disp(' '); disp(' '); disp('Example 1: PFE with distinct real roots');
Example 1: PFE with distinct real roots
This case will consider only distinct real roots.
Define numerator and denominator polynomial.
n=[1 1]; %n=s+1 d=conv([1 0],[1 2]); %Use "conv" to multiply polynomial disp(['Numerator = ' poly2str(n,'s')]); disp(['Denominator = ' poly2str(d,'s')]);
Numerator = s + 1 Denominator = s^2 + 2 s
Now use "residue" command to do inverse transform. r = magnitude of expansion term p = location of pole of each term k = constnat term (k=0 except when numerator and denominator are same order (m=n)).
[r,p,k]=residue(n,d)
r =
0.5000
0.5000
p =
-2
0
k =
[]
Note that the function is implicitly defined only for t>0. Some texts show the time domain function multiplied by the unit step. We will keep our expressions simpler by making that relationship implicit.
- -
disp(' '); disp(' '); disp('Example 2: PFE with repeated real roots');
Example 2: PFE with repeated real roots
Define numerator and denominator polynomial.
n=[1 0 1]; %n=s^2+1 d=conv([1 0 0],[1 2]); %Use "conv" to multiply polynomial disp(['Numerator = ' poly2str(n,'s')]); disp(['Denominator = ' poly2str(d,'s')]); [r,p,k]=residue(n,d)
Numerator = s^2 + 1
Denominator = s^3 + 2 s^2
r =
1.2500
-0.2500
0.5000
p =
-2
0
0
k =
[]
Note, second order pole (s^2) comes second in list.
- -
disp(' '); disp(' '); disp('Example 3: PFE with complex conjugate roots.');
Example 3: PFE with complex conjugate roots.
We also did this example in class.
Define numerator and denominator polynomial.
n=[1 0 1]; %n=s^2+1 d=conv([1 0 0],[1 2 5]); %Use "conv" to multiply polynomial disp(['Numerator = ' poly2str(n,'s')]); disp(['Denominator = ' poly2str(d,'s')]); [r,p,k]=residue(n,d)
Numerator = s^2 + 1
Denominator = s^4 + 2 s^3 + 5 s^2
r =
0.0400 - 0.2200i
0.0400 + 0.2200i
-0.0800
0.2000
p =
-1.0000 + 2.0000i
-1.0000 - 2.0000i
0
0
k =
[]
Note, first two roots are complex conjugate roots.
Get magnitude and phase from magnitude and phase of pfe Refer to notes from last class
M=2*abs(r(1)) %Magnitude of cosine phi=angle(r(1)) %Phase of cosine
M =
0.4472
phi =
-1.3909
Get frequency and decay rate from location of pole
omega=imag(p(1)) alpha=-real(p(1))
omega =
2
alpha =
1
Plot the response
t=0:0.1:4; f=M*exp(-alpha*t).*cos(omega*t+phi) + r(3) + r(4)*t; plot(t,f); xlabel('Time'); ylabel('f(t)');
- -
disp(' '); disp(' '); disp('Example 4: PFE: order of num=order of den');
Example 4: PFE: order of num=order of den
We also did this example in class.
Define numerator and denominator polynomial.
n=[3 2 3]; %n=3s^2+2s + 3 d=[1 3 2]; %d=s^2+3s+2=(s+1)(s+2) disp(['Numerator = ' poly2str(n,'s')]); disp(['Denominator = ' poly2str(d,'s')]); [r,p,k]=residue(n,d)
Numerator = 3 s^2 + 2 s + 3
Denominator = s^2 + 3 s + 2
r =
-11
4
p =
-2
-1
k =
3
Note this time that k is not empty, k=3.
