Potentiometric circuit arrangement

A potentiometric circuit arrangement 30 is based on a resistive potentiometer tarck 11 with a capacitively coupled `wiper` 14', the resistive track being driven by switched alternating voltages of +/- V at 12 and -/+(X-V) at 13, where X is a reference voltage and V is the output of an integrator 15 to which the voltage sensed by the elecrode 37, and rectified at 36, is applied. A null voltage point establishes itself at the position of the electrode and stabilizes the integrator output at a d.c. voltage proportional to the distance of the `wiper` from the track end 12. Any `wiper` displacement taps a non-null signal which integrated applies new voltages to the track until the null point is re-established at the `wiper` position. The capacitive pick-up enables a potentiometer-type voltage output with less wear of the track, less drag against motion and less electrical noise, and electronic differentiation (27) of the position signal to give a rate signal. The arrangement is suited to use as an angle and rate sensor in a gyroscope 70 (FIG. 6).

This invention relates to potentiometric circuit arrangements which provide 
an electrical signal relating the position of a "movable" part relative to 
a "fixed" part and to position measuring apparatus based on such 
potentiometric circuit arrangements. The invention is particularly 
concerned with providing a potentiometric circuit arrangement suitable for 
measuring position determination within a sensitive precision instrument 
for which potentiometer based systems are normally considered 
unsatisfactory, not least by a sometimes requirement to provide a signal 
representing rate of change of determined position. 
There are many examples of apparatus wherein it is necessary to determine 
the relative positions of moving parts and sometimes the rate of relative 
motion. One such apparatus is a gyroscope in which an inertial mass spins 
within cages or gimbals pivotally mounted on gimbal axes for rotation in 
different planes relative to each other and to a gyroscope housing. 
Such a gyroscope may be employed in determining motion of a vehicle about 
an axis, the motion being completely described in terms of angular 
position of the vehicle and rate of rotation. Whereas it is possible in 
principle to derive orientation rate from an orientation-sensing gyroscope 
by differentiating its output signal with respect to time electronically, 
the differentiation amplifies any electrical noise present in the 
orientation signal and can result in excessive wear and power consumption 
in any mechanical devices controlled by the rate signal. 
Such electrical noise usually originates not in the gyro itself but in the 
transducer used to measure the gyroscope orientation, the most common 
transducer form in gyroscopes being the resistive potentiometer in which a 
wiper contacts, and slides over, a resistive track. Although simple in 
concept this component has to satisfy criteria other than low noise, 
requiring good linearity, a low wear rate and good immunity to shock and 
vibration. Furthermore, to avoid degrading gyroscope performance it must 
have a very low operating torque. However there is conflict in satisfying 
these criteria as, for instance, a high contact pressure between wiper and 
resistive track which provides good shock and vibration performance also 
gives a high operating torque and possibly excessive wear. 
Numerous attempts have been made to produce alternative types of 
transducer, frequently using magnetic and optical techniques, but often 
obtaining low noise at the expense of accuracy and linearity provided by 
the resistive potentiometer and an increased space requirement. 
It is frequently preferable to provide the rate signal by means of a 
separate rate gyroscope, demonstrating that alternatives to the resistive 
potentiometer for simply and compactly providing displacement and rate 
signals still leave a requirement for a potentiometer-like arrangement 
that will satisfy the above outlined conflicting criteria. 
Clearly a potentiometer, or a potentiometer-based circuit arrangement, 
which is able to satisfy these criteria is suited to use in other 
situations where displacements, suitably detected as movement of a 
potentiometer wiper or its physical analogue, is required to be manifested 
as an electrical signal or a desired electrical signal level is to be 
achieved by effecting such displacement, and with or without any or all of 
the above discussed criteria applying, and in keeping with the generality 
of the invention it is a first object of the present invention to provide 
a potentiometer circuit arrangement which emulates a resistive 
potentiometer in which at least one of the above-outlined disadvantages of 
conventional resistive potentiometer are mitigated. 
It is a second object of the present invention to provide a potentiometric 
circuit arrangement able to provide a useful electrical signal 
differentiated with respect to time or suitable for such differentiation. 
Further objects of the present invention are to provide position 
determining, and particularly angular position determining, arrangements 
including a potentiometric circuit arrangement. 
Yet further objects of the present invention are to provide velocity 
determining, and particularly angular rotation rate determining, 
arrangements including a potentiometric circuit arrangement. 
According to a first aspect of the present invention a potentiometric 
circuit arrangement comprises a resistive element extending between two 
fixed terminals, a signal-tapping member movable along the resistive 
element to derive a signal from the resistive element and characterised by 
integration means operable to integrate with respect to an integration 
datum a signal derived from the signal-tapping member and provide an 
integrated signal and feedback means including means, including a source 
of reference voltage of amplitude in excess of the maximum integrated 
signal amplitude and algebraic summing means, operable to apply to one of 
said fixed terminals a voltage related to the amplitude of the reference 
voltage reduced by the amplitude of the integrated signal and means 
operable to apply to the other of said fixed terminals a voltage related 
to the amplitude of the integrated signal but of opposite polarity, with 
respect to the integration datum, to the voltage applied to said one of 
the fixed terminals such that a voltage having a peak amplitude equal to 
the reference voltage exists between the fixed terminals and a null 
voltage, with respect to said integration datum, exists at some position 
between the fixed terminals, said integration means being responsive to 
variation of the input signal thereto by displacement of the 
signal-tapping member from said null voltage position towards either fixed 
contact to vary the amplitudes of the signals applied to the fixed 
terminals relative to each other to restore the null voltage to the 
position of signal-tapping member and to provide said integrated signal 
amplitude as a function of the position of the signal-tapping member with 
respect to the fixed terminals. 
According to a second aspect of the present invention a potentiometric 
circuit arrangement as defined in the preceding paragraph includes 
differentiation means responsive to the integrated signal, representing 
the position of the signal-tapping member with respect to the fixed 
terminals, to provide a signal representative of the rate of change of 
position of the signal-tapping member. 
According to a third aspect of the present invention an angular position 
determining arrangement for determining for a body, mounted for rotation 
about an axis within a housing, the angular position of the body about 
said axis with respect to the housing, includes a potentiometric circuit 
arrangement in which the resistive element is carried by the housing and 
formed as an arc of a circle centred on said axis and the signal-tapping 
member is rotatable with the body about said axis. 
According to a fourth aspect of the present invention an angular rotation 
rate determining arrangement for determining for a body, mounted for 
rotation about an axis within a housing, the rate of change of angular 
position of the body about said axis with respect to the housing, includes 
a potentiometric circuit arrangement as defined in the last-but-one 
paragraph in which the resistive element is carried by the housing and 
formed as an arc of a circle centred on said axis and the signal-tapping 
member is rotatable with the body about said axis.

Referring to FIG. 1 a potentiometric circuit arrangement 10 in accordance 
with the present invention comprises a resistive element 11 extending 
between two `fixed` terminals 12 and 13, a signal-tapping member 14, 
movable along the resistive element to derive a signal from the resistive 
element, integration means 15 for the derived signal and feedback means 
16. 
The signal-tapping member may make contact with the resistive element 
(corresponding to the wiper of a conventional resistive potentiometer) or 
be insulated therefrom and require a.c. coupling as described below. The 
tapped signal, representing a proportion of the voltage difference between 
the fixed terminals in dependence upon the position of the tapping member, 
is applied to the integration means 15. Integration is performed on the 
tapped signal in relation to an integration datum potential on bus 17, 
conveniently at zero volts or ground potential, and involves polarity 
reversal of the signal relative to the datum. The signal provided by the 
integration means at 18 is also presented as the circuit arrangement 
output at terminals 19 relative to datum bus terminal 20. 
The integrated signal is also applied to the feedback means 16. The 
feedback means 16 includes means 21, to apply a signal having an amplitude 
related to the integrated signal to the fixed terminal 12, this being 
shown as a simple connecting line 21' which applies the integrated signal 
itself. The feedback means 16 also includes means 22 which applies to the 
fixed terminal 13 a signal having an amplitude related to the algebraic 
sum of said integrated signal and a reference signal of opposite polarity, 
with respect to the integration datum, to said integrated signal. The 
means 22 comprises a reference voltage generator 23, algebraic summing 
means 24, arranged to receive an input from the generator 23 comprising a 
predetermined reference voltage with respect to the integration datum and 
an input on line 25 from the integration means comprising the integrated 
signal, and a line 26 connecting the output of the summing means to the 
fixed terminal 13. In practice the algebraic summing means 24 may be 
arranged to subtract from the integrated signal a reference signal which 
is of the same polarity as the integrated signal or to subtract the 
integrated signal from said reference signal of same polarity with 
polarity inversion. 
As will become apparent, the polarity of the integrated signal, and thus 
the output signal of the arrangement, is in fact determined by the 
polarity chosen for the reference voltage. 
Considering operation of the circuit arrangement, if the generator 23 
produces a reference voltage which is positive with respect to the 
integration datum and of amplitude X volts, in excess of the maximum 
amplitude of signal produceable by the integration means, and the 
integration means initially produces an integrated signal of zero 
amplitude with respect to the datum, then fixed terminal 12 is initially 
at zero volts and terminal 13 at -X volts. The signal-tapping member, if 
not at terminal 12, will tap a signal voltage negative with respect to the 
integration datum and supply this to the integration means which will, in 
turn, reverse polarity and provide a positively increasing integrated 
signal. The voltage on fixed terminals 12 and 13 will thus go more 
positive with respect to the integration datum and the zero volts, or 
null, point will travel along the track from terminal 12 towards the 
position of the signal-tapping member. As it approaches, the signal tapped 
will fall in magnitude and the output of the integration means, the 
integrated signal, will stabilise at some positive level +V. The voltage 
levels at fixed terminals 12 and 13 will thus be +V and -(X-V) 
respectively. 
It will be appreciated that with a reference voltage negative with respect 
to the integration datum, that is, -X, the integrated signal will 
stabilise at -V. 
If the tapping member is displaced from the position at which the 
integrated signal/output signal has stabilised towards one of the fixed 
terminals then a non-null voltage, which may have either positive or 
negative polarity with respect to the integration datum, is input to the 
integration means with the result that the amplitude of the output changes 
from V and the distribution of voltage between fixed terminals 12 and 13 
will be such that the null voltage point moves in relation to the 
terminals. At some integrated signal level V' the distribution of voltage 
between the terminals is such that the null voltage point coincides with 
the signal-tapping member, at which time the signal applied to the 
integration means is reduced to zero and the integrated signal becomes 
settled at V'. 
It will be appreciated that if the signal-tapping member is caused to take 
up a position adjacent the fixed terminal 12 the amplitude of the 
integrated signal (and circuit output between terminals 19 and 20) will be 
zero volts whilst if caused to take up a position adjacent the fixed 
terminal 13 the amplitude of the integrated signal will be X volts. 
Assuming a length L of resistive elements between the fixed terminals then 
at any position of the signal-tapping member intermediate the fixed 
terminals, say at displacement d from terminal 12, that gives an output 
signal amplitude V, the null point will also stabilise at said distance d 
from terminal 12 whereby 
EQU d/(L-d)=V/(X-V) 
EQU and 
EQU d=V.(L/X) (1) 
EQU and 
EQU V=d.(X/L) (2) 
X and L are constants and their ratio may be accommodated by a scaling 
factor of the integration means or any subsequent circuitry. 
The circuit arrangement thus behaves like a resistive potentiometer in that 
it can provide an output signal level which is directly proportional to 
the position of the signal-tapping member between the fixed terminals, 
whether the output signal level V is a measure of an unknown 
tapping-member position d, equation (1), or has a value generated by 
deliberately positioning the tapping member, equation (2). 
It will be appreciated that when the signal-tapping member is displaced 
from any position at which a voltage null is established, the integrated 
signal/output signal is inaccurate until the null establishes itself at 
the new position of the signal-tapping member. Clearly, with extremes of 
long integration times and rapid frequent displacement the integrator may 
not settle to provide an accurate output signal. However by choosing 
integration times appropriate to the mechanical bandwidth associated with 
physical means of displacing the tapping member it can provide a suitable 
output signal. What is valuable in practice and compensates for the above 
is that the motion of the null point follows displacement of the 
signal-tapping member so that the tapping point deals only with low level 
signals and is less likely to be a source of high level noise signals 
induced in the resistive element or caused by poor contact with, or 
imperfections in, the resistive element, and any noise signals which are 
present are furthermore reduced by the integration. 
Such potentiometric circuit arrangement 10 is suited to incorporating, or 
supplying the output signals at 19 and 20 to, a differentiation circuit as 
shown at 27 for providing between further output terminal 28 and terminal 
20 a signal which is the derivative with respect to time of the integrated 
signal and representing the rate of change of position of the 
signal-tapping member. 
It will be appreciated that the integration means may function without 
polarity inversion if such inversion is achieved between its output and 
the terminals 12 and 13, in the latter case by removing existing inversion 
achieved by the algebraic summing means 24 and/or reference generator 23. 
Notwithstanding the options available, the fixed terminals polarities are 
opposite to each other, with respect to the integration datum, and there 
are advantages to energising the resistive element with fixed terminal 
voltages whose instantaneous values with respect to the integration datum 
vary together cyclically as a function of time at a higher rate than 
variations permitted of relative movements of the signal-tapping member 
along the resistive element. Such variation may compare the varying of the 
fixed terminal voltages either by controlling the levels of voltage from 
antiphase a.c. generators in accordance with the above defined terminal 
voltages V and -(X-V) and the integration datum level or by `chopping` the 
aforementioned fixed terminal voltages V and -(X-V) between the current 
relatively slowly varying levels and the integration datum. Preferably 
such variation includes alternating the polarities of voltages applied to 
the two fixed terminals. Such polarity alternation may be effected by 
controlling the levels of voltage from antiphase a.c. generators in 
accordance with the above defined terminal voltages V and -(X-V) or by 
`chopping` the above defined terminal voltages between positive and 
negative values of V or -(X-V) respectively. Such `chopping` of an 
erstwhile d.c. voltage between its d.c. volume and its polarity inverse 
derives an a.c. energising signal which enables the instantaneous voltage 
between the two fixed terminals to be the reference voltage X, giving 
maximum sensitivity to signal-tapping member displacement throughout the 
energising variation cycle, whilst also eliminating from the resistive 
element an average steady, or d.c., voltage relative to the integration 
datum that leaves the signal-tapping member to respond only to changes or 
`errors` in energising voltage. 
Referring to FIG. 2, this shows a schematic potentiometric circuit 
arrangement 30 similar to 10 of FIG. 1 but including modifications 
associated with a.c. operation, that is, the application of energising 
voltages of alternating polarity to the resistive element. Parts 
corresponding to those shown in, and described above in relation to, FIG. 
1 are given the same reference numbers. 
The potentiometric circuit arrangement 30 comprises resistive element 11 
having fixed terminals 12 and 13, a signal-tapping member 14', described 
further hereinafter, integration means 15, feedback means 16', output 
terminals 19, 20 and optional differentiation means 27 employing further 
output terminal 28 and terminal 20. 
The feedback means 16' includes a.c. generating means 31 in the form of a 
first a.c. generator 32 connected to supply the fixed terminal 12 and a 
second a.c. generator 33 connected to supply the fixed terminal 13. Both 
generators produce signals of alternating polarity with respect to the 
integration datum 17, produce signals in antiphase and are maintained in 
relative phase by a synchronising signal on line 34 from an integral or 
external synchronisation generator 35. 
The algebraic summing amplifier 24' is arranged to receive a d.c. reference 
voltage (.+-.X) from reference generator 23' of the same polarity as the 
integrated signal (.+-.V) and substract that from the integrated signal on 
line 25 providing the difference .+-.(X-V) on the line 26. 
The lines 21' and 26 control the amplitudes of the a.c. generators 32 and 
33 respectively. It will be seen that by having generator 33 produce a 
signal in antiphase to generator 32, the generator is in effect inverting 
the polarity of its amplitude-controlling signal from algebraic summing 
means 24' so that the controlling signal is in effect the sum of the 
integrated signal and a reference signal of opposite polarity to the 
integrated signal described for arrangement 10. 
Any non-null signal tapped from the resistive element by the member 14' 
will be an a.c. signal and the arrangement 30 further includes 
rectification means 36 between the signal-tapping member and the input to 
integration means 15 in order to restore a d.c. or unipolar input level 
with respect to the integration datum. The rectification means 36 
preferably comprises a phase sensitive rectifier connected to line 34 and 
under the control of synchronisation generator 35. 
It will be appreciated that the waveforms of the signals produced by the 
a.c. generators 32 and 33 is a matter of choice. However there are 
practical advantages in terms of circuit simplicity, from using a square 
waveform signal derived from the essentially d.c. signal fed back from the 
integration means on lines 21' and 26, the generators 32 and 33 then 
comprising switches of the polarities of the d.c. feedback signal voltages 
under the control of a square wave synchronisation generator 35. 
A.c. energisation of the resistive element 11 enables transformer isolation 
(not shown) of the resistive element 11 and signal-tapping member, 
permitting its safe use in explosive environments or high voltage 
applications. 
A further and more useful advantage of using a.c. energisation of the 
resistive element 11 is enabling the signal-tapping member 14' to be 
non-contact-making with the element and take the form of a tapping 
electrode 37 capacitively coupled to the resistive element by way of a 
dielectric material such as air or suitable solid material. 
The tapping electrode 37 senses the spatial average of voltage over the 
region of resistive element to which it is adjacent. When displaced, the 
electrode produces a sensed signal which is less susceptible than a 
conventional point-contact-making wiper member to microscopic 
imperfections in the resistive element and in their contact and is thus 
relieved of spatial noise and non-linearity associated with wiper 
position, and change of position, caused thereby. 
The signal that the electrode senses from the resistive element is 
amplified by amplification means 38 before being fed to rectification 
means 36. The amplification means includes a preamplifier, described 
hereinafter, coupled to the tapping electrode 37 and that electrode, and 
preferably the preamplifier, are provided with an electrostatic screen 39 
to further reduce electrical pick up from anything but the resistive 
element. Interference by noise from any nearby switching circuits, such as 
switch-mode power supplies, may also be minimised by choosing the 
switching frequency of synchronisation generator 35 different from such 
circuit. 
There are other benefits and improvements which are more readily seen and 
appreciated in FIG. 3, which represents a more practicable embodiment 40 
of the arrangement 30 described above. Again, parts which are common 
thereto are given like reference numbers. 
Referring to FIG. 3 the integration means 15, algebraic summing means 24' 
and optical differentiation means 27 all comprise conventional 
constructions of operational or the like amplifiers with passive input and 
feedback components and, as such, require no further description. 
The reference voltage generator 23' comprises a simple potential divider 
including voltage defining zener diode 41, connected between a positive 
supply rail 42 and the integration datum 17. 
The a.c. generator 32 comprises a two-state switching means 43 arranged in 
a first state (shown) to connect the integrated signal on line 21' to the 
fixed terminal 12 directly and in a second state by way of inverting 
amplifier 44. 
The a.c. generator 33 comprises a similar two-state switching means 45 
arranged in a first state (shown) to connect the signal on line 26 to the 
fixed terminal 13 by way of inverting amplifier 46 and in a second state 
to connect the signal on line 26 to the fixed terminal 13 directly. 
The switching means 43 and 45 are `ganged` or switched between states in 
synchronism by means of the square wave switching signal from 
synchronisation generator 35 on line 34. 
It will be appreciated that the switching means 43 and 45 may be provided 
conventionally by semiconductor switching devices. It will also be 
appreciated that inverting amplifiers 44 and 46 are not in operation 
simultaneously and that with the use of more complex switching means a 
single inverting amplifier having suitable operating characteristics may 
be employed instead. 
The phase sensitive rectifier 36 comprises a simple switch, which may be a 
semiconductor device, in combination with storage capacitor 36' and 
capacitor charging resistors 36" and 36"'. The switch is opened and closed 
by synchronisation generator 35 in synchronism with the polarity reversal 
of the energising voltages at the fixed terminals 12 and 13 of the 
resistive element, being opened when the energisation polarity of fixed 
terminal 12 corresponds to the output of the integration means and closed 
when inverted. The time constant of capacitor 36' and resistor 36" 
combination is approximately equal to the switching period whereas the 
time constant of capacitor 36' and resistor 36" combination is very much 
greater. Thus when the polarity change of the synchronisation generator 
causes any signal picked up from the resistive track to change between a 
negative and a positive value with respect to the integration datum the 
capacitor is charged by the magnitude of the change, although it only 
remains so until the switch is next closed. Apart from a small discharge 
by way of the resistor 36"' the voltage-time product per cycle of the 
rectifier switch providing the input to the integration means is 
equivalent to the amplified signal derived from the signal-tapping member 
applied throughout the cycle and the integrated signal reflects the level 
of the tapped signal. 
The amplification means 38 comprises a pre-amplifier 47, which is carried 
with the signal-tapping electrode 37, and a amplifier section 48. The 
pre-amplifier is configured such that it both receives operating power and 
provides amplified electrode signals by way of the same conductor pair 49, 
50. Operating current is derived by way of resistor 51 from supply line 52 
and the amplified electrode signals are manifested as modulation of the 
operating current, the signals being decoupled therefrom by capacitor 53 
at the input to amplifier part 48. 
The pre-amplifier may be constructed as a miniature hybrid circuit readily 
screened with the electrode 37 and the provision of the signals by way of 
supply current modulation removes the need to screen further connecting or 
amplifying circuitry. 
Depending upon the shape of the resistive element 11 and the extent and 
manner of motion required for tapping electrode 37, the form in which the 
signals are carried and the use of only two interconnecting conductors 49, 
50 makes it readily feasible to include sliding contacts as shown at 54 
and 55. For instance, the resistive element may comprise an arc of a 
circle such that the electrode 37 and pre-amplifier 47 are displaced by 
rotation about an axis, the conductors 49 and 50 then including sliding 
contacts 54 and 55 in the form of slip rings. Such slip rings may used to 
provide with less resistance to motion than a flexible conductor link or 
because any constant frictional drag introduced by the slip rings is 
preferable to a restoring torque, caused by rotation of a conductor link, 
which tends to increase with deflection from a datum position, or to 
enable a shaft carrying the signal-tapping electrode to make complete 
revolutions between orientations at which potentiometric measurements are 
possible. 
It will be appreciated that the structure and form of the resistive 
material and non-contact-making electrode may take many forms depending on 
the use to which the potentiometric circuit arrange is put. 
In its simplest form and/or where minimum resistance to electrode 
displacement is needed, the electrode 37 may be spaced from the surface of 
the resistive element by an air gap, as shown in FIG. 4(a) which is a 
sectional elevation through a resistive element 11 in the form of a flat 
`track` deposited on substrate 11'. The resistive element 11 may be coated 
with a layer of dielectric material to protect it and provide a different 
capacitance between electrode and resistive element. 
FIG. 4(b) shows an alternative construction in which the resistive element 
is coated with a layer 56 dielectric material against which the electrode 
37 bears, possibly with a bias force to ensure a constant separation from 
the resistive element and constant capacitance. The dielectric material 
may be hard wearing and/or offer low friction to the electrode. 
FIG. 4(c) shows yet another construction in which the electrode 37 is in 
the form of a spindle on which is mounted a roller 57 of dielectric 
material such that the electrode can be biased towards the resistive 
member but maintain a constant separation equal to the thickness of the 
roller which permits displacement of the electrode with low resistance to 
motion and little abrasive wear to the resistive element. 
As shown in FIG. 4(d) combinations may be effected, such as the provision 
of both a roller 57' or other dielectric coating on the electrode 37 and a 
dielectric coating 56' on the resistive element 11. 
The resistive element may take a form other than of a flat track, exemplary 
configurations of resistive element and signal-tapping electrode being 
illustrated in FIGS. 5(a) and 5(b). 
In FIG. 5(a) a resistive element 58 is formed on the exterior surface of a 
hollow cylindrical substrate 59. The signal-tapping member, in the form of 
a cylindrical electrode 60 and preamplifier 61, is moved on shaft member 
62 for reciprocal translation within, and along the axis of, the substrate 
59. 
In FIG. 5(b) the resistive element 63 is formed on the exterior of a 
cylindrical substrate 64, or possibly on the interior if it is hollow, and 
the signal-tapping member takes the form of a saddle-like or ring-like 
electrode 65 which with its pre-amplifier 66 is mounted for reciprocal 
translation provided to the longitudinal axis of the substrate. 
Clearly the resistive material may conform to any path between the fixed 
electrodes provided the signal-tapping member is suitably mounted to 
translate it. 
Similarly, it will be understood that in any configuration as above, the 
motion between resistive track and signal-tapping member is relative and 
either the resistive track or the signal-tapping member may be `fixed` 
with respect to other apparatus whilst the other part undergoes motion. 
The most common form of conventional potentiometer is that used in 
connection with angular motion wherein the resistive element is in the 
form of a flat track describing an arc of a circle centred on the 
rotational axis of a shaft carrying the normally contact-making wiper. 
Many of the constructional details of such a device may be employed in an 
arrangement according to the invention in its preferred form, that is, 
based upon the arrangement of FIG. 3. As illustrated at 67 in FIG. 5(c), a 
flat arcuate resistive element 11 has associated therewith instead of a 
contact making wiper, a capacitively coupled signal-tapping electrode 37 
with a pre-amplifier, 47, being mounted on rotatable shaft 67', and slip 
rings, 67", 67"' provided for supplying current to, and signals from, the 
pre-amplifier. 
An alternative angular rotation arrangement is shown at 68 in FIG. 5(d) 
wherein the resistive element 11 conforms to part of a cylindrical surface 
and the signal-tapping electrode 37 moves within, and is to some extent 
shielded mechanically and electrically by, the resistive element and/or 
more extensive substrate. 
As mentioned above, and illustrated in FIG. 5(e) at 69, a potentiometric 
circuit arrangement for angular rotation may have its resistive element 11 
carried by rotatable shaft 69' with its fixed contacts 12, 13 energised by 
way of slip rings 69" and 69"' and the signal-tapping electrode 37 
disposed on a relatively fixed part of external apparatus. In such 
arrangement, the small signal from the electrode does not have to pass by 
way of slip ring contacts and the amplification means 69"" may comprise 
single amplification means without the facilities offered by pre-amplifier 
47. 
As indicated hereinbefore, the benefits of a.c. operation, namely the use 
of a non-contact-making signal-tapping member and, if desired, transformer 
coupling, may be realised with a cyclically varying, conveniently an 
intermittent, or `chopped`, d.c. energisation of the resistive element. 
However, it will be appreciated that by alternating the fixed terminal 
voltage between opposite polarities with respect to the integration datum 
the average, that is, d.c., level of voltage at any point on the resistive 
element is zero so that any variation in capacitance between tapping 
electrode or resistive element will not result in generation of a noise 
signal to contaminate the a.c. signal produced by displacement of the 
tapping electrode from an established null signal position. This may be of 
particular value when the signal-tapping electrode 37 is spaced from the 
resistive element by an air gap and the parts are subjected to vibrational 
or other forces on the signal-tapping electrode, varying the air gap and 
thus the capacitance. 
It will be appreciated that the signal-to-noise performance of the a.c. 
potentiometer control arrangement improves with the frequency at which the 
fixed terminal energising voltages are alternated and the capacitance at 
the tapping electrode by which `error` signals are coupled to the 
integration means. The former is a matter of choice, bearing in mind the 
desirability of avoiding a frequency near to any other potentially 
interfering equipment. The latter is a compromise between the use of a 
low-capacitance air gap between signal-tapping electrode and resistive 
track that offers no resistance to motion and a higher-capacitance solid 
dielectric with which the electrode makes contact but with potential 
stiction-induced non-linear response. The arrangement 40 of FIG. 3 
demonstrates that the disadvantage of a low-impedence air dielectric can 
be mitigated by a suitable preamplifier design of low input capacitance. 
As mentioned above the present invention is concerned with the use of the 
potentiometric circuit arrangement, as an alternative to a conventional 
potentiometer, in position measuring apparatus which gives an output 
signal amplitude, often a voltage, related to the relative position of two 
members or even the rate of change of relative position. 
By way of example, FIG. 6 shows a cut-away view of a gyroscope 70 
comprising a rotor 71, carried in a cage or gimbal 72 itself mounted on a 
shaft 73 for rotation about the longitudinal axis of the shaft within a 
housing 74. It is required to produce a voltage proportional to the 
angular rotation of the gimbal with respect to the housing from a fixed 
datum orientation and to produce a voltage proportional to the rate of 
rotation. Furthermore any transducer which provides such voltages must not 
place significant drag on the gimbal motion to interfere with accurate 
gyro operation. The potentiometric circuit arrangement 40 of FIG. 3 is 
employed and as described hereinbefore provides these voltage between 
terminals 19 and 20 and 28 and 20 respectively. 
The resistive element is carried by the housing 74 and formed as an arc of 
a circle centred on the rotational axis of shaft 73 in the manner shown in 
FIG. 5(c). A signal-tapping member, in the form of electrode 37, is 
mounted on the shaft 73 to overlay the resistive element during rotation 
of the shaft and a pre-amplifier 47 is also mounted on the shaft, coupled 
to the electrode and screened with it by electrostatic screen 39. The 
pre-amplifier 47 is connected to the remaining circuitry by way of slip 
ring contacts 49 and 50. It will be seen that as well as providing the 
necessary electrical and signal requirements, the arrangement also 
operates without the friction that is required within normal 
potentiometers to give good signal levels, the only contacts being the 
slip rings. It will be appreciated that the use of slip rings is not 
mandatory and if the rotation of the shaft is limited the slip rings may 
be replaced by flexible conductors of low restoring force. Also, the form 
of the resistive track and its disposition in relation to the 
signal-tapping electrode may be varied, such as taking on of the form 68 
or 69 shown in FIGS. 5(d) and 5(e) respectively. 
The integrated signal provided by integration means that provide a measure 
of angular position of the gimbal is, by use of the features described 
above, sufficiently stable and free of electrical noise to be 
differentiated electronically in 27 to prove a rate of gimbal rotation 
signal. 
It will be appreciated that in all of the above embodiments, the integrated 
signal need not be presented at an accessible terminal 19 if the 
arrangement is dedicated to providing just a rate signal. 
All of the embodiments of potentiometric circuit arrangements in accordance 
with the invention described hereinbefore have included a resistive 
element having two distinct ends at which the resistive terminals are 
located and between which the signal-tapping member can move in a 
rectilinear or acuate path. 
It will be appreciated that there are occasions when in an instrument 
similar to that of FIG. 6 a shaft is capable of making continuous 
rotations before settling at any particular shaft orientation and there is 
a requirement to ascertain the shaft orientation and the rate of shaft 
rotation. 
In accordance with the present invention and as illustrated in the 
schematic circuit of FIG. 7, a 360.degree. angular position and/or rate 
measuring potentiometric circuit arrangement 80 has, extending between two 
fixed terminals 81 and 82, a resistive element 83 which extends as a 
semi-circular arc about a shaft 84 rotatable about its longitudinal axis 
and supporting a signal-tapping member 85, preferably, but not 
necessarily, of the non-contact-making types described above in 
operational pick-off relationship with the resistive element 83. 
Amplification means 86, phase sensitive rectification means 87, 
integration means 88 and feedback means 89 correspond to those items 36, 
38, 15 and 16' described above with reference to FIG. 2 or 3. 
The arrangement 80 includes a further resistive element 90 extending also 
between the two fixed terminals 81, 82, in a complementary arcuate form 
and along which the signal-tapping member 85 can move separately from 
motion along the resistive element 83, that is, the resistive and further 
resistive elements combine to form a 360.degree. cylinder resistive track 
and the signal-tapping member moves along each for separate 180.degree. 
rotations of the shaft 84. 
It will be appreciated that in operation the resistive element 83 and 
further resistive element 90 are in parallel and the same voltages appear 
on each at the same distance from the fixed terminal 81 so that an 
integrated signal giving an energising voltage V at terminal 81 may 
correspond to the shaft 84 holding the signal-tapping member at an angle 
.theta..degree. clockwise from terminal 81 or at an angle 
(360-.theta.).degree. clockwise or at an angle .theta..degree. 
anti-clockwise from terminal 81 or (360-.theta.).degree. anti-clockwise. 
The measurement is clearly subject to ambiguity as to direction of shaft 
displacement, if both directions are possible, and degree of displacement 
in any particular direction. 
To this end the arrangement 80 also includes ambiguity resolving means 91 
to resolve ambiguity in the integrated signal produced by integration 
means 88. The means 91 comprises two further fixed terminals 92, 93 
connected one each to the resistive elements 83 and 90 each at a point 
between the two fixed terminals 81 and 82, conveniently mid way, 
representing a 90.degree. rotational displacement from terminals 81 and 
82, duplicate integration means 94, duplicate feedback means 95 associated 
with the two further fixed terminals and the signal-tapping member 85 (and 
amplification means 86 and phase sensitive rectifier 87), to receive 
signals from the signal-tapping member and provide energising voltages to 
the two further fixed terminals 92 and 93, and time division multiplexing 
means 96. The multiplexing means 96 may be controlled by synchronisation 
generator 35 to direct signals from the signal-tapping member to either 
the original integration means 88 or duplicate integration means 94 and 
direct energising voltages from the corresponding original feedback means 
89 or duplicate feedback means 95 to the respective two fixed terminals 
81, 82 or two further fixed terminals 92, 93. 
The integration means 88 and further integration means 94 produce 
integrated signals which, in addition to controlling energisation of the 
two fixed terminals and further fixed terminals uniquely define the 
position of the signal-tapping member with respect to fixed terminal 81. 
Referring to FIG. 8 this shows the voltage levels relative to the 
integration datum applied to the fixed terminals 81, 82, 92 and 93 as a 
function of rotation angle of shaft 82 and assuming a 0.degree. rotation 
angle when the signal-tapping member is adjacent fixed terminal 81. 
Although the arrangement of FIG. 7 is shown with the feedback means having 
the preferred form of a.c. generating means effected by polarity switching 
of the integrated signals by synchronisation generator 35, the waveform 
diagram is drawn as if d.c. energised to show only the states of the 
feedback means in which the actual integrated signals (V) output from the 
integration means 88 and 94 are applied to fixed terminals 81 and 92 
respectively and the difference between reference voltage X and the output 
from the integration means (-(X-V)) are applied to respective fixed 
related terminals 82 and 93, the waveforms of voltages V.sub.81 and 
V.sub.92 in the upper part of the diagram thus corresponding to variations 
in the integrated signals of the arrangement with shaft angle .theta.. The 
resistive track energising voltages are shared between the two waveforms 
by the time division multiplexing although the integrated signals 
corresponding thereto are, of course, continuous. 
Because of the time division multiplexing the outputs from original and 
duplicate integration means derived by using fixed terminal points 
90.degree. displaced are also displaced 90.degree. in phase and enable the 
positional ambiguity to be resolved using techniques applicable to other 
types of angular measuring systems. 
As will be seen from the waveforms of FIG. 8(a) the advantageous 
characteristic of the non-contact-making signal-tapping member picking off 
a signal that is a spatial average of the voltage about a particular 
position causes a rounding of the signal peaks and troughs of each 
integrated signal produced when the signal-tapping member is in the 
immediate vicinity of the fixed terminals associated with the integrated 
signal and detracts from the accuracy of relationship between integration 
signals and rotational position. However, it will also be seen that when 
the shaft position places the signal-tapping member in the vicinity of one 
fixed terminal it is mid way between the terminals of the other pair and 
the integrated signal associated with that other pair of terminals is at 
its most linear part between peak and trough. 
Thus it is convenient to derive an output signal from the potentiometric 
circuit arrangement 80 by switching between, or combining with appropriate 
weighting, the integrated signals from the integration means 88 and 
further integration means 94 in accordance with amplitudes thereof that 
define the relationship between angular position of the signal-tapping 
member and the fixed terminals. 
One method by which angular shaft position ambiguity can be resolved will 
be briefly explained with reference to FIGS. 8(b) and 8(c). 
FIG. 8(b) shows a graphical representation of the variation of the 
energisation voltages V.sub.81 and V.sub.92 of FIG. 8(a) in terms of 
integrated signals V.sub.88 and V.sub.94, from which they are derived, as 
a function of shaft angle .theta. in degrees. 
For convenience the reference level of the integrated signals has been 
restated as mid-way between their peak values, rather than the integration 
datum, as indicated by the multiply labelled ordinate scale. 
It will be found convenient to refer to the integrated signal V.sub.88 
derived from integration means 88 (corresponding to V.sub.81 in FIG. 8(a)) 
as A, where A=V.sub.88 -X/2, and to refer to the integrated signal 
V.sub.94 derived from integration means 94 as B, where B=V.sub.94 -X/2. 
Thus in relation to the restated ordinate origin, A and B have values 
between .+-.X/2 as the shaft moves through 180.degree. sectors of rotation 
as marked on the second ordinate scale. The third ordinate scale of FIG. 
8(b) expresses the values of A and B in terms of angular displacements 
from the restated origin whereby the values of A and B which vary between 
.+-.X/2 can be expressed or used directly in terms of angles between 
.+-.90.degree.. 
It is convenient to divide the rotational cycle of the shaft into sectors, 
in this case each extending 45.degree. centred on the peaks and crossovers 
of the individual waveforms and denoted by the regions marked 1 to 8 
repetitively in FIG. 8(b). 
It will be recalled that one of the aims of utilising both signals is to 
maximise the contribution made by each signal to the part of its waveform 
away from the rounded peaks. 
To this end it will be seen that in shaft rotation sectors 1 and 5 waveform 
B is at its optimum linearity and in sectors 3 and 7 waveform A is at its 
optimum. In sectors 2, 4, 6 and 8 both waveforms progress between the 
optimum linear region and the rounded peak regions but in opposite sense. 
It will be readily seen that given the "measured" values of A and B, in 
sector 1 the shaft angle .theta. (between 0.degree. and 360.degree.) is 
given by 
EQU .theta.=-B (3), 
ignoring for the moment that sector 1 straddles the 360.degree./0.degree. 
boundary; 
##EQU1## 
For the even number sectors the angle is derived from a combination of both 
signals. Given that signal B corresponds to signal A but shifted in phase 
by 90.degree. a signal D can be derived therefrom for creating a weighted 
sum with signal A. 
The signal D, shown graphically in FIG. 8(b) by the chain dotted line 
superimposed on waveform A, also, of course, has non-linearities due to 
the rounded peaks of B and these manifest as a discontinuities at the 
zero-crossing points. The signal D is defined by 
EQU D=[90.degree.-.vertline.B.vertline.].multidot.sign A (7) 
A weighing factor K is defined by 
EQU K=(67.5-.vertline.A.vertline.)/45 (8) 
The weighted signal A' that replaces A is given by 
EQU A'=(K.A+(1-K).D) (9) 
so that in sectors 2 and 4 adjacent sector 3 
EQU .theta.=90+A', (10) 
and in sectors 6 and 8 adjacent sector 7 
EQU .theta.=270.degree.-A' (11) 
These even number sectors relationship may be expressed together as 
EQU .theta.=(90.degree.+A').(-sign B) (12) 
although for digital processing they may be dealt with separately as 
described below with reference to FIG. 8(c) which shows a flow chart of 
the procedural steps in processing the signals A and B to provide a 
smoothly varying value for shaft angle .theta. which obviates the 
non-linear effects of the rounded peaks of waveforms A and B. 
It will be seen that by testing the levels of A and B in relation to the 
67.5.degree. linearity boundary at 100 and 101 the even and odd sectors 
are distinguished and, if an odd sector is formed, whether it is 1 or 5 or 
3 or 7. Depending upon the sign of the signal tested at 102, 103 
respectively for each sector pair branch, the odd sectors angle is derived 
directly from the values of A and B at 104-107 in accordance with 
equations (3)-(6). 
It will be appreciated that the signal B may represent an angle 
.theta.=0.degree. to 22.5.degree. or an angle 337.5.degree. to 
360.degree., the former being characterised by B having a negative value 
and the latter by B having a positive value. The flow chart responds at 
108 to such a positive value of B, by .theta.=-B being less than 0, to 
derive at 109 a value representing (360-B).degree.. 
If an even sector is distinguished at 101 the sign of A is tested at 110 to 
distinguish between sector 4 and 6 (positive A) and sectors 2 and 8 
(negative A) in order to derive expression D at 111, 112 respectively. The 
weighting factor K is computed for A at 113 and the weighted value of A, 
A', is computed at 114. In accordance with equations (10) and (11) and the 
value of A' effectively substituted for A, the test for sign B is 
performed at 103 and in accordance therewith the stages 105 or 107 derive 
.theta. as (90+A').degree. or (270-A').degree. as appropriate. 
It will be appreciated that in a similar manner values of A and B 
differentiated with respect to time may also be combined in such a way as 
to mitigate the effects of non-linearities in the signals A and B due to 
shaft position. 
It will be appreciated that the quantisation noise present in an angle 
measurement digitally derived as above precludes producing a rate signal 
therefrom and it is preferred to derive rates of change of A and B, that 
is, A and B, and to process the rate signals digitally and in accordance 
with the instantaneous shaft position determining the forms of measured 
signals A and B. 
By defining the aforementioned sectors 1 to 8 the rate 
EQU for sector 1 is .theta.=-B=B.sign A (13) 
EQU for sector 3 is .theta.=A=-A.sign B (14) 
EQU for sector 5 is .theta.=B=B.sign A (15) 
EQU for sector 7 is .theta.=-A=-A.sign B (16) 
and in the even numbered sectors .theta. is derived from a weighted sum 
EQU .theta.=(1-K).[B.sign A]+K.multidot.[-A.sign B] (17) 
when weighting factor K=(67.5-.vertline.A.vertline.)/45 as specified in 
equation (8). 
The flow chart of FIG. 8(d) illustrates the comparable steps taken to 
derive a continuous shaft rotation rate output from the two signals A and 
B and their individual rates A and B. 
The steps 120 and 121 distinguish between odd and even sectors of angular 
position. For sectors 1 and 5, the sign of A is tested at 122 and then 
value of .theta. computed at 123 or 124. Similarly for sectors 3 and 7 the 
sign of B is tested at 125 and the value of .theta. computed at 126 or 
127. For even sectors the weighting factor K is computer at 128 and then 
the value of .theta. computed at 129 in accordance with equation (17). The 
values of .theta. derived separately by the different paths are delivered 
sequentially to output 130 as a continuous rate output. 
It will be understood that the above is only exemplary of a particular form 
of deriving unambiguous signals from two phase-displaced components 
combined with a mechanism for preserving linearity despite the rounding of 
waveform peaks caused by the tapping of a spatial average signal. 
An alternative form of ambiguity resolving means is shown at 91' in FIG. 9, 
the remaining components of the potentiometric circuit arrangement 
corresponding to, and being given the same reference numbers as in, FIG. 
7. 
The ambiguity resolving means 91' comprises a duplicate 140 of the 
combination of resistance element 83 and further resistance element 90 
with two fixed terminals 141 and 142 and disposed parallel to the original 
combination such that a signal-tapping member 85' moves along, and derives 
signals from both the original and duplicate resistive element and further 
resistive element combinations. The fixed terminals 141 and 142 on the 
duplicate combination 140 are offset 90.degree. about shaft 84 from the 
fixed terminals 81, 82 of the original combination and receive energising 
voltages from duplicate feedback means 143 controlled by duplicate 
integration means 144 which receives signals from duplicate signal-tapping 
member 85' by way of amplification means 86 and phase sensitive rectifier 
87. 
The separate integrated signals from integration means 88 and 104 together, 
by virtue of their 90.degree. phase shift, uniquely define the position of 
the signal-tapping members 85 and 85' with respect to the two fixed 
terminals 81, 82 of the original combination of resistive elements and 
further resistive elements; the signals may be continuously derived 
without any requirement for multiplexing the energisation of the resistive 
tracks and may also be subjected to differentiation at 97 and 98 before 
any combination of the signal is made. 
Alternatively, the arrangement may be provided with a single signal-tapping 
member and amplifier common to both original and duplicate resistive 
elements as shown in FIG. 10. 
This corresponds generally to that shown in FIG. 9 except that the 
resistive tracks are also shown as coplanar concentric rings with the 
single signal-tapping member 85 bridging both rings. 
The synchronisation generator 35 is arranged to define separate energising 
frequencies for the original and duplicate resistive track combinations 
and a simple filter 86', conveniently associated with the pick-off 
amplification means 86, is disposed between the amplification means and 
integration means to separate error signals associated with either or both 
of the original or duplicate resistive track combinations and deliver them 
to the appropriate integration means. 
It will be appreciated that although a 360.degree. rotation is the most 
common form of a continuous resistive element path from which ambiguity of 
signal-tapping member needs resolving the techniques for resolving 
ambiguities are applicable to other configurations and different phase 
relationships. 
In addition to deriving from the basic linear potentiometric circuit 
arrangements one which copes with continuous linear motion about a closed 
path there may be derived a potentiometric circuit arrangement 145, FIG. 
11, which is able to define the position of the signal-tapping member in a 
two dimensional region defined by a resistive element. 
Referring to FIG. 11 a resistive element 146 extends between two fixed 
terminals 147, 148 in a longitudinal direction, denoted by arrow 149, and 
is extensive in a direction, denoted by arrow 150, orthogonal to said 
longitudinal direction to define a two-dimensional resistive region 151. 
The signal-tapping member 152 is movable with respect to the resistive 
element in both the longitudinal and orthogonal directions, conveniently 
freely movable with components of the motion in those directions. 
The signal-tapping member 152 provides signals by way of amplification 
means 153 and phase sensitive rectifier 154 to integration means 155, the 
latter providing an integrated signal to feedback means 156 which 
energises fixed terminals 147 and 148. The arrangement 145 also includes 
coordinate resolving means shown generally at 157 operable to determine a 
point in the resistive region 151 by the disposition of the signal-tapping 
member with respect to the fixed terminal 147 in both the longitudinal and 
orthogonal directions. 
As shown in FIG. 11 the coordinate resolving means 157 comprises a 
duplicate resistive element 158 substantially parallel to the original 
resistive element 146 such that the signal-tapping member 152 moves along 
both original and duplicate resistive elements in both longitudinal and 
orthogonal directions. The duplicate resistive element has two fixed 
terminals 159, 160 separated in the orthogonal direction and duplicate 
integration means 161 and feedback means 162 associated with, and operable 
to provide signals representative of the position of the signal-tapping 
member 152 between, fixed terminals 159 or 160. In a manner described 
above in relation to FIG. 10 the original and duplicate resistive elements 
are energised at different frequencies and a filter 153' associated with 
the pick-off amplification means 153 directs the error signals tapped by 
the pick-off to the appropriate integration means. 
The integration means 155 and duplicate integration means 161 produce 
integrated signals which together uniquely define the position of the 
signal-tapping member 112 in the two-dimensional resistive region. The 
position signals may also be differentiated before use or combination in 
differentiation means 162, 163. 
If desired, and in a manner similar to that described with reference to 
FIG. 9 a duplicate signal-tapping member may be provided to separate the 
tapped signals completely. 
As indicated above in relation to resolving positional ambiguity from two 
such component signals, the resolving of position from two such signals 
involves corresponding techniques and will not be described further. 
An alternative form of coordinate resolving means is shown at 157' in FIG. 
12 and comprises a pair of subsidiary fixed terminals 165, 166 contacting 
the resistive element 146 defining region 151 at points between the fixed 
terminals 147, 148 at opposite sides of the element separated in the 
orthogonal direction 150. Duplicate integration and feedback means 167 and 
168 respectively are associated with the subsidiary fixed terminals to 
provide energising signals thereto representative of the position of the 
signal-tapping member 152 between terminals 165 and 166. Time division 
multiplexing means 169 is operable to direct signals from the 
signal-tapping member alternately to either the original integration means 
155 or duplicate integration means 167 and synchronously direct energising 
voltages from corresponding original feedback means 156 or duplicate 
feedback means 168 to the respective terminals 147, 148 or subsidiary 
terminals 165, 166. The separate integrated signals delivered by the 
original integration means 155 and the duplicate integration means 167 
together uniquely define the position of the signal-tapping member in the 
resistive region 151. 
It will be appreciated that one but not both of the fixed terminals pair 
147, 148 or subsidiary pair 165, 166 may extend along the edge of the 
resistive element to provide equipotentials between those terminals which 
extend in the direction of separation of the other terminal pair, but the 
terminals of the other pair must make only point contact. In such 
arrangement there is an inherent non-linear relationship between the 
integration means outputs, which define null position and the actual 
position in terms of distances between the fixed terminals. However, such 
a device is capable of computing a correction factor derived from the 
values of the two signals or calibration with such a correction being 
built into a readily accessed look-up table. 
The signals may be combined or used in any known manner and, as indicated 
above, the signals may be differentiated at 162, 163 to provide signals 
defining the velocity of signal-tapping member displacements. 
It will be appreciated that if the signal-tapping member 152 of FIG. 11 is 
caused to translate so as to describe a circular path adjacent the 
two-dimensional resistive element 146 the integrated signal from 
integration means 155 will vary sinusoidally. Furthermore, the signal from 
integration means 161 associated with the orthogonally disposed resistive 
element 158 of the resolving means will vary 90.degree. out of phase, the 
whole arrangement providing both sine and cosine outputs for resolving 
ambiguities or to perform other functions. 
The signal-tapping member 152 may be carried by, and moved with, an arm 170 
as shown and undertake circular motion by being coupled to a rotating 
device by a crank or eccentric. Alternatively, and more practicably, the 
signal-tapping member 152 is carried in the manner illustrated in FIG. 13 
on a radial arm 171 fixed to a shaft 173 which rotates about an axis 
within the boundary of the resistive element 146. A duplicate resistive 
element corresponding to, and energised in the manner of, 151 in FIG. 11, 
or possibly a duplicate signal-tapping member (not shown) displaced 
90.degree. about the shaft, enables both analogue sine and cosine signal 
outputs to be used directly or for deriving rate signals. 
To ensure a true sinusoidal relationship between shaft angle and energising 
voltage it is necessary to have a uniform voltage gradient between fixed 
terminals and equipotential which extend truely orthogonal to them within 
the plane of the element. To this end the resistive element should 
preferably not be apertured for the shaft to pass therethrough, although 
it is believed that in practice if the radius of the locus of 
signal-tapping member is large in relation to any aperture in the 
resistive element an accurate sinusoidal relationship between shaft angle 
and integrated signal is obtainable. 
Whereas the structure of FIG. 13 may appear somewhat clumsy in relation to 
the many purpose built sine/cosine potentiometers already available, given 
the problems traditionally associated with depositing uniform areas of 
resistive materials for potentiometric devices, it is to be realised that 
by virtue of the non-contact-making relationship possible between 
signal-tapping member and resistive element, the latter can be formed of 
suitable materials chosen solely for the resistivity and uniformity of 
deposition, not the ability to withstand contact wear nor even contact 
with the atmosphere. The resistive element may thus be formed a thin film 
of pure metal or alloy requiring minimal trimming to give uniform 
resistivity and, if the metal or alloy is prone to oxidation, a protective 
dielectric coating may be provided.