real part of Z

42 AC circuit 4 (AC source L+R+C): Z = R 2 + (ωl 1 ωc )2 tan φ = ωl 1 ωc R φ phase difference between current and voltage Z = R + i(ωl 1 ωc ) Impedance = Voltage ሚCurrent RLC circuit reactance is imaginary part of Z

43 AC circuit 4 (AC source L+R+C): Z = R 2 + (ωl 1 ωc )2 tan φ = ωl 1 ωc R φ phase difference between current and voltage Z = R + i(ωl 1 ωc ) Impedance = Voltage ሚCurrent RLC circuit Inductive reactance

44 AC circuit 4 (AC source L+R+C): Z = R 2 + (ωl 1 ωc )2 tan φ = ωl 1 ωc R φ phase difference between current and voltage Z = R + i(ωl 1 ωc ) Impedance = Voltage ሚCurrent RLC circuit capacitive reactance

45 AC circuit 4 (AC source L+R+C): Z = R 2 + (ωl 1 tan φ = ωl 1 ωc R ωc )2 Impedance = Voltage amplitude Current amplitude φ phase difference between current and voltage RLC circuit I t = I 0 cos ωt V t = Z I 0 cos ωt + φ

46 AC circuit 4 (AC source L+R+C): I 0 φ ω Z I 0 I t = I 0 cos ωt RLC circuit V t = Z I 0 cos ωt + φ

47 AC circuit 4 (AC source L+R+C): I t = I 0 cos ωt V t = Z I 0 cos ωt + φ Power: P t = I t V t = I 0 cos ωt Z I 0 cos ωt + φ = Z I 0 2 cos ωt cos ωt + φ = Z I 0 2 cos ωt [cos ωt cos φ + sin ωt sin φ ] Average power: തP = 1 T න 0 T P t dt = 1 2 Z I 0 2 cos φ RLC circuit = Z I rms 2 cos φ

48 Some examples (of RLC circuits):

49 What happens at high frequencies in a R circuit with an AC voltage source? A. The AC current does not depend on frequency B. The AC current will go to zero at high frequencies C. The AC current will increase at high frequencies

50 What happens at high frequencies in a L circuit with an AC voltage source? A. The AC current does not depend on frequency B. The AC current will go to zero at high frequencies C. The AC current will increase at high frequencies

51 Is a low pass filter ( Z = ωl)

52 What happens at high frequencies in a C circuit with an AC voltage source? A. The AC current does not depend on frequency B. The AC current will go to zero at high frequencies C. The AC current will increase at high frequencies

53 Is a high pass filter ( Z = 1 ωc )

Handout 11: AC circuit. AC generator

Handout 11: AC circuit. AC generator Handout : AC circuit AC generator Figure compares the voltage across the directcurrent (DC) generator and that across the alternatingcurrent (AC) generator For DC generator, the voltage is constant For