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Announcements

Announcements. Assignment 0 due now. solutions posted later today Assignment 1 posted, due Thursday Sept 22 nd Question from last lecture: Does V TH =I N R TH Yes!. Lecture 5 Overview. Alternating Current AC Components. AC circuit analysis. pure DC. V. pulsating DC. V.

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Announcements

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  1. Announcements • Assignment 0 due now. • solutions posted later today • Assignment 1 posted, • due Thursday Sept 22nd • Question from last lecture: • Does VTH=INRTH • Yes!

  2. Lecture 5 Overview • Alternating Current • AC Components. • AC circuit analysis

  3. pure DC V pulsating DC V pulsating DC V AC V -V Alternating Current • pure direct current = DC • Direction of charge flow (current) always the same and constant. • pulsating DC • Direction of charge flow always the same but variable • AC = Alternating Current • Direction of Charge flow alternates

  4. Why use AC? The "War of the Currents" • Late 1880's: Westinghouse backed AC, developed by Tesla, Edison backed DC (despite Tesla's advice). Edison killed an elephant (with AC) to prove his point. • http://www.youtube.com/watch?v=RkBU3aYsf0Q • Turning point when Westinghouse won the contract for the Chicago Worlds fair • Westinghouse was right • PL=I2RL: Lowest transmission loss uses High Voltages and Low Currents • With DC, difficult to transform high voltage to more practical low voltage efficiently • AC transformers are simple and extremely efficient - see later. • Nowadays, distribute electricity at up to 765 kV

  5. AC circuits: Sinusoidal waves • Fundamental wave form • Fourier Theorem: Can construct any other wave form (e.g. square wave) by adding sinusoids of different frequencies • x(t)=Acos(ωt+) • f=1/T (cycles/s) • ω=2πf (rad/s) •  =2π(Δt/T) rad/s •  =360(Δt/T) deg/s

  6. RMS quantities in AC circuits • What's the best way to describe the strength of a varying AC signal? • Average = 0; Peak=+/- • Sometimes use peak-to-peak • Usually use Root-mean-square (RMS) • (DVM measures this)

  7. i-V relationships in AC circuits: Resistors Source vs(t)=Asinωt vR(t)= vs(t)=Asinωt vR(t) and iR(t) are in phase

  8. 2 2 Complex Number Review Phasor representation

  9. i-V relationships in AC circuits: Resistors Source vs(t)=Asinωt vR(t)= vs(t)=Asinωt vR(t) and iR(t) are in phase Complex representation: vS(t)=Asinωt=Acos(ωt-90)=real part of [VS(jω)] whereVS(jω)= A[cos(ωt-90)-jsin(ωt-90 )]=Aej (ωt-90) Phasor representation: VS(jω) =A(ωt-90) IS(jω)=(A/R) (ωt-90) Impedance=complex number of ResistanceZ=VS(jω)/IS(jω)=R Generalized Ohm's Law: VS(jω)=ZIS(jω) http://arapaho.nsuok.edu/%7Ebradfiel/p1215/fendt/phe/accircuit.htm

  10. Capacitors What is a capacitor? Definition of Capacitance: C=q/V Capacitance measured in Farads (usually pico - micro) Energy stored in a Capacitor = ½CV2 (Energy is stored as an electric field) In Parallel: V=V1=V2=V3 q=q1+q2+q3 i.e. like resistors in series

  11. Capacitors In Series: V=V1+V2+V3 q=q1=q2=q3 i.e. like resistors in parallel No current flows through a capacitor In AC circuits charge build-up/discharge mimics a current flow. A Capacitor in a DC circuit acts like a break (open circuit)

  12. Capacitors in AC circuits Capacitive Load "capacitive reactance" • Voltage and current not in phase: • Current leads voltage by 90 degrees (Physical - current must conduct charge to capacitor plates in order to raise the voltage) • Impedance of Capacitor decreases with increasing frequency http://arapaho.nsuok.edu/%7Ebradfiel/p1215/fendt/phe/accircuit.htm

  13. Inductors What is an inductor? Definition of Inductance: vL(t)=-LdI/dt Measured in Henrys (usually milli- micro-) Energy stored in an inductor: WL= ½ LiL2(t) (Energy is stored as a magnetic field) • Current through coil produces magnetic flux • Changing current results in changing magnetic flux • Changing magnetic flux induces a voltage (Faraday's Law v(t)=-dΦ/dt)

  14. Inductors Inductances in series add: Inductances in parallel combine like resistors in parallel (almost never done because of magnetic coupling) An inductor in a DC circuit behaves like a short (a wire).

  15. Inductive Load Inductors in AC circuits (back emf ) from KVL • Voltage and current not in phase: • Current lags voltage by 90 degrees • Impedance of Inductor increases with increasing frequency http://arapaho.nsuok.edu/%7Ebradfiel/p1215/fendt/phe/accircuit.htm

  16. AC circuit analysis • Effective impedance: example • Procedure to solve a problem • Identify the sinusoid and note the frequency • Convert the source(s) to complex/phasor form • Represent each circuit element by it's AC impedance • Solve the resulting phasor circuit using standard circuit solving tools (KVL,KCL,Mesh etc.) • Convert the complex/phasor form answer to its time domain equivalent

  17. Example

  18. Top: Bottom:

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