Summary Week 9
 
1. Review of Exam 2
-Objective: to obtain dynamic relationship between the input and the output of a system using
    sinusoidal input ( black box modeling ) of different frequencies.

-Motivation:

  • The system may be too complex to model.
  • The frequency domain provides a powerful stability criterion and tools for robust design and filter designs.
    • - Experiment:
    1. Introduce a sinusoidal input :
    u = A sin (w t) if w is in rad/sec
    or
    u = A sin(2pwt) if w is in cycles/sec or Hz
    2. Observe the output variable after it attains periodic behavior, i.e.
        y = B sin(w t + f) if w is in rad/sec
             or
             y = B sin(2pwt + f) if w is in cycles/sec or Hz
    3. Perform the experiment for a range of w values and record the corresponding values of amplitude, B and time shift, tshift and calculate
    a) magnitude ratio,  MR = B/A
    b) phase shift,  f =  - (tshift/P)*360o  (degrees)  =  - (tshift/P)*2p  (radians)  = - tshift*w(radians)
             ( Note:   P=1/w if w is in cycles/sec or P=2p/w if w is in rads/sec )
           - Representations:
    a) Bode Plots:
    (1) Log Modulus : LM (in decibels)    vs  w ( in log10 scale)
    (2) Phase Shift:  f (in degrees)   vs  w (  in log10 scale)
                           where LM = log modulus = 20 log10 (B/A)
    b) Nyquist Plots
    ReG = B/A cos(f) in x-axis
    ImG = B/A sin(f) in y-axis
    4. Short Review of Complex Numbers
     
    - Polar Form:

        Let      z = Re(z) + i Im(z)     then    z = |z| exp( i arg(z) ),

        where

        a)  |z| = magnitude of z = (Re(z)2 + Im(z)2 )1/2
        b)  arg(z) = Arctangent( Im(z)/Re(z) )
        c) Re(z) = |z| cos( arg(z) )
        d) Im(z) = |z| sin( arg(z) )

    - Properties and Functions of Complex numbers:

        a) Conjugate Function:  conj(  a + i b ) = a - i b
        b) Re(z) = (1/2)(z + conj(z) )
        c) Im(z) = (1/2i)(z - conj(z) )
        d) Functions of complex numbers yield complex numbers
        e) ( to be continued next week ) Function of the conjugate is the conjugate of the function,
            i.e.  Let G(a+ib) = ReG + i ImG,  then  G(a-ib) = ReG - i ImG