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The Bode plots are shown in Fig.
3
and from these it is found
that the phase margin is 20" approximately and the gain
30r
Fig.
3 Loopfrequency response plots
a
Without compensator
h
With compensator
crossover frequency is 0.68 rad/s. Suppose a minimum phase
margin of
45"
is required without detriment to the gain cross
over frequency.
The procedure outlined above is applied as follows:
(i) From Fig.
3,
the frequency at which
I
Guw)
I
=

3
dB is
0.836rad/s. This will be the new gain crossover frequency.
Therefore
T
=
1/0.836
=
1.2s
(ii) At 0.836rad/s,
~ww)
is found from Fig.
3
to be

170".
The amount of phase lead required is therefore
35".
(iii)
a
is determined using eqn.
11.
a
=
tan
(45"

35")
=
tan IOc
=
0.176
Thus the compensator transfer function is
1
+
1.2s
G,(s)
=

1
+
0.21s
The compensated loop frequency response is shown in Fig.
3
where it is seen that the phase margin is now
46"
and the gain
crossover frequency is 0.826 rad/s.
Conclusion:
A method has been proposed for phase lead com
pensator design for loops with varying gain and phase fre
quency response characteristics. It does not require an
iterative process and has been seen to work effectively in a
chosen example. It has also been tested successfully on a
variety of other systems having similar varying gain and
varying phase frequency response characteristics, in many
cases yielding nearoptimal design values for
T
and
U.
Even in
the case of the constant phase characteristic, a working design
will be obtained but it will not be optimal. The method always
yields a predictable gain crossover frequency.
An obvious limitation of the method is the maximum
amount of phase lead angle available. Theoretically this is
45"
for
a
=
0. In practice, it is probably about
40"
corresponding
to a minimum value for
U
of
0.09.
Where the amount of phase
lead required is greater than
40".
Two identical lead networks
can then be used in cascade and the method can readily be
extended to the design of these.
In cases where the phase varies rapidly with frequency, as
with nonminimum phase systems, e.g., those containing time
delay elements, lead compensation itself is not effective and
the proposed method will obviously not apply.
E. P. McCARTHY
Department of Electronic and Electrical Engineering
University of Salford
Salford M5 4WT, United Kingdom
6th September 1990
References
1
2
3
KUO, B.
r.:
'Automatic control systems' (PrenticeHall Inc., 1987)
~ORF.
R.
r.:
'Modern control systems' (Addison Wesley, 1974)
n'uzo.
J.
J.,
HOUPIS,
c.
H.:
'Linear control system analysis and
design' (McGrawHill. 1988)
HIGHLYLINEAR TRANSCONDUCTOR CELL
REALISED BY DOUBLE MOS TRANSISTOR
DIFFERENTIAL PAIRS
Indexing terms
'
Transconductors, Circuit theory and design.
MOSFETC filters. Tunina
A
novel scheme for tunable transconductance cells with high
linearity
is
described. The proposed configuration is realised
by double MOS transistor ditlerential pairs with two inde
pendent linear outputs, which can
be
employed separately as
well as in a parallel or crosscoupled connection.
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