1. 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!

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

3. 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 pure DC pulsating DC AC V V V V -V

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 P L =I 2 R L : 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 v s (t)=Asinωt v R (t)= v s (t)=Asinωt v R (t) and i R (t) are in phase

8. Complex Number Review 2 2 Phasor representation

9. i-V relationships in AC circuits: Resistors Source v s (t)=Asinωt v R (t)= v s (t)=Asinωt v R (t) and i R (t) are in phase Complex representation: v S (t)=Asinωt=Acos(ωt-90)=real part of [V S (j ω) ] where V S (j ω)= A[cos(ωt-90)-jsin(ωt-90 )]=Ae j (ωt-90) Phasor representation: V S (jω) =A  ( ωt -90) I S (jω)=(A/R)  ( ωt -90) Impedance=complex number of Resistance Z=V S (jω)/I S (jω)=R Generalized Ohm's Law: V S (jω)=ZI S (jω) http://arapaho.nsuok.edu/%7Ebradfiel/p1215/fendt/phe/accircuit.htm

10. Capacitors In Parallel: V=V 1 =V 2 =V 3 q=q 1 +q 2 +q 3 What is a capacitor? Definition of Capacitance: C=q/V Capacitance measured in Farads (usually pico - micro) Energy stored in a Capacitor = ½CV 2 (Energy is stored as an electric field) i.e. like resistors in series

11. Capacitors In Series: V=V 1 +V 2 +V 3 q=q 1 =q 2 =q 3 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) i.e. like resistors in parallel

12. Capacitors in AC circuits Capacitive Load 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 "capacitive reactance"

13. Inductors What is an inductor? Definition of Inductance: v L (t)=-LdI/dt Measured in Henrys (usually milli- micro-) Energy stored in an inductor: W L = ½ Li L 2 (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. Inductors in AC circuits Voltage and current not in phase: Current lags voltage by 90 degrees Impedance of Inductor increases with increasing frequency Inductive Load http://arapaho.nsuok.edu/%7Ebradfiel/p1215/fendt/phe/accircuit.htm (back emf ) from KVL

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

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