APPENDIX
D
TABLE OF TWO-SINUSOID-INPUT
DESCRIBING FUNCTIONS (TSIDFs)
The TSIDF can be represented by either of the following integral expressions (cf. Sec. 5.1)
where J, and J, are the Bessel functions of orders 0 and 1, respectively. Use of this table
is facilitated by application of the relationship
NA ( A3 ) = NB(B,A)
All entries are for the case of nonharmonically related sinusoids, corresponding to the above
integral formulations.
TABLE O F T W O - S I N U S O I D - I N P U T DESCRIBING F U N C T I O N S (TSIDFs) (Continued)
Nonlinearity
Comments
,Fl is the gaussian
hypergeometric function.
See Gibson and Sridhar,
Ref. 10 of Chap. 5 .
!
!
n 2
(
m cos 2m sin- J1 -
gj
(-lyr(m+n)
.=O
n ! tm
-
n ) !U n
+ 1) ( 'Br 2 F l [ - n ,
- n ; 2;
]I:(
or, in integral form,
3. Relay with dead zone
See Fig. D.l
K(k) and E(k) are the elliptic
integrals of first and second
kind, respectively.
~7r2BB
! ( . ( 3 - [ l - ( z ] ] K ( $ ) ]
B
for A <I
4. Ideal relay
See Fig. D.2 and Sec. 5.1
See Gibson and Sridhar
(op. cit.)
7. Limiter
See Fig. D.3
20
-
(7
sin
sin-.
n6 m=o
or, in integral form,
S
++
(-1)"Um
n)
n ! (rn - n ) ! r ( n 1 ) ("A[-.,
B
- n ; 2;
(f)]
560
TWO-SINUSOID-INPUT
DESCRIBING F U N C T I O N S
Normalized amplitude B / S
Figure D.1 Normalized relay with dead zone TSIDF. (Gibson and Sridhar,
Ref. I0 of Chap. 5.)
TWO-SINUSOID-INPUT
0
1
2
3
4
DESCRIBING F U N C T I O N S
5
Amplitude B
Figure 0 . 2 Nornza/ized a - r e TSZDF.
6
7
561
8
562
TWO-SINUSOID-INPUT
O
. I
2
DESCRIBING F U N C T I O N S
3
4
5
6
7
8
Normalized amplitude B / 6
Figure 0 . 3 Normalized limiter TSZDF. (Gibson and Sridhar, Ref. 10 of Chap.
TWO-SINUSOID-INPUT
0
1
2
3
DESCRIBING F U N C T I O N S
4
5
Amplitude B
Figure D.4
Cubic characrerisric TSIDF.
6
7
563
8
564
TWO-SINUSOID-INPUT
Figure D.5
DESCRIBING F U N C T I O N S
Normalized harmonic nonlinearity TSZDF.