Phys 345 Electronics for Scientists - Delaware Physics
Announcements Assignment 0 due now. solutions posted later today Assignment 1 posted, due Thursday Sept 22nd Question from last lecture: Does VTH=INRTH Yes! Lecture 5 Overview Alternating Current AC Components. AC circuit analysis Alternating Current pure DC
V pure direct current = DC Direction of charge flow (current) always the same and constant. pulsating DC V pulsating DC Direction of charge flow always the same but variable AC = Alternating Current pulsating DC V
V Direction of Charge flow alternates -V AC 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 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(t+) f=1/T (cycles/s) =2f (rad/s) =2(t/T) rad/s =360(t/T) deg/s 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)
I rms Ip 2 , Vrms Vp 2 , Pave I rmsVrms i-V relationships in AC circuits: Resistors Source vs(t)=Asint vR(t)= vs(t)=Asint iR (t )
vR (t ) A sin t R R vR(t) and iR(t) are in phase Complex Number Review Phasor representation 2 2 i-V relationships in AC circuits: Resistors Source vs(t)=Asint vR(t)= vs(t)=Asint
iR (t ) vR (t ) A sin t R R vR(t) and iR(t) are in phase Complex representation: vS(t)=Asint=Acos(t-90)=real part of [VS(j)] where VS(j)= A[cos(t-90)-jsin(t-90 )]=Aej (t-90) Phasor representation: VS(j) =A(t-90) IS(j)=(A/R) (t-90) Impedance=complex number of Resistance Z=VS(j)/IS(j)=R Generalized Ohm's Law: VS(j)=ZIS(j) Capacitors What is a capacitor? Definition of Capacitance: C=q/V Capacitance measured in Farads (usually pico - micro)
Energy stored in a Capacitor = CV2 (Energy is stored as an electric field) In Parallel: V=V1=V2=V3 q=q1+q2+q3 q q1 q2 q3 Ceq C1 C2 C3 V V i.e. like resistors in series Capacitors In Series: V=V1+V2+V3 q=q1=q2=q3 1 V V1 V2 V3 1
1 1 Ceq q q C1 C2 C3 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 Capacitors in AC circuits Capacitive Load
vC A sin t qC CvC dqC CA cos(t ) dt VC ( j ) A(t 90) iC I C ( j ) CA(t 0) ZC VC ( j ) 1 90 C I C ( j ) cos( 90) j sin( 90) j
j j. j 1 C jC jC "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 Inductors What is an inductor? Definition of Inductance: vL(t)=-LdI/dt Measured in Henrys (usually milli- micro-) Energy stored in an inductor: WL= 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)=-d/dt) 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). Inductors in AC circuits Inductive Load
vS A sin t vL L diL dt A sin t L (back emf ) diL dt from KVL A A iL sin tdt cos t L L
A A iL sin(t 90) cos(t 180) L L VL ( j ) A(t 90) A (t 180) L V ( j ) ZL L L90 I L ( j ) cos(90) j sin(90) j Z L j L I L ( j ) 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 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
Example ( Z R1 Z C ) I1 ( j ) Z C I 2 ( j ) VS ( j ) Z C I1 ( j ) ( Z C Z L Z R 2 ) I 2 ( j ) 0 VS ( j ) ZC 0 ZC Z L Z R2 ( Z C Z L Z R 2 )VS ( j ) I 1 ( j ) 2 Z R1 Z C ZC ( Z R1 Z C )( Z C Z L Z R 2 ) Z C
ZC ZC Z L Z R2 1 1 66.7 66.7 j () 6 jC j1500 10 j Z L jL j1500 0.5 750 j () ZC (75 683 j )150 I 1 ( j ) (100 66.7 j )(75 683 j ) 4450 (75 683 j )150 I 1 ( j )
(100 66.7 j )(75 683 j ) 4450 Top: Bottom: (75 683 j )150 68783.7 150 tan 1 b 683 tan 1 83.7 a 75 A a 2 b 2 687 (100 66.7 j )(75 683 j ) 4450 7500 45600 5000 j 683 j 4450 57550 63300 j 8550047.8
Computational PhysicsNumerical Differentiation. Dr. Guy Tel-Zur. Clouds. Picture by Peter Griffin, publicdomainpictures.net. MHJ Chapter 3: Numerical Differentiation. Should be f'c "2" stands for two points. forward/backward 1st derivative: ±h . f(x)=a+bx^2.
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