SINUSOIDAL WAVEFORMS
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1 SINUSOIDAL WAVEFORMS The sinusoidal waveform is he only waveform whose shape is no affeced by he response characerisics of R, L, and C elemens. Enzo Paerno
![CIRCUIT ELEMENTS R [ Ω ] Resisance: Ω: Ohms Georg Simon Ohm Capaciance: C [ F ]](/docs-images/72/66643818/images/2-5.jpg "+ F: Farads Michael Faraday Inducance: L [ H ] L: Henries Joseph Henry Enzo")
2 CIRCUIT ELEMENTS R [ Ω ] Resisance: Ω: Ohms Georg Simon Ohm Capaciance: C [ F ] + F: Farads Michael Faraday Inducance: L [ H ] L: Henries Joseph Henry Enzo Paerno
3 SINUSOIDAL FUNCTION πf ; π rad 360 ; π 3.45;rad A Sine wave A θ 0 T f π v( Asin( + θ Asin(πf + θ Asin( + θ T A peak ampliude [vols] f frequency [cycles/s, herz] angular frequency [rad/s] T period [sec] θ phase [radians or degrees] ime [sec] ac source Alernaing Curren Creaes an alernaing (+, -, varying volage Enzo Paerno 3
4 SINE WAVE GEOMETRY hp:// JSP-Apple: Sine Wave Enzo Paerno 4
5 SINE WAVE Degrees Vs Radians r r r C π 360 πr π A radian is defined by a quadran of a circle where he disance subended on he circumference equals he radius of he circle 360 π 3π π radians π degrees radians rad 57.3 Enzo Paerno 5
6 SINE WAVE Degrees Vs Radians π π 3π π π Radians x Degrees x π degrees radians degrees radians Enzo Paerno 6
7 SINE WAVE Angular Velociy The velociy wih which he radius vecor roaes abou he cener, called he angular velociy, can be deermined by: Angular Velociy α α π π f T α angle disance[ radians ] ime[s] f, T Noe : π f α Enzo Paerno 7
8 General Forma for I & V SINUSOIDALS A 0 A Asin v( α sinα i( α I p V p A sinα Peak ampliude, α uni of v( p V p sin ac measure for he horizonal axis α π πf T sinα i( I sin ac curren α volage α sin sin v V Enzo Paerno i I p p
9 RELATIVE PHASES OF SINUSOIDALS A θ 0 f( 0 Asin Asin θ phase 0 (no verical axis shif ( ± θ phase θ ( θ shif from verical axis +v Asin ( +θ Shif Lef 0 α 0 θ Shif Righ Asin ( θ α Shif wave lef -v Shif wave righ Enzo Paerno 9
10 RELATIVE PHASES OF SINUSOIDALS v v a b ( ( 0sin( 0sin( va and vb have same frequency bu differen phases: 8 6 v b v a vb 60 º 0 va 4 - v b leads v a by 60º (leads in ime -4-6 leads lags v a lags v b by 60º (lags in ime º Enzo Paerno 0
11 RELATIVE PHASES OF SINUSOIDALS +cos leads +sin by 90º +sin lags +cos by 90º - sine leads +cos by 90º - cos lags +sin by 90º -sin +cos -cos CCW: + angles +sin CW: - angles cos sin sin cos sin( cos( sin cos sin( cos( sin cos( sin( cos sin( cos( ± 80 ± 80 Enzo Paerno Odd Funcion Even Funcion
12 RELATIVE PHASES OF SINUSOIDALS 90 º 0 cos sin( cos leads +sin by 90º 90 +cos CCW: + angles 0 -sin +sin -cos CW: - angles Enzo Paerno
13 Oscilloscope Frequency Measuremens A T T H sensiiviy no. of H div ; f div T Enzo Paerno 3
14 Oscilloscope Phase Measuremens Noe: possible answers i.e. θº or 360º θº Ɵ e leads i by 44º Enzo Paerno 4
15 Oscilloscope Phase Measuremens Phase and Time Measuremens Coninued Phase and ime measuremens are relaed as shown by he equaion: θ 360 T The ime from he sar of a waveform o a given phase angle can be found using: θ T 360 Enzo Paerno 5
16 Oscilloscope Phase Measuremens Enzo Paerno 6
17 RELATIVE PHASES OF SINUSOIDALS v( i( 0 sin( sin( Example: + 70 Noe ha v( and i( are boh sin, hus: i leads v by 40º v lags i by 40º 40 v( i( Example: 0 sin( 0 5sin( Noe ha v( and i( are boh sin, hus: i leads v by 80º v lags i by 80º Enzo Paerno 7
18 PHASOR REPRESENTATION Whenever an arbirary phase angle, θ, is associaed wih a sine or cosine funcion of ampliude A, we can skech his funcion using a phasor diagram. We skech a phasor of lengh A a an angle θ degrees wih respec o he appropriae axis. f ( Asin( + θ f ( A θ Phasor Diagram A θ We only skech he magniude and phase. We have in effec frozen he wave in ime. If we desire o plo sine and cosine funcions ono he same phasor diagram, we eiher conver he sine funcions o is cosine funcion equivalen or conver he cosine funcions o is sine funcion equivalen. Enzo Paerno 8
19 RELATIVE PHASES OF SINUSOIDALS Example: v( 3sin( i( cos( º Relaive o cos 00º Relaive o sin i( cos( + 0 sin( + 00 OR v( 3sin( 0 3cos( 00 00º i leads v by 0º v lags i by 0º Enzo Paerno 9 3 0º
20 i( RELATIVE PHASES OF SINUSOIDALS Example: v( sin( + 0 sin( º -50º Alhough, one can use ± 80, use -80 because he resuling angle gives: θ 80 i( sin( + 30 sin( 50 v leads i by 60º i lags v by 60º Enzo Paerno 0
21 We know: +cos -sin PHASOR REPRESENTATION Example: Express e 5 cos ( + 30º in erms of he sine funcion. We know: Thus: +sin cos sin( cos( sin( + 0 e 30º 5 Relaive o cos 0º Relaive o sin -cos Enzo Paerno
22 PHASOR REPRESENTATION Example: Deermine he phase of e wih respec o e when e 5 sin ( - 30º and e -0 sin ( - 40º We know: Thus: We know: +cos -sin -cos +sin sin sin( ± 80 e 0 sin( 40 0 sin( + 40 e 0 40º 5 40º 30º e Δθ 70º Enzo Paerno
23 f ADDITION OF SINUSOIDS WITH SAME FREQUENCY ( A sin( + θ + A sin( + θ + + A n sin( + θn f ( A θ + A θ + + Sinusoidal form f f means same velociy Conver o Phasor form A n θ n A sin θ j A θ A cos θ f ( f j ( A ( A cosθ + sinθ + Conver o Recangular form A A cosθ sinθ ( Asin( + θ A A Conver o Sinusoidal form n n cosθ n sinθ n + Enzo Paerno 3
24 ADDITION OF SINUSOIDS Example: Deermine e( e( + e( + e3( for e 0 sin ( º, e -0 cos (500-45º, e3 30 cos ( º e 0 sin ( º, e 0 sin (500-35º, e3 30 sin ( º We know: +cos -sin +sin e e e 3 ( 0 30 ( ( Sinusoidal form Conver o Phasor form j5 j5.98 j4.4 Conver o Recangular form -cos e( e( e( sin( cos( Conver o Sinusoidal form Enzo Paerno 4
25 AVERAGE VALUE dc VALUE Wha do we mean by he average value? For example we wan he average heigh of he mound of sand: Heigh Mound of sand Compac Sand in he Sandbox Average Heigh Compaced sand Heigh d If d is increased: d Mound of sand d Average Heigh decreases Compaced sand d Enzo Paerno 5
26 AVERAGE VALUE dc VALUE Wha do we mean by he average value? If a depression exiss: Heigh Mound of sand Average Heigh decreases even furher Compaced sand d Average Area Average Heigh x d If we knew he mahemaical equaion for he curve represening he original mound of sand: Average Area Area under he curve Toal disance Enzo Paerno 6 d
27 AVERAGE VALUE dc VALUE The average value (i.e. dc value of a periodic funcion f( is defined as; Average value and is given by: Area under he curve for cycle Period F T T 0 f ( dc d If f( is defined differenly over various regions of he cycle, hen he Average value will be several inegrals, one for each region Remark: Average value Alegbraic sum of areas for cycle Period A rue dc volmeer or ammeer reads he average value of a periodic Waveform when se o dc measuremen. Enzo Paerno 7
28 AVERAGE VALUE dc VALUE Example 30: 8 I( [A] Find he average value of he curren waveform: T8 F dc [( 5 x + ( 8 x + ( -4 x ] / 8.75 A Enzo Paerno 8
29 AVERAGE VALUE dc VALUE 0 sin π Example 3: π Find he average value of he sine wave: F F F F dc dc dc dc 0 π π sin d π 0 cos π (cos π cos0 π ( (0 π π 0 Enzo Paerno 9
30 If he ac source is a sinusoidal signal: P P ac ac ( i I ac EFFECTIVE / RMS VALUES Quesion: Wha is he equivalen dc (average power o an ac power for a paricular load R when an ac source is applied? This equivalen dc power is called he effecive power or he rms power (Roo Mean Square. mr R ( I I m sin R mr cos v ac V I mr I mr Pav ( ac cos m sin I m i ac 0 (since average power I sin ( cos R sin ( cos mr Enzo Paerno 30 I m
31 EFFECTIVE / RMS VALUES I m I R I m dc We wan: I I dc I m dc R I P av( ac m I eff Anoher word, he equivalen dc value of a sinusoidal curren or volage is of is maximum value. This equivalen dc value is called he effecive value of he sinusoidal quaniy. I P dc rms Enzo Paerno 3
32 EFFECTIVE / RMS VALUES The effecive value of any quaniy ploed as a funcion of ime, f(, can be found using he formula: RMS F rms Square he funcion Find he Mean (average 3 Take he square roo Remark: T 0 f ( d A rue rms volmeer or ammeer reads he effecive value of a periodic waveform when se o ac measuremen. T The average power due o a periodic (ac source is: P P E R R Enzo Paerno 3 av av I rms rms
33 EFFECTIVE / RMS VALUES Example: 8 I( [A] Find he rms value of he curren waveform: T (5x + (64x 8 + (6x Irms A Enzo Paerno 33
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