enee204Lectures_05_06_Gomez

enee204Lectures_05_06_Gomez - Welcome to ENEE 204: Basic...

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1 9/13/2004 1 Welcome to ENEE 204: Basic Circuit Theory Lecture 5 Chapter 3: AC signals in steady state (HW2, in front of room please) - AC quantities - Sinusoidal (Harmonic functions) - Complex numbers - Phasors - Solving circuits using Phasors 9/13/2004 2 AC sources in general • It is used in practically ALL forms of electrical devices. AC circuits WORK very BROAD range of operating frequencies. Specific applications require specific frequencies: Electrical power distribution - 60 Hz Communications (RF 10kHz - 100MHz) Microwaves 100MHz - 100GHz • AC – alternating current, are voltages and currents that vary in a PERIODIC manner with TIME • It is represented by sinusoidal function of the forms: ) sin( ) ( ) cos( ) ( φ ω + = + = t A t g t A t f 9/13/2004 3 Understanding sinusoidal functions For example: v (t) = V m cos ( t + v ) Amplitude V m Phase v Parameters describing sinusoids: 1. amplitude 2. period 3. radial frequency 4. phase period = Τ T Radial frequency T ω = v =0 v <0 9/13/2004 4 AC Quantities + = T t t RMS o o dt t f T f 2 ) ( 1 v (t) = V m cos ( t + v ) Peak Voltage, aka V p Voltage Phase Root mean square of f(t), f RMS i (t) = I m cos ( t + I ) Peak Current, aka I p Current Phase Example: 3 V 4 / 0 , 4 ) ( T/4 at peak wave Triangular RMS m V T t t T V t v = < < =
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2 9/13/2004 5 A -A -A A -A A Physical interpretation of periodic oscillation T = period, time for each cycle A = amplitude 9/13/2004 6 Physical interpretation of periodic oscillation T = period, time for each cycle A = amplitude A -A -A A -A A 9/13/2004 7 Physical interpretation of periodic oscillation T = period, time for each cycle A = amplitude t t T ω ϑ π = = ) ( 2 Can be regarded as a rotation with constant angular frequency: A -A -A A -A A 9/13/2004 8 Physical interpretation of ‘periodic oscillation’ ) sin( : example t A y = time +A +A +A -A -A -A Note: could obtain cos( θ ) if we “ measured θ from y-axis”. In other words, cos( )=sin( θ+π/2 ) y example of “phase shift”
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3 9/13/2004 9 Variation in amplitude Variation in frequency small ω, low frequency large ω , high frequency small V m large V m Physical interpretation of periodic oscillation v(t) = V m cos ( t + φ v ) Example: NOTE: Obvious that DC (direct current) is a special case of AC with ω=0 . 9/13/2004 10 Variation of phase Understand: range of phase: +/- π/2 Common terms: ‘quarter cycle’: φ=π/2 ; ‘half cycle’: φ=π ; ‘full cycle’: φ=2π Physical interpretation of periodic oscillation v(t) = V m cos ( t + v ) Example: v(t) = V m cos ( t + 0 ) v = 0 ‘in-phase’ v(t) = V m cos ( t + v ) v > 0 ‘phase is ahead’ v(t) = V m cos ( t + v ) v < 0 ‘phase is lagging’ 9/13/2004 11 Understanding Phase π /2 : sine function is the same as cosine function shifted to the right by quarter of a cycle cos ( t) sin ( t) t = 2 T -sin( t) sin ( t)=cos ( t- /2) -sin ( ω t)=cos ( t + /2) cos ( t)=sin ( t+ /2) -cos ( t)=sin ( t- /2) 9/13/2004 12 Review elementary trigonometry* (harmonic functions) cos( π/ 2)= 0 sin( /2) = 1 cos( ) = -1 sin( ) = 0 cos(3 2) = 0 sin(3 /2) = -1 cos(0) = cos(2 ) = 1 sin(0)= sin(2 ) = 0 θ 0,2 π/2 3π/2 sin ( Α + Β ) = sinA cosB + cosA sinB cos ( ) = cosA cosB – sinA sinB sin ( Α − Β ) = sinA cosB – cosA sinB cos ( ) = cosA cosB + sinA sinB Remember (memorize) trigonometric identities: ) sin( ) 2 sin( ) sin( ) 2 cos( ) cos( ) 2 cos( t t t t = + = Example: Show that cos( t- /2)=sin( t)
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This note was uploaded on 04/07/2008 for the course ENEE 204 taught by Professor Gomez during the Fall '04 term at Maryland.

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enee204Lectures_05_06_Gomez - Welcome to ENEE 204: Basic...

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