Robert_W_Brown;_Yu-Chung_N_Cheng;_E_Mark_Haacke;(z-lib.org).pdf - Adobe ReaderFile Edit View Window Help(77 of 978)Fill & SignToolsCommentОpen43100%Sign InExport PDF3.2. The Rotating Frame for an RF Field43Create PDFEquation (3.21) displays a constant term plus two oscillating terms whose effects averageto zero over times that are half multiples of the rf period or, more relevantly, for all timeslarge compared to the rf period. The definition for an average over a time interval T is givenbyEdit PDFAdobe Acrobat ProEasily edit text and images in PDFdocuments(3.22)< f(t) >dt f (t)Start NowWith the details left as aproblem for the reader, the primed-frame time averaged value of(3.21) isSend Files1primedStore Files(3.23)for constant binProblem 3.2f (t)Show that the average (3.22) over time interval T for f(t) = sin (nwt)cos (nwt) for any positive integer n is zero whenora) T 2mT/w for any positive integeror T =mT/w for even n)or when the following limit is takenb) Т> 2п/wThe reduced amplitude bn in (3.23) is noteworthy. It implies that only half of theoriginal linearly polarized field (3.15) amplitude is available in the rotating reference framefor tipping the spin. Also, (3.23) is still valid for a time dependent bn(t), provided it changesonly over time scales much larger than the rf period 2T/w. It has been noted earlier thatLlin(.

Question
Asked Sep 26, 2019
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Problem 3.2
Show that the average (3.22) over time interval T for f(t) = sin (nωt) or f(t) =
cos (nωt) for any positive integer n is zero when
a) T = 2mπ/ω for any positive integer m (or T = mπ/ω for even n)
or when the following limit is taken
b) T ≫ 2π/ω

 

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3.2. The Rotating Frame for an RF Field
43
Create PDF
Equation (3.21) displays a constant term plus two oscillating terms whose effects average
to zero over times that are half multiples of the rf period or, more relevantly, for all times
large compared to the rf period. The definition for an average over a time interval T is given
by
Edit PDF
Adobe Acrobat Pro
Easily edit text and images in PDF
documents
(3.22)
< f(t) >
dt f (t)
Start Now
With the details left as a
problem for the reader, the primed-frame time averaged value of
(3.21) is
Send Files
1
primed
Store Files
(3.23)
for constant bin
Problem 3.2
f (t)
Show that the average (3.22) over time interval T for f(t) = sin (nwt)
cos (nwt) for any positive integer n is zero when
or
a) T 2mT/w for any positive integer
or T =
mT/w for even n)
or when the following limit is taken
b) Т> 2п/w
The reduced amplitude bn in (3.23) is noteworthy. It implies that only half of the
original linearly polarized field (3.15) amplitude is available in the rotating reference frame
for tipping the spin. Also, (3.23) is still valid for a time dependent bn(t), provided it changes
only over time scales much larger than the rf period 2T/w. It has been noted earlier that
Llin(.
help_outline

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Robert_W_Brown;_Yu-Chung_N_Cheng;_E_Mark_Haacke;(z-lib.org).pdf - Adobe Reader File Edit View Window Help (77 of 978) Fill & Sign Tools Comment Оpen 43 100% Sign In Export PDF 3.2. The Rotating Frame for an RF Field 43 Create PDF Equation (3.21) displays a constant term plus two oscillating terms whose effects average to zero over times that are half multiples of the rf period or, more relevantly, for all times large compared to the rf period. The definition for an average over a time interval T is given by Edit PDF Adobe Acrobat Pro Easily edit text and images in PDF documents (3.22) < f(t) > dt f (t) Start Now With the details left as a problem for the reader, the primed-frame time averaged value of (3.21) is Send Files 1 primed Store Files (3.23) for constant bin Problem 3.2 f (t) Show that the average (3.22) over time interval T for f(t) = sin (nwt) cos (nwt) for any positive integer n is zero when or a) T 2mT/w for any positive integer or T = mT/w for even n) or when the following limit is taken b) Т> 2п/w The reduced amplitude bn in (3.23) is noteworthy. It implies that only half of the original linearly polarized field (3.15) amplitude is available in the rotating reference frame for tipping the spin. Also, (3.23) is still valid for a time dependent bn(t), provided it changes only over time scales much larger than the rf period 2T/w. It has been noted earlier that Llin(.

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Step 1

Formula for average of any function f(t) is shown below

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-(1) ()dt Javg T=T ime period of fuction

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Step 2

For part (a) of the given problem,  function and time period is shown below

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f(t)= sin (not )or cos (not 2тя 2тл for evenn time period T or T

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Step 3

 plugging sinusoidal function(sine term) and time period in...

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2mr sin (not)dt lavg 2m7 2mr 1 cos no t - avg 2mn7 2πηmη -1 1 1 (2) COs avg 2πηη although 2mn multiplying integerof 2n hence, cos 27mn =1 plugging value of cos2umnin equation(2) -1-1)] avg 2πmn =0 Javg

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