### How they work

A resolver is an electromechanical transducer that converts shaft angle to an absolute analog signal. Extremely rugged, it can be mounted anywhere an angle (down to 0.1 arcminute) needs to be measured without regard to dust, oil, temperature, shock, and vibration.

Essentially a voltage transformer, a resolver consists of a one or two-phase primary and a two-phase secondary. For a given input (a sinusoid of magnitude *V*), the output varies with the sine and cosine of the shaft angle θ. The outputs *V* sinU and *V cos* θ are simply the components of the input vector, *V*.

Exciting both input phases, say with voltages *A* and *B*, produces two outputs whose magnitudes are of the form:

*Y = A sin θ + B cos θ X = A cos θ - B sin θ*

For applications where conventional commutation with slip rings and brushes is undesirable, brushless versions are available. However, they normally require more power and introduce more phase shift into the control loop.

There are several ways to read a resolver. If the goal is to produce a digital output, consider the “phase analog” technique. Here, two sinusoids (in phase quadrature) are placed across the primary (stator) windings. The phase of the induced voltage on the secondary (rotor) winding varies with shaft angle, while amplitude and frequency remain constant. Accuracy is determined by the exactness at which zero crossing intervals are measured.

Resolvers take an input voltage from 0.5 to 115 Vrms (60 Hz to 100 kHz). Although they can operate at voltages and frequencies other than those specified, it’s important not to exceed the recommended input current; otherwise the iron core may saturate. A convenient way of avoiding excessive current, while changing frequency, is to change the voltage proportionally with frequency. Doubling the frequency means that the voltage can safely be doubled; halving the frequency means that the voltage should also be halved – and so on.

Input current is the current, in amps, flowing through the primary winding at rated voltage and frequency. It is typically very low (less than 100 mA). Input power, usually less than 1 W, is the product of the square of the input current and the real (resistive) portion of the input impedance.

### General sizing procedures

Of all the parameters associated with resolvers, accuracy is probably the most important. It can be measured several ways. One way is to look at each winding separately, comparing actual and theoretical voltage values.

Another way of specifying accuracy is interaxis error, the angular deviation of null positions at 0, 90, 180, and 270°. It is expressed in arcminutes or arcseconds. The lower the interaxis error, the more accurate the resolver.

Linearity error (associated with linear transformers) is also used. It’s a measure of the nonconformity of the secondary voltage over the entire range of rotation. It is expressed as a percentage of the secondary voltage at the maximum excursion. In general, the more linear, the more accurate.

Another measure of accuracy is velocity error. Many resolver applications operate in a virtual static mode – with low rotational velocities – but the increased use of brushless resolvers in high-speed motor control makes velocity error tests one of the most common methods for measuring accuracy.

Velocity errors are rotationally generated voltages (the result of conductors moving through a magnetic field) that couple into the output. Although they are difficult to measure directly, a handy formula can tell you where they’re likely to occur.

Synchronous velocity *S _{c}* is a function of system frequency:

*S _{c}* = 120

*f /p*rpm

where *f* = system excitation frequency (Hz) and *p* = number of poles.

As a rule-of-thumb, speeds above onequarter synchronous speed cause serious velocity errors and should be avoided.

Voltage sensitivity, also important, is the output voltage expressed as a function of the shaft angle in mV/degree. This parameter (also referred to as voltage gradient) is usually specified at a shaft angle of 1°. It can be calculated by multiplying the output voltage at maximum coupling by the sine of 1°.

Other parameters of significance include transformation ratio, phase shift, and null voltage. Transformation ratio (TR) is the ratio of output voltage to input voltage when the output is at maximum coupling. Practical TRs are usually between 0.1 and 1.0. TRs greater than 1.0 are possible, depending on the design of the unit. Common values for TR are 0.454, 0.5 and 1.0.

Phase shift (expressed in electrical degrees) is the time-phase difference between the primary and secondary voltages at maximum coupling. Generally, single-speed synchros and resolvers have leading phase shifts between 0 and 20°. At 400 Hz, the nominal phase shift can be approximated as the arctangent of the ratio of the primary winding dc resistance *R _{dc}* to its reactive component

*X*.

_{L}Null voltage is the residual voltage remaining when the inphase component of the output voltage is zero. When primary and secondary windings are perpendicular (as at electrical zero), there should be no voltage induced in the secondary winding. However, mechanical imperfections, winding errors, and distortions in the magnetic circuit (such as grinding smear), cause some voltage to appear in the output winding at the minimum coupling position.

The null voltage comprises three components: in-phase fundamental, quadrature fundamental, and harmonics. The in-phase fundamental component is an angular inaccuracy that can be canceled by re-nulling the rotor, thereby introducing an error (sometimes called null spacing error). Quadrature voltage is 900 out of phase with the in-phase component and cannot be nulled by rotor rotation. The harmonic voltages consist predominantly of the third harmonic, which is three times the excitation frequency.

Null voltages are usually specified as total null voltage, which is the total of the quadrature fundamental and harmonics. Depending on size, input voltage, and input frequency, the total null voltage is approximately 1 to 3 mV/V of input voltage. The fundamental null voltage is usually slightly less than or equal to the total null voltage.

### Common applications

Besides measuring shaft angle – radial position and velocity – resolvers are often used to commutate brushless motors. Usually, the rotor winding is excited by an ac signal and the output is taken from the two stator windings. Both outputs have nearly the same time-phase angle as the original signal, but their amplitudes are modulated by the sine and cosine of the rotor position angle as the shaft rotates. The outputs are fed to either a receiving- type resolver (resolver chain) or an analog-to-digital converter.

Compared to standard commutation components – Hall effect switches and magnetic code wheels – resolvers withstand higher temperatures, operating at over 180°C. Hall devices are usually limited to 125°C. And they can‘t give you accurate position and velocity readings.

### Failure modes

Since they are inductive devices, synchros and resolvers have a very long life expectancy. The shortest lifetimes would be expected for units with brushes and slip rings, the longest for true brushless units using rotary transformers.

An average value for mean time between failures (MTBF) cannot be stated because of the many factors that influence the calculation. The best guide available for MTBF calculations is in the U.S. Department of Defense Handbook, MIL-HDBK-217. The book lists many factors that affect synchro and resolver failure rates, and defines a method of estimating them, taking into account ambient temperature, temperature rise, type of winding insulation, size of unit, number of brushes, and type of application.

As an example, for a Size 18 synchro with two brushes operating in a ground fixed service environment, with a frame temperature of 70°C, the failure rate calculates to 0.06 failures/million hours. The MTBF is thus 16.7 million hours or 1,906 years.

In electromagnetic brushless devices, energy is transferred to or from the rotor by means of a circular rotary transformer mounted in tandem with the synchro or resolver. Since there are no physical connections to the rotor, the life of the unit is limited solely by the life of the bearings. Tens of thousands of hours of operational life at high rotational speeds are within reach with this type of unit.

Other things to consider regarding lifetime and failure include dielectric withstanding voltage and insulation resistance.

Dielectric withstand voltage testing consists of applying a voltage higher than the rated voltage, for a specific time, between mutually-insulated windings and between insulated windings and ground. This is also called a high potential, Hi- Pot, or dielectric strength test. This test is used to prove that a component can operate safely at its rated voltage and withstand momentary over-potentials due to switching surges. It determines whether the insulating material and the separation of the component parts is adequate.

Insulation resistance is a measure of the resistance provided by insulating components to an applied direct voltage. Inadequate resistance lets current leak through. Usually, insulation resistance values in synchros and resolvers are very high (greater than 50 MΩ). Excessive leakage currents can breakdown insulation (through heat and electrolysis) and destroy sensitive circuits.

Insulation resistance measurements should not be considered the equivalent of dielectric withstanding voltage. Clean, dry insulation may have a high insulation resistance, yet have a mechanical fault that could cause failure in a dielectric withstanding voltage test. Conversely, dirty, deteriorated insulation with a low insulation resistance may pass a dielectric voltage test.

### Installation and troubleshooting

Resolvers are generally checked for mounting dimensions and (on housed units) for conformance to shaft end play, shaft radial play, and starting friction requirements.

Shaft end play is the total axial motion of the shaft when an eight-ounce reversing load is applied along the axis. Shaft radial play is the total side-to-side motion of the shaft measured as close to the bearing as possible, with a four-ounce reversing load applied radially to the shaft within 0.25 in. of the bearing. Starting friction is the torque necessary to overcome the internal friction of the bearings and brushes and initiate rotor rotation.

Since most resolvers come in standard sizes and are electrically coupled to the rotor by slip rings, hairsprings, or transformers, the mounting considerations are minimal. The two major areas of concern are the housing mounting and the shaft coupling.

Normally the housing is mounted into a pilot diameter and then secured by means of synchro clamps or screws entering the mounting face. If clamps are used, two or four point contact is a must.

Coupling arrangements include solid couplings, bellows-type couplings, and gear coupling. Be careful not to apply too much radial loading to the shaft.

During installation, unhoused units obviously require the most care. Stators and rotors should be positioned precisely. In no case should any part be pressed into holes or onto shafts, as this could cause physical or magnetic damage.

Improper mounting – whether it’s caused by system tolerance buildup or defective system hardware – comes at a price. Axial offset, for example, increases electrical error, phase shift, and power consumption. It also affects the null voltage and transformation ratio. Radial offset also decreases the electrical accuracy and increases null voltage, but has a lesser effect on TR, phase shift, and input power.

Rotor or stator tilt is another concern. If it’s slight – less than 0.002 in. with respect to each other – very little change will occur in any parameter. Tilts greater than this must be avoided in units with small air gap clearances to prevent rotor and stator from making contact.

### Temperature effects

Resolver error is determined during design and manufacturing and, theoretically, shouldn’t change. However, mounting stresses and differential expansion in the lamination stack, windings, and hardware due to temperature changes, may cause variations.

In general, single-speed resolvers are more sensitive than multispeed units. The magnitude can be anywhere from a few arcseconds to a few minutes, depending on the device.

Even though there may be no change in the shape of the error curve, there may be a dc bias. Minute mechanical changes due to temperature fluctuations may cause the electrical zero to shift. The socalled EZ shift cannot be predicted or calculated, and varies from unit to unit.

Temperature induced resistive changes in the copper magnet wire have an effect on phase shift. The coefficient of resistance of copper is approximately 0.4%/°C; phase shift changes by the same percentage. Heat increases resistance and phase shift; cold does the opposite.

Transformation ratio changes, related to changes in the phase shift, must also be considered. They are proportional to the ratio of the cosine of the phase shift at the new temperature to that at room ambient. Higher temperatures mean lower transformation ratios, and vice versa. The new value may be calculated as follows:

*TR _{2}* =

*TR*cosΦ

_{1}_{2}/ cosΦ

_{1}

where *TR _{1}* = transformation ratio at room ambient (25°C)

Φ

_{1}= phase shift at room ambient

Φ

_{2}= calculated phase shift

Impedances, input current, and input power are also affected by changes in the resistance of the copper magnet wire. So is the null voltage. As temperatures change, mounting stresses build as do the effects of differential expansion in the lamination stack, winding, and hardware. The changes vary with unit type and from unit to unit. As a rule-ofthumb, the maximum change (in null voltage) is less than 1%/°C.

### Cost saving strategies

One way to save money with resolvers is to remember that an R/D converter is not necessarily required when commutating brushless motors. If the only requirement is to rotate the motor, accurate position data are probably not that important. In that case, a synchronous demodulator can be used to process sensor feedback. The significant parameter is the zero crossing frequency, and careful mechanical design can make this very accurate.

Another cost-cutting tip has to do with the excitation electronics. Normally, the carrier frequency should be ten times greater than the highest frequency of the motor. Because the shape of the carrier has little effect on circuit performance, a square wave is worth considering because it is the least expensive to generate.

A sine wave carrier is optimum for system noise reduction, but there are other ways of dealing with noise like using a tracking converter. Modern converters are inexpensive, accurate, and immune to noise.

Tracking converters operate ratiometrically, using only the ratio of the sine and cosine stator outputs of a rotor excited resolver. Since the resolver acts as a transformer, any carrier distortion or amplitude variation has little effect on accuracy. Frequency variation and incoherent noise also have little impact because they’re filtered out by a phase demodulator.

### Did you know?

Resolvers can be used to transform coordinates from one system to another. Spacecraft and aircraft usually require pitch, yaw, and roll to be transformed back to earth references. One resolver readily handles a two-axis transformation; three are needed for three axes.

Resolver chains are also employed to solve trigonometric problems and for phase shifting. Using a balanced *RC* network and a stable frequency source, resolver- based phase shifters can achieve 0.25° accuracy or better.

*This month’s handy tips are brought to you by Joe Spetzer and Bill Ekhaml of Litton Poly-Scientific, Blacksburg, Va. Joe, a Senior Engineer, can be reached at (610) 328-4000, ext. 339. or [email protected]. Bill is a Senior Design Engineer and can be reached at (800) 336-2112 ext. 325 or [email protected].*

*For more information on resolvers and other feedback devices, visit www.motionsystemdesign.com.*