Last lecture
Today’s menu
Loading effects:
Thévenin equivalent circuits.
Norton equivalent circuits.
Transfer functions and the Laplace transform.
First-order elements.
Second order elements.
Identification of element dynamics.
Step-responses.
Sinusoidal responses.
Two-port networks:
Generalized effort and flow variables.
Process loading.
Bilateral transducers.
Discussions about the lab.
Dynamic errors.
Dynamic error compensation.
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Thévenin equivalent circuits
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Thévenin equivalent circuits (cont’d...)
From circuit theory we know that (Thévenin’s theorem) that any network of
linear impedances and voltage sources can be replaced by an equivalent
circuit consisting of a voltage source in series with an impedance.
The Thévenin voltage, ET h is the open-circuit voltage across the
output terminals.
The Thévenin impedance, ZT h is the impedance seen at the output
terminals when all voltage sources in the networks are set to zero.
i
i
ZL
=
ETh
ZL
VL
ZTh
Linear
network
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Thévenin equivalent circuits (cont’d...)
Thévenin equivalent circuits (cont’d...)
Example
Example (cont’d...)
Doing the calculations we get:
i
Rp(1-x)
A
VS
Rpx
Load
=
RL
B
ET h
RT h
ETh
RL
VL
VL
RTh
N (x)
= Vs x
= Rp x (1 − x)
1
x (1 − x) Rp /RL + 1
Rp /RL x2 − x3
= Vs
Rp /RL x (1 − x) + 1
= Vs x
The potentiometer can be seen as a displacement sensing element and
the load can be seen as a presentation element
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Thévenin equivalent circuits (cont’d...)
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Thévenin equivalent circuits (cont’d...)
Example (cont’d...)
Some comments on the example
The output of the sensor is non-linear due to a loading effect between
the sensing element and the presenting element.
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The non-linearity can be reduced by using a loading element with
higher input impedance.
Ideal straight line
Rp/RL = 1
0.9
0.8
0.7
L
V /V
s
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
Fractional displacement, x
1
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Norton equivalent circuits
Norton equivalent circuits (cont’d...)
The Norton current, IN is the short-circuit current flowing through the
output terminals.
Norton’s theorem states that any linear network can be replaced by an
equivalent current source in parallel with an impedance.
The Norton impedance, ZN is the impedance seen at the output
terminals when all sources in the networks are set to zero.
The loading of a Norton equivalent circuit is given by
i
i
ZL
=
iN
ZN
VL = iN
ZL
ZN · ZL
.
ZN + ZL
Linear
network
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Norton equivalent circuits (cont’d...)
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Norton equivalent circuits (cont’d...)
Example
Example (cont’d...)
The transfer function of the system, including the piezoelectric crystal, a
capacitive cable, and a recorder is thus:
iN
CN
CC
G (s) =
RL
RL
VL
=
.
sRL (CN + Cc ) + 1
iN (s)
Thus the effect of electrical loading in this example is the introduction of a
first-order transfer function.
Piezoelectric
crystal
Capacitive Recorder
cable
This affects the dynamic accuracy of the force measurement system.
The piezoelectric crystal generates a current, proportional to the velocity
of a force acting on its surface.
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Two-port networks
Two-port networks (cont’d...)
Equivalent mechanical and electrical systems
kx
F
Mass
m
i
Mechanical system:
L
Spring k
kx_
Equivalent mechanical and electrical systems
F
C
V
R
Damper l
dx
d2 x
+ kx
+λ
dt2
dt
dẋ
+ λẋ + k ẋ dt.
= m
dt
= m
Electrical system:
x
x= 0
Mechanical
V =L
Electrical
1
di
+ Ri +
dt
C
i dt.
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Two-port networks (cont’d...)
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Two-port networks (cont’d...)
Equivalent mechanical and electrical systems
Comments on the analogies
We see that m corresponds to L, λ corresponds to R, and k to 1/C.
In the Laplace domain, this becomes:
Mechanical system:
This means that mechanical systems can also be replaced by Norton
or Thévenin equivalents.
k
F (s) = ẋ(s) ms + λ +
s
The same goes for other types of linear systems. Read the text book and
make sure you understand this!
Electrical system:
1
V (s) = i(s) Ls + R +
sC
By replacing for example mechanical or thermal systems by their electrical
equivalents, we can now study the loading effects when connecting these
in series.
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Two-port networks (cont’d...)
Two-port networks (cont’d...)
Equivalent mechanical and electrical systems
Example of a two-port network
This gives the two equivalent impedance transfer functions:
ZM (s) =
ZE (s) =
V (s)
i̇(s)
= Ls + R +
i
x_
k
F (s)
= ms + λ +
s
ẋ(s)
1
sC
F
ZM
iN
ZN
V
Two-port network
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Process loading
Bilateral transducers
By representing both the process and sensor, and other system
elements as Thévenin or Norton equivalent circuits (using electrical
analogies), we can quantify process and system loading effects.
Bilateral transducers are associated with reversible physical effects. Such
a device can (for example):
Convert mechanical energy to electrical energy and vice versa. An
example is a piezoelectric crystal. Forces acting on the surface
generates voltage changes. Correspondingly, applying a varying
voltage to the crystal will cause it to vibrate. This principle is used in
most ultrasound transmitters/receivers.
Another such reversible effect is the electromagnetic effect. Moving a
mass through a magnetic field will generate a current, just as
alternating a current can cause a mass to move.
Read Section 5.2.3 on your own and derive all intermediate steps to
make sure you understand the reasoning.
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Next lecture
Summary
Loading effects:
Thévenin equivalent circuits.
Norton equivalent circuits.
Signals and noise in measurement systems.
Statistical representation of random signals.
Probability density functions, mean and variance.
Autocorrelation.
Cross-correlation.
Power spectral density.
Two-port networks:
Generalized effort and flow variables.
Process loading.
Bilateral transducers.
Effects of noise in measurement systems.
Noise sources and coupling mechanisms.
Noise reduction techniques.
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Recommended exercises
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Questions?
5.1, 5.3
5.5, 5.8, 5.9.
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