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Chapter 31

Chapter 31. An alternator. The Great Divide: 60 Hz vs 50 Hz. w is an angular frequency. w =2 p f where f is the frequency in Hertz (Hz) In the US and the rest of the Western Hemisphere, the frequency is 60 Hz w =2 p (60)=377 s -1 or rad/s In Europe and Asia, 50 Hz or 314 s -1.

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Chapter 31

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  1. Chapter 31

  2. An alternator

  3. The Great Divide: 60 Hz vs 50 Hz • w is an angular frequency. • w=2pf where f is the frequency in Hertz (Hz) • In the US and the rest of the Western Hemisphere, the frequency is 60 Hz • w=2p(60)=377 s-1 or rad/s • In Europe and Asia, 50 Hz or 314 s-1

  4. 3 Simple Circuits-- Resistor This is sometimes called a “resistive load” on the circuit VR

  5. Phasor Diagrams • Phasor Diagrams • A method to analyze the relative phase between voltage and current • Of great concern since power maximizes when voltage and phase are in phase • A convention is that the driving frequency, wd , is measured from the +x axis and rotates counterclockwise around the origin • In the case to the right, the phase angle between VR and IR is zero.

  6. 3 Simple Circuits– Capacitive Load For symmetry reasons, we introduce “capacitive reactance”, XC VC Vmax=ICXC Capacitive reactance has units of ohms

  7. ICE—For capacitors, current leads EMF If you monitor iC, the current will reach its maximum before the voltage across the capacitor, VC

  8. 3 Simple Circuits– Inductive Load For symmetry reasons, we introduce “inductive reactance”, XL VL Vmax=ILXL Inductive reactance has units of ohms

  9. ELI—For inductors, the EMF leads current If you monitor iL, the voltage reaches its maximum value before the current in the circuit.

  10. ELI the ICEman • For resistors, current and EMF are in phase • For inductors, the EMF leads the current (ELI) • For capacitors, the current leads the EMF (ICE) • ELI the ICEman will always help you from getting killed…

  11. VR VL VR VL-VC VC RLC Circuit wdt

  12. VR VL-VC RLC Phasor From Pythagorean theorem, V

  13. Impedance, Z • The impedance of the circuit is the effective resistance • Has units of ohms • Consists of • Total resistance, R • Total capacitance, C • Total inductance, L • Frequency of the electric field, wd • V=IZ, amplitude of voltage across AC circuit

  14. Something missing? • If there is no resistor, set R=0 and solve for Z and f • If there is no inductance, set L=0 and solve for Z and f • If there is no capacitor, set C=∞, and solve for Z and f • If DC then Z=R

  15. Resonance Driving Frequency, wd Natural frequency of swing is w=1/(LC) When the driving frequency is equal to the natural frequency, then energy transfer is maximized. wd=w or XC= XL

  16. Power in AC circuits • P=i2R where i=I sin(wdt-f) • P=I2R* sin2(wdt-f) • Since P=i2R, then average power is • P=i2R/2 and if R is constant, • Irms=I/sqrt(2) • Or Vrms= V/sqrt(2) • RMS means root mean square value • Typical AC voltage values measured by voltmeter • Pav=IrmsVrms cosf • For pure resistive load, f=0 • For pure inductive load, f=+900 • For pure capacitive load, f=-900 • For RLC load, cosf=R/Z

  17. Transformers • A method of stepping up or stepping down the voltage or current in an AC circuit • A wire is wrapped around an iron core, the current in the wire induces a magnetic flux in the iron core • A second wire is wrapped around the core. The flux in the core is then induces EM in the second wire. Symbol

  18. Transformers cont’d 1 is called “Primary” and 2 is called “Secondary” The current, I1 is the current drawn by a load, R, placed on transformer which has a turns ratio of N2/N1 with primary voltage V1

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