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and where

φ
˜
˜
˜
V = A e j tot . = V + V (6.68)
tot . tot . 1 2
Preparatory Exercise

Pb. 6.35 Write the analytical expression for A tot. and φ in Eq. (6.68) as func-
tot.
tions of the amplitudes and phases of signals 1 and 2.

The above result can, of course, be generalized to the sum of many signals;
specifically:

N
) =
V = A cos(ω t + φ tot. ∑ A cos(ω t + φ )
tot. tot. n n
n=1
(6.69)
 N   N 
= Re ∑ A exp( j t +ω jφ ) = Re e jt ∑ A e j n 
φ
ω
 n n   n 
n  =1   n=1 
and

N
˜
=
V ˜ . ∑ V (6.70)
tot n
n=1
⇒ A = V ˜ (6.71)
tot. tot.

˜
φ = arg( V ) (6.72)
tot. tot.
That is, the resultant field can be obtained through the simple operation of
adding all the complex numbers (phasors) that represent each of the individ-
ual signals.

Example 6.8
φ
Given ten signals, the phasor of each of the form Ae j n , where the ampli-
n
1
tude and phase for each have the functional forms A = and φ = n , write
2
n n
n
a MATLAB program to compute the resultant sum phasor.
© 2001 by CRC Press LLC