Electrical engineering control stream

1. ECEg535:-Instrumentation Eng’g
By Sintayehu Challa 1
DYNAMIC CHARACTERISTICS OF INSTRUMENTS
The static characteristics of any electrical instrument are described for relatively long
periods of time, i.e. after the transient responses have settled down, and the steady state
conditions have been attained. The basic static features like zero drift and linearity
considerations are specified to provide accuracy measures with appropriate error limits.
Furthermore, during the very short time interval in which a measured signal is applied to the
input terminals of a given instrument, and the different units or sections have been energized
to respond to step-like or fast periodic signals with unknown amplitudes and time variations,
the instrument is said to have specified dynamic characteristics. Thus, without being
restricted to a single instrument, or to groups of similar instruments, it will be very useful to
treat the dynamic performance characteristics of instruments in the most general terms.
As in so many other areas of studies in electrical engineering (e.g applications of circuit
and control theory), the most widely used mathematical model for a study of the dynamic
response of a measurement system is the ordinary linear differential equation with
constant coefficients: that is to say, resort has to be made to differential equations with
some reasonable assumptions and approximations. We can assume that the relation
between any particular input (desired, interfering, or modifying), and the output can, by
application of suitable simplifying assumptions, be put in the form
i
i
m
i
m
mm
i
m
mn
n
nn
n
n qb
dt
dq
b
dt
qd
b
dt
qd
bqa
dt
dq
a
dt
q
da
dt
qd
a 011
1
100
0
11
01
1
0
...... ++++=++++ −
−
−−
−
−
where qo, qi and t are output quantity, input quantity (i.e. a sample of the quantity under
measurement) and time, respectively. The combinations of system physical parameters,
a's, b's, are all assumed constant. The terms dn
qo/dt n
and d m
q i/dt m
are the n th
and m th
time derivatives of qo and qi, respectively. These can be replaced with the D operator as
D n
q o and D m
q i with the operator D = d/dt. We can thus write the output-input
relationship as
i
n
m
m
m
n
nn
n
n qbDDDbDbqaDaDaDa )...()...( 01
1
1001
1
1 ++++=++++ −
−
−
−
Similarly, it should be noted that the differential equation can be changed into an
algebraic equation using Laplace Transform techniques.
The complete solution will be
qo = qoc + qop
Where
qoc = complementary-function part of the solution.
qop = particular-integral part of the solution.
The solution qoc is obtained from the characteristic equation
anDn
+ qn-1 Dn-1
+ . . . . + a1D + qo = 0
we can then obtain the transfer function
01
1
1
01
1
1
)(
)(
)(
asasasa
bsbsbsb
sQ
sQ
sH n
n
n
n
m
m
m
m
i
O
++−−−++
++−−−++
== −
−
−
−

2. ECEg535:-Instrumentation Eng’g
By Sintayehu Challa 2
By taking mathematical models and by classifying instruments into three broad
categories, namely:
(a) Zero-order instruments
(b) First-order instruments
(c) Second-order instruments
As followed in the study of responses of liner electrical circuits, it is possible to gain a
broad understanding of the responses of the above types of instruments to three standard
signals: the step, impulse and ramp signals.
Zero-Order Instrument
The zero-order instrument is described by
aoqo = boqi
from which one obtains
0
0
0
0
0
a
b
Kwhere
Kqq
a
b
q ii
=
==
be defined as static sensitivity.
With qo = Kqi, it is clear that, no matter how qi might vary with time, the instrument
output (reading) follows it perfectly with no distortion or time lag of any sort. Thus a
zero-order instrument represents an ideal or perfect dynamic performance as illustrated in
Fig. 2-22(a). If the response is not ideal, the general non-linear response illustrated in
Fig. 2-22 (b) is to be expected.

3. ECEg535:-Instrumentation Eng’g
By Sintayehu Challa 3
It should however be noted that xi(t) actually varies non-linearly in Figs. 2-22(a) and(b).
The readings xi(t) and hence eo(t) will be varying non-linearly with time for two basic
reasons:
(i) The assumption that a potentiometer is a pure resistance is not correct, and
(ii) A pure resistance is a mathematical model, because a real resistance has inductive
and capacitive components, although a potentiometer is normally called a zero-
order instrument.
Still a zero-order response can be approached by an instrument if
(a) Reactive effects, i.e. due to inductive and capacitive effects can be
ignored,
(b) The speeds ("frequencies") of time variation to be measured are not high
enough to make the inductive or capacitive effects noticeable.
In a practical situation, it is clear that the presence of mechanical loading effects must
also be taken into account. By mechanical loading effect is simply meant here the energy
loss due to the friction between sliding and the potentiometer. Note also that this effect is
different in kind from the inductive and capacitive phenomena mentioned earlier since the
affected relation is
L
E
Kwhere
KXE
L
X
e i
i
=
==0
and whereas the mechanical loading has no effect on this relation, but makes xi different
from the undisturbed case.

4. ECEg535:-Instrumentation Eng’g
By Sintayehu Challa 4
First-order Instrument
For such an instrument, all a's are made equal to zero except a1 and ao, and thus we can
set
iqbqa
dt
dq
a 000
0
1 =+
Step Response of First Order Instruments
As shown in Fig. 2-23, one can set the operational transfer function as
0
0
0)1(
a
b
KwhereKqqD i ==+τ
static sensitivity and ς denotes time constant.
We thus obtain the operational transfer function
1
0
+
=
D
K
q
q
i τ
where K = bo/ao is the static senstivity and τ denotes time constant. Initially, it is
assumed that with qi = 0, also qo = 0. At time t = 0+
, the input quantity increases instantly
by an amount qi as shown in Fig. 2-23(a). As illustrated in Fig. 2-23(b), it can then be
easily noted that
(i) the complementary-function solutionis
τ
t
oc eCq
−
=
and
(ii) the particular solution is qop = Kqis giving
is
t
KqeCq +=
−
τ
0
Using the initial conditions qo(0+
) = 0, then
τ
t
is eKqq
−
−= 1(0
To generalize the above results, we can finally put
τ
t
is
e
Kq
q −
−= 10
which is illustrated as a universal step response curve for a first order instrument, as
shown in Fig. 2-24.
Suppose we also define a measurement error em by

5. ECEg535:-Instrumentation Eng’g
By Sintayehu Challa 5
τ
ττ
t
is
m
t
is
t
isisim
e
q
e
thatfollowsthereforeIt
eqeqq
K
q
qe
−
−−
=
=−−=−= )1(0
The step response of the first-order instrument leads to two dynamic characteristics useful
in characterizing the speed of response of any instrument. These are:
(i) The rise-time, tr = 2.22τ , which is the time that it takes for the response to rise
from 10% to its maximum value; and
The settling time ts which is defined as the time (after application of a s step input) for
the instrument to reach and stay within a stated range tolerance found around its
final value.
A small numerical value of ts is indicatives of fast response. As an example, it can be
shown that for a 5 percent settling time, ts = 3τ because qo/qis = 0.95 when t/τ = 3
as shown in Fig. 2-25.

6. ECEg535:-Instrumentation Eng’g
By Sintayehu Challa 6
Ramp Response of First-Order Instruments
Another useful signal for testing the response of a first-order instrument is the ramp
signal which has a constant slope qis as illustrated in Fig. 2-26(a).
The ramp response of a first-order instrument can again be described by
(a.D + qo) qo = boqist
or
(τD + 1) qo = Kqist
where τ = a1/ao is the time-constant, and K=bo/ao is the static sensitivity. A ramp input
signal as illustrated in Fig. 2-26(a) is simply a straight line waveform of slope qis. In
terms of the performances and characteristics of practical instruments one can here cite
analog instruments with damped oscillations , the cathode-ray oscilloscope and graphical
recorders .
Applying zero initial conditions, it follows that
)(0 ττ τ
−+=
−
teKqq
t
is
from which we can also set a measurement error em defined by
τ
ττ
t
isisim eqq
K
q
qe
−
−=−= 0
The measurement error has thus transient and steady state components with a lag equal to
qisς It can be seen that this error disappears more quickly if ς is small, but increases if qis
increases.
Further, we can define a non-dimensional error function

7. ECEg535:-Instrumentation Eng’g
By Sintayehu Challa 7
τ
t
ssm
m
e
e
e −
−= 1
,
which again can be seen to approach zero at t = 0+
, but steadily rises with increasing t as
shown in (Fig. 2-26b).
Frequency Response of First-Order Instruments
Next to the step and ramp test signals, the sinusoidal signal is of prime importance for
testing the steady state response of a measuring instrument. With most practical
instruments, such signals are either pre-conditioned or rectified into d-c signals (i.e. step
signals) before activating the display mechanisms. However, there are still some
instruments (e.g. electromechanical and electronic meters, CROs, graphical recorders,
etc.) whose frequency responses can be evaluated by applying sinusoidal signals directly.
Suppose qi is a sinusoidal signal input. Then qo must also be a sinusoidal output signal,
and it can be shown that for a first order instrument, the transfer function is given by
1)(
)(
1)(
)(
)(
2
0
+
=
+
==
τω
ω
τωω
ω
ω
K
jH
and
j
K
jq
jq
jH
i
For a zero-order instrument, the ideal sinusoidal response would be
00
0
)(
)(
)( ∠== K
jq
jq
jH
i ω
ω
ω
Thus a first-order instrument approaches perfection if its response approaches that of a
zero-order instrument. The best example of this approximation is again the response of
the cathode ray oscilloscope. This can occur if the product ωτ is sufficiently small, and
hence for any τ there will be some input frequency ω below which measurement is
accurate. Alternatively, if a qi of high frequency ω must be measured, the instrument
must have a sufficiently small τ
Second-order Instruments
A second-order instrument is defined as one that satisfies the equation
iqbqa
dt
dqa
dt
qda
000
01
2
0
2
2
=++
where a 2, a 1, a o, and b o are again assumed to be constants as determined by the internal
components of the instrument. Dividing by ao, one obtains
iq
a
b
q
dt
dq
a
a
dt
qd
a
a
0
0
0
0
0
1
2
0
2
0
2
=++
We define the following parameters:

8. ECEg535:-Instrumentation Eng’g
By Sintayehu Challa 8
),dim.(,
2
,
10
1
2
0
0
0
ensionlesseiratiodamping
aa
a
andfrequencynatural
a
a
ysensitivitstatic
a
b
K
n
==
==
==
ς
ω
to obtain
1
2
)(
2
2
0
++
=
nn
i DD
K
D
q
q
ω
ε
ω
For a step input of amplitude kqis, and with the initial conditions
+
=== 000
0 tfor
dt
dg
q
it can be shown that there are three possible solutions based on the roots of the
characteristic equation. These are grouped as follows:
Case (1) : ς > 1 (overdamped response),
tt
is
nn
ee
Kq
q ωξξωξξ
ξ
ξξ
ξ
ξξ )1(
2
2
)1(
2
2
0
22
12
1(
12
1( −−−−+−
−
−−
+
−
−+
−=
Case (ii) : ς = 1 (critically damped response),
10
)1( +−
+−= t
n
is
n
et
Kq
q ω
ω
Case (iii) : ς < 1 (underdamped or oscillatory
response),
)1(sin
1)1sin(
1
21
2
2
0
ςφ
φως
ς
ως
−=
++−
−
=
−
−
where
t
e
Kq
q
n
t
is
n
The graphs of the above responses are shown plotted against ωnt Fig. 2-29. The
overdamped response (ς = 1.5) is seen to rise very slowly towards the normalized steady
state value, and hence it has a long settling time. For ς = 1.0 (critical damping), there is
no oscillation in the response, but the settling time is still relatively long. When values of
ς approach zero, the oscillatory components of the response are clearly very significant,
and correspondingly the settling times become very long again. In standard instruments,
values of ς can range from about 0.6 to 0.7 give the shortest settling times. The optimum
values of ς are thus typically chosen between 0.6 and 0.7. Within this optimum range,
most common instruments have found to provide reasonably accurate responses over
wide frequency ranges.

9. ECEg535:-Instrumentation Eng’g
By Sintayehu Challa 9