The upper part of the figure illustrates the major functional blocks of a direction-sensitive analog tachometer circuit based on the use of an unexcited two-phase brushless dc motor as a rotation transducer. The primary advantages of this circuit over many older tachometer circuits include the following:

- Its output inherently varies linearly with the rate of rotation of the shaft.
- Unlike some tachometer circuits that rely on differentiation of voltages with respect to time, this circuit relies on integration, which results in signals that are less noisy.
- There is no need for an additional shaft-angle sensor, nor is there any need to supply electrical excitation to a shaft-angle sensor.
- There is no need for mechanical brushes (which tend to act as sources of electrical noise).
- The underlying concept and electrical design are relatively simple.

This circuit processes the back-electromagnetic force (back-emf) outputs of the two motor phases into a voltage directly proportional to the instantaneous rate (sign · magnitude) of rotation of the shaft. The processing in this circuit effects a straightforward combination of mathematical operations leading to a final operation based on the well known trigonometric identity (sin *x*)^{2} + (cos *x*)^{2} = 1 for any value of *x*. The principle of operation of this circuit is closely related to that of the tachometer circuit described in "Tachometer Derived From Brushless Shaft-Angle Resolver" (MFS- 28845), *NASA Tech Briefs*, Vol. 19, No. 3 (March 1995), page 39. However, the present circuit is simpler in some respects because there is no need for sinusoidal excitation of shaft-angle resolver windings.

The two back-emf signals are *kθ̇*sin *θ* for phase A and *k*θ̇cos θ for phase B, where *k* is a constant that depends on the electromagnetic characteristics of the motor, θ is the instantaneous shaft angle, and the overdot signifies differentiation with respect to time. Note that θ̇ is the quantity that one seeks to measure.

Each back-emf signal is fed to one of two inputs of a multiplier circuit of gain *k*_{2} dedicated to its respective phase. Each of these signals is also integrated with a suitable time constant and gain to obtain a voltage of *k*_{1}sin θ for phase A and *–k*_{1}cos θ for phase B (where *k*_{1} is a constant that incorporates the combined effects of the gain and the time constant). The output of the integrator for phase B is inverted to obtain a voltage *k*_{1}cos θ. Each of these signals is fed to the other input terminal of the multiplier circuit for its respective phase.

The multiplier circuit for phase A thus generates an output signal proportional to both of its inputs; namely *k*_{3}θ̇(sin θ)^{2}, where *k*_{3} = *k*_{1}*k*_{2}. In a similar manner, the multiplier circuit for phase B generates an output signal of *k*_{3}θ̇(cos θ)^{2}. These signals are fed to an adder circuit. By virtue of the identity (sin θ)^{2} + (cos θ)^{2} = 1, the output of the adder is simply *k*_{3}θ̇ .

The lower part of the figure illustrates the major functional blocks of a direction-insensitive analog tachometer that, except for its lack of directionality, offers the same advantages as does the analog tachometer described above. However, this circuit is conceptually simpler in that it does not contain integrators.

This circuit processes the back-emf outputs of the two motor phases into a voltage directly proportional to magnitude of the instantaneous rate of rotation of the shaft. As in the circuit described above, the processing in this circuit effects a straightforward combination of mathematical operations leading to a final operation based on the identity (sin *x*)^{2} + (cos *x*)^{2} = 1 for any value of *x*.

Further as in the circuit described above, the two back-emf signals are *k*θ̇sin θ for phase A and *k*θ̇cos θ for phase B, where *k* is a constant that depends on the electromagnetic characteristics of the motor. In the present case, the quantity that one seeks to measure is ⎪θ̇⎪.

Each back-emf signal is fed to a dedicated squaring circuit. The outputs of the squaring circuits for phases A and B are thus proportional to (θ̇sin θ)^{2} and (θ̇cos θ)^{2}. The outputs of the squaring circuits are fed to an adder. By virtue of the identity (sin θ)^{2} + (cos θ)^{2} = 1 the output of the adder is proportional to θ̇^{2}; this output is fed to a square-root circuit to obtain a final output proportional to ⎪θ̇⎪.

*This work was done by David E. Howard and Dennis A. Smith of Marshall Space Flight Center. *

*This invention has been patented by NASA (U.S. Patent No. 6,084,398). Inquiries concerning nonexclusive or exclusive license for its commercial development should be addressed to*

###### Sammy Nabors

MSFC Commercialization Assistance Lead

at This email address is being protected from spambots. You need JavaScript enabled to view it..

*Refer to MFS-31142/3.*

##### NASA Tech Briefs Magazine

This article first appeared in the November, 2007 issue of *NASA Tech Briefs* Magazine.

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