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.