Exercises with Maple

Calculate Amplitude and Phase of a Harmonic Oscillation

Write a Maple Procedure that calculates numerically the resulting amplitude and phase of a sum of sin-oscillations,

which all have the same frequency but possibly different amplitudes and phases, i.e., calculate A and ,

so that for a given number N and given   it holds:

(w is a fixed angular frequency, t a time parameter).

A Solution:

The array a[1] .. a[N] contains the amplitudes of the given superposition,

the array phi[1]..phi[N ] the corresponding phase shifts. The procedure  result

gives the resulting amplitude A and the corresponding phase  of the given

superposition. We calculate the resulting phase in three different ways.
Explain the different phase results in the examples

>

restart:

>	result:=proc(a,phi)

>	local N::integer,s1,s2; global Amplitude,Phase;

>	N:=linalg[vectdim](a);

>	s1:=evalf(sum(a[k]*cos(phi[k]),k=1..N)):print('s1'=evalf(s1));

>	s2:=evalf(sum(a[k]*sin(phi[k]),k=1..N)):print('s2'=evalf(s2));

>	Amplitude:=evalf(sqrt(s1^2+s2^2)):print('Amplitude'=Amplitude);

>	#Now you can calculate the phase using the arccos

>	#We first consider the cases of phase 0 and phase Pi,

>	#then the other cases

>	if Amplitude=0 or s1=Amplitude then Phase:=0 elif s1=-Amplitude then Phase:=Pi

>	else Phase:=evalf(signum(s2)*arccos(s1/Amplitude))

>

end if:

>	print('Phase'=Phase);

>	# alternatively you can use the arctan using the theorem: arccos(x)=P/2-arctan(x/sqrt(1-x^2))

>	if Amplitude=0 or s1=Amplitude then Phase:=0 elif s1=-Amplitude then Phase:=Pi

>	else Phase:=evalf(signum(s2)*(Pi/2-arctan(s1/sqrt(Amplitude^2-s1^2))))

>	end if:  print('Phase'=Phase);

>	# 3rd formula again using the arctan

>	if Amplitude=0 then Phase:=0 elif s1=0 then Phase:=signum(s2)*Pi/2

>	elif s1>0 then Phase:=evalf(arctan(s2/s1))

>	elif s2>=0 then Phase:= evalf(arctan(s2/s1)+Pi)

>	else Phase:= evalf(arctan(s2/s1)-Pi)

>

end if:

>	print('Phase'=Phase);

>

end;

  At first,   test the cases A=0, phi=0, phi=Pi, Phi=Pi/2, phi=-Pi/2 :

>	a:=vector([1,-1]);phi:=vector([1,1]); #given amplitudes and phases in the superposition

>	result(a,phi);                 #Amplitude and Phase of the resulting sine wave

>

>	a:=vector([1,1]);phi:=vector([0,0]); #given amplitudes and phases in the superposition

>	result(a,phi);                 #Amplitude and Phase of the resulting sine wave

>	a:=vector([1,4]);phi:=vector([Pi,Pi]); #given amplitudes and phases in the superposition

>	result(a,phi);                 #Amplitude and Phase of the resulting sine wave

>

>	a:=vector([1,1]);phi:=vector([Pi/2,Pi/2]); #given amplitudes and phases in the superposition

>	result(a,phi);                 #Amplitude and Phase of the resulting sine wave

>

>	a:=vector([1,1]);phi:=vector([-Pi/2,-Pi/2]); #given amplitudes and phases in the superposition

>	result(a,phi);                 #Amplitude and Phase of the resulting sine wave

  Then test other examples

  Example 1:   We calculate the resulting amplitude and phase for

               2*sin(w*t)+2.5*cos(w*t+Pi/3)-4*sin(w*t-Pi/4).

To apply the above procedure we note that  cos(w*t+Pi/3) = sin(w*t+5*Pi/6) .

>	a:=vector([2,2.5,-4]);phi:=vector([0,5*Pi/6,-Pi/4]);

>	result(a,phi);

>	Amplitude;Phase;

 Example 2:   We calculate the resulting amplitude and phase for

                       cos(w*t)+sqrt(3)*cos(w*t+Pi/2).

>	b:=vector([1,sqrt(3)]);psi:=vector([Pi/2,Pi]);

>	result(b,psi);

  Example 3:     We calculate the resulting amplitude and phase for

                        sin(w*t+Pi/4)+3*sin(w*t-3*Pi/4).

>	c:=vector([1,3]);alpha:=vector([Pi/4,-3*Pi/4]);

>	result(c,alpha);

>