basic concepts signal waveforms. continuous/discontinuous
TRANSCRIPT
BASIC CONCEPTS
•Signal Waveforms
Continuous/Discontinuous
SINUSOIDS
tXtx M sin)(
(radians)argument
(rads/sec)frequency angular
valuemaximumor amplitude
t
XM
Adimensional plot As function of time
tTtxtxT ),()(2
Period
)(cycle/sec Hertz infrequency
2
1
Tf
f 2
"by leads"
"by lags"
BASIC TRIGONOMETRY
)cos(2
1)cos(
2
1coscos
)sin(2
1)sin(
2
1cossin
sinsincoscos)cos(
sincoscossin)sin(
IDENTITIES DERIVED SOME
cos)cos(
sin)sin(
sinsincoscos)cos(
sincoscossin)sin(
IDENTITIES ESSENTIAL
)sin(sin
)cos(cos
)2
cos(sin
)2
sin(cos
tt
tt
tt
tt
NSAPPLICATIO
)90sin()2
sin( tt
CONVENTION EE ACCEPTED
RADIANS AND DEGREES
(degrees) 180
(rads)
degrees 360 radians
2
ktA
ktA
dttA
tAtAdt
da
tAa
)2
sin()cos()sin(
)2
sin()cos(
)sin(
Integral and Derivative
)45cos( t
)cos( t
Leads by 45 degrees
)45cos( t)18045cos( t
Leads by 225 or lags by 135
)36045cos( t
Lags by 315
Example
Example
VOLTAGESBETWEEN ANGLEPHASE ANDFREQUENCY FIND
)301000cos(6)(),601000sin(12)( 21 ttvttv
Frequency in radians per second is the factor of the time variable1sec1000 HzHzf 2.159
2)(
To find phase angle we must express both sinusoids using the sametrigonometric function; either sine or cosine with positive amplitude
)180301000cos(6)301000cos(6 tt
)180cos()cos( with sign minus of care take
)90sin()cos( with cosine into sine Change
)902101000sin(6)2101000cos(6 tt
We like to have the phase shifts less than 180 in absolute value)601000sin(6)3001000sin(6 tt
)601000sin(6)(
)601000sin(12)(
2
1
ttv
ttv 120)601000()601000( tt120by leads 21 vv
120)601000()601000( tt120by lags 12 vv
Using the unit step to build a pulse function
Unit step
)()()( tutftf functions time positiveFor
Using square pulses to approximatean arbitrary function
Pulse width = T, Pulse height = 1
THE IMPULSE FUNCTION
Representation of the impulse
Height is proportionalto area
Approximationsto the impulse
(Good model for impact, lightning, and other well known phenomena)
Periodic Functions
Square Waveforms
Triangle Wavefrom
Sawtooth Waveform
...2
3TotTofor A v
Tot2
Tofor A - v
2
Tot0for A v
etc
EFFECTIVE OR RMS VALUES
)(ti
R
Rtitp )()( 2
power ousInstantane
Tt
t
Tt
tav dtti
TRdttp
TP
T
0
0
0
0
)(1
)(1 2
period with periodic iscurrent If
2
)(
dcdc
dc
RIP
Iti
then )( DC iscurrent If dcaveff PPI :
Tt
teff dtti
TI
0
0
)(1 22
Tt
teff dtti
TI
0
0
)(1 2
The effective value is the equivalent DCvalue that supplies the same average power
Definition is valid for ANY periodicsignal with period T
2
)cos()(
Meff
M
XX
tXtx
is valueeffective the
signal sinusoidal aFor
If the current is sinusoidal the averagepower is known to be
RIP Mav2
2
1
22
2
1Meff II
)cos(2
1ivMMav IVP case sinusoidalFor
)cos( iveffeffav IVP
square) mean(root rmseffective
EXAMPLE Compute the rms value of the voltage waveform
One period
32)2(4
210
104
)(
tt
t
tt
tv
3T
3
2
21
0
2
0
2 ))2(4()4()( dttdttdttvT
The two integrals have the same value
3
32
3
162)(
1
0
33
0
2
tdttv
)(89.13
32
3
1VVrms
Tt
trms dttx
TX
0
0
)(1 2
)(ti
R
EXAMPLE Compute the rms value of the voltage waveform and use it todetermine the average power supplied to the
resistor
2R
)(4 sT
40;16)(2 tti
Tt
trms dttx
TX
0
0
)(1 2
)(4 AIrms
)(322 WRIP rmsav