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switches at input values of ± A to output levels of ± 1, then
is increased for gain-guided modes.
Acknowledgment: T. Li contributed to this letter with stimulating discussions and helpful suggestions.
D. MARCUSE
and
14th September 1982
Bell Laboratories
Crawford Hill Laboratory
Holmdel, NJ 07733, USA
Y(z) = l - z - 1 + z - 2 - z -
l[Z)
PATZAK, E.: 'Comment on "Spontaneous emission factor of
narrow-stripe gain-guided diode lasers"', Electron. Lett., 1982, 18,
pp. 278-279
KRESSEL, H., and BUTLER, J. K.: 'Semiconductor lasers and heterojunction LEDs' (Academic Press, New York, 1977)
MARCUSE, D.: 'Computer model of an injection laser amplifier',
IEEE J. Quantum Electron., to be published
MARCUSE, D.: 'Light transmission optics' (Van Nostrand Reinhold,
New York, 2nd edn., 1982), pp. 275-282
PETERMANN, K.: 'Calculated spontaneous emission factor for
double-heterostructure injection lasers with gain-induced waveguiding', IEEE J. Quantum Electron., 1979, QE-15, p. 566
2
3
4
5
6
+ ... = l/(l
Substituting these values in eqn. 1, taking
References
1
3
~
1 + b 1 z ~ 1 +b2z~2
+ ...
and balancing the terms in the steady state gives
(B, - B2)A = A2-A1
(2)
where
A
\ = aO + a2 + fl4 + •••
A2 = ax + a3 + as + . . .
STREIFER, w., SCIFRES, D. R., and BURNHAM, R. D.: 'Spontaneous
emission factor of narrow-stripe gain-guided diode lasers', Electron. Lett., 1981,17, pp. 933-934
MARCUSE, D. : 'Principles of quantum electronics' (Academic Press,
New York, 1980), p. 179
7
and
B2 = bx + b3 + b5 + ...
Thus, for example, if G(s) = K(l - s)/s(l + s),
0013-5194/82/210920-03SL50/0
'Mi
K[jT -2 + 2g- 7 >- 1 + (2 - 2e'T +e~Tz
so that
Al=2-2e-T
LIMIT CYCLES IN RELAY SYSTEMS
-Te~T
A2 = T-2 + 2e~T
Indexing terms: Control theory, Limit cycles, Relay systems, z
transforms
A method using the z transform is presented for evaluating
the frequency of limit cycles in relay systems. The technique
as described is only applicable when the relay has no dead
zone and the plant transfer function no time delay. An extension of the approach to these systems is possible but requires
use of the modified z transform; this makes the calculations
more cumbersome and the method loses its simplicity.
As in the well known methods of Hamel and Tsypkin1"3 the
starting point of the analysis is to assume that the limit cycle
output from the relay of Fig. la is a square wave of period Tp.
One then notes that if two fictitious synchronous samplers
with sampling period T = Tp/2 and a zero order hold are
introduced into Fig. la, as shown in Fig. Ib, the behaviour of
the two systems will be identical when the samplers operate at
the relay switching instants. Now if X(z) and Y{z) are the z
transforms of the sampled signals, then
X(z) + Y(z)G1(z) = 0
B2=
- l - e ~
T
Substituting these values in eqn. 2 gives
7/(1 + e~T) - 4(1 - e~T) = (1 + e~T)2A/K
from which for given values of A and K the limit cycle frequency w = n/T can be found. In particular, if A = 0 the
equation reduces to
7/4 = tanh (T/2)
relay
Gfe)
relay
[936/31
(1)
Fig. 2 Two-variable systems
where
Gt(z) = (1 - z"1)5f'[G(s)/s].
relay
Assuming
the
relay
In the two variable system shown in Fig. 2 it is known that
when G(s) is symmetrical limit cycles may exist where the relay
output square waves are either in phase or antiphase.4 The
frequencies of these limit cycles can again be calculated by the
above procedure. For G{s) symmetrical:
Gfe)
Gl2is)
i
- ®-^-
relay
Gfe)
(3)
Gxl(s)
and the transfer function feeding the relay output signals back
to both relay inputs is
Fig. 1 Relay systems and sampled equivalent
922
ELECTRONICS LETTERS 14th October 1982
Vol. 18 No. 21
RESOLUTION OF MONOCHROMATIC
SIGNAL SOURCES USING A SIMPLE
RECEIVING ANTENNA ARRAY
for an in-phase oscillation, and
for an antiphase oscillation.
Thus for the method presented the required expressions for
Gx(z) in eqn. 1 are:
Indexing terms: Antennas, Antenna arrays
A simple method to isolate the amplitude and the direction
of m incident rays on a linear antenna array of 2m elements
is described. This method differs from other similar proposals by its ease of hardware implementation at microwave
frequency.
for a n in-phase oscillation and
1
G 1 (z) = ( l - z " ) .
for a n antiphase oscillation.
Taking as a n example
1
fl-s
5(1 + 5)
1-5
we have
(1-z-1)
1
(T-2
+ (2 - 2e~T -
-(1
Te~T)z
Methods of resolving the relative strength and the direction of
arrival of incident rays on an antenna array have been investigated by Moody 1 and Hackett. 2 These methods rely on the
determination of voltages induced in each individual antenna
element. However, at microwave frequencies, hardware implementation of such systems can be very difficult, and often
an elaborate frequency conversion scheme is employed to
enable the signal processing to be done at a more manageable
intermediate frequency.
To make the processing of signal at microwave frequencies
more practicable, the receiving antenna system described by
Leavitt 3 is an attractive option. It requires only a 2-bit phase
shifter for each receiving antenna element and a single power
measuring device which is readily available commercially. A
block diagram of this receiving antenna system is shown in
Fig. 1.
e 'z
and
signal source
processor
{\ - e~T — (1 + e
T
)z
l
+ e Tz~
giving
T -2 + 2e~T + fi(T + e
z" 2 [2 - 2e~T - Te~T +
- 1)]
- e~T -
for an in-phase oscillation, and
7
[T - 2 + 2e~~TT + n(l - e ~T - T e " ) ]
- 1)]
z~ 2 [2 - 2e~T - Te~T
power
detector
Fig. 1 Block diagram of receiving antenna system
As shown by Leavitt, the relative phase i/^n of the induced
signals at the nth antenna elements can be evaluated by
making only eight separate power measurements for the first
element and six other power measurements for each of the
subsequent elements of the antenna array.
The phase setting 0,j for the ith phase shifter during the ;th
power measurement for the determination of \j/n is:
for an antiphase oscillation.
Using these values in eqn. 2 for ideal relays with A = 0 gives
<pij = 0
T(l+e-T)-4(l-e-T)
= ± //[T(l +e~T)-
2(1 - e " r ) ]
for half the oscillation period T, for the in-phase and antiphase oscillations, respectively.
D. P. ATHERTON
School of Engineering & Applied Sciences
The University of Sussex
Faimer, Brighton, Sussex BN1 9QT, England
j =2
-n/2
j =3
TT/2
; =4
<t>i,j+A = <f>i,j
13th September 1982
7 = 1, 2, 3, 4
ij/n can be expressed in terms of the power measurements a s
follows:
- P(<D7) =
GILLE, J. G., PELEGRIN, M. }., and DECAULNE, p.: 'Feedback control
systems' (McGraw-Hill, New York, 1959), pp. 446-483
2
n
and
References
1
for ; = 1 o r i• £ «
GELB, A., and VANDER VELDE, W. E.: 'Multiple-input describing
functions and nonlinear system design' (McGraw-Hill, New York,
1968), pp. 185-199
3 ATHERTON, D. P.: 'Nonlinear control engineering' (Van Nostrand
Reinhold, Wokingham, 1981, Student edn.), Chap. 6
4 ATHERTON, D. p.: 'Oscillations in relay systems', Trans. Inst. Meas.
& Control, 1981, 3, pp. 171-184
where
It follows that the amplitude of the induced signal a t t h e nth
a n t e n n a element can also be expressed as
0013-5194/82/210922-02$!. 50/0
ELECTRONICS LETTERS 14th October 1982
(P(<D3)
Vol. 18 No. 21
- P((D7) - P(O 4 )) 2 } 1 / 4 C
923
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