A circuit generates an analog voltage proportional to an angle, in response to two sinusoidal input voltages having magnitudes proportional to the sine and cosine of the angle, respectively. That is to say, given input voltages proportional to sin(*ωt*)sin(θ) and sin(*ωt*)cos(θ) [where θ denotes the angle, *ω* denotes 2π × a carrier frequency, and *t* denotes time], the circuit generates a steady voltage proportional to θ. The output voltage varies continuously from its minimum to its maximum value as θ varies from –180° to 180°. While the circuit could accept input modulated sine and cosine signals from any source, it must be noted that such signals are typical of the outputs of shaft-angle resolvers in electromagnetic actuators used to measure and control shaft angles for diverse purposes like aiming scientific instruments and adjusting valve openings.

In effect, the circuit is an analog computer that calculates the arctangent of the ratio between the sine and cosine signals. The full-circle angular range of this arctangent circuit stands in contrast to the range of prior analog arctangent circuits, which is from slightly greater than –90° to slightly less than +90°. Moreover, for applications in which continuous variation of output is preferred to discrete increments of output, this circuit offers a clear advantage over resolver-to-digital integrated circuits.

The figure depicts the main functional blocks of the arctangent circuit. In addition to the aforementioned input signals proportional to sin(*ωt*)sin(θ) and sin(*ωt*)cos(θ), the circuit receives the carrier signal proportional to sin(*ωt*) as an auxiliary input. The carrier signal is fed to a squarer (block 7) to obtain an output square-wave or logic-level signal, LL[sin(*ωt*)]. The demodulator (block 1) uses LL[sin(*ωt*)] to demodulate input signal sin(*ωt*)cos(θ), generating an output proportional to cos(θ).

The carrier signal sin(*ωt*) is also fed to an integrator and inverter (block 8) to obtain a signal proportional to cos(?t). The cos(*ωt*) signal is fed to a squarer (block 9) to obtain a logic-level signal LL[cos(*ωt*)]. The cos(θ) and cos(*ωt*) signals are fed to a multiplier (block 2) to obtain a signal proportional to cos(θ)cos(*ωt*). This signal and the input sin(*ωt*)sin(θ) signal are fed to an inverter and adder (block 3) to obtain a signal proportional to –[cos(T)cos(*ωt*) + sin(T)sin(*ωt*)], which, by trigonometric identity, equals –cos(*ωt*– θ). This signal is processed by an inverter and squarer (block 4) to obtain a logic-level signal LL[cos(*ωt*– θ)].

The signal LL[cos(*ωt*)] from block 9 and the signal LL[cos(*ωt*– θ)] from block 4 have the same frequency but differ in phase by θ. These signals are fed as inputs to block 5, which contains logic circuitry that determines the magnitude and trigonometric quadrant of the phase difference, and generates a logic-level pulse width modulated signal, PWM(θ), in which the pulse width varies continuously with θ. The quadrant-detection function eliminates the difficulty, encountered in prior analog arctangent circuits, caused by the discontinuity of the tan(θ) at θ = ±90°.

PWM(θ) is fed to block 6, which responds by generating a PWM waveform that switches between precise reference voltage levels of +10 and –10 V. This waveform is processed by a two-pole, low-pass filter (block 10), which filters out the carrier-frequency component. The output of block 10 is a DC potential, proportional to θ, that ranges continuously from –10 V at θ = –180° to +10 V at θ = +180°.

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

This invention has been patented by NASA (U.S. Patent No. 6,138,131). Inquiries concerning nonexclusive or exclusive license for its commercial development should be addressed to

###### Sammy Nabors, MSFC Commercialization Assistance Lead, at (256) 544-5226 or This email address is being protected from spambots. You need JavaScript enabled to view it. .

*Refer to MFS-31219.*

##### NASA Tech Briefs Magazine

This article first appeared in the April, 2005 issue of *NASA Tech Briefs* Magazine.

Read more articles from the archives here.