8.2 Control Synchros, Resolvers & Inductive Sensors

8.2 Control Synchros, Resolvers & Inductive Sensors

While torque synchro systems (TX-TR) are ideal for driving light instrument pointers, they cannot deliver significant mechanical power. Attempting to position a heavy flight control surface, an autopilot hydraulic power control unit (PCU), or an airborne weather radar antenna pedestal directly with a torque receiver causes massive positional lag, excessive stator heating, and loss of synchronization. In high-power aerospace positioning, aircraft design relies on Control Synchros, Resolvers, and solid-state Inductive/Capacitive Transmitters.

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Control Synchro Systems: CX & CT

A control synchro system replaces the direct mechanical drive of a torque receiver with an electromechanical closed-loop servo system consisting of a Control Transmitter (CX), a Control Transformer (CT), a high-gain Servo Amplifier, and an AC/DC Servomotor coupled to a mechanical gear train:

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1. Control Transmitter (CX)

The physical construction of a Control Transmitter (CX) is virtually identical to a Torque Transmitter (TX). It consists of a laminated stator with three wye-connected single-phase windings spaced $120^\circ$ apart (S1, S2, S3) and a salient-pole dumbbell rotor energized through two slip rings (R1, R2) by the aircraft $400\text{ Hz}$ reference bus ($115\text{ V}$ or $26\text{ V AC}$). The CX converts its shaft mechanical angle into a set of three single-phase stator voltages that establish the direction of the system command vector.

2. Control Transformer (CT) Constructional Differences

Although a Control Transformer looks externally similar to other synchros, its internal design differs fundamentally from a torque receiver (TR):

1. Cylindrical (Non-Salient) Drum Rotor: The CT rotor core is completely cylindrical with slots housing a single distributed winding. Unlike the salient dumbbell rotor of a TX/TR, a cylindrical rotor presents a completely uniform air gap and constant magnetic reluctance to the stator flux at all rotor angles. Consequently, the stator magnetic flux exerts zero magnetic pull or reaction torque on the CT rotor shaft. The CT rotor can be positioned by delicate instrument gear trains without reflecting mechanical loading back into the system.
2. High-Impedance Stator Windings: The CT stator coils are wound with many turns of finer-gauge wire, resulting in a much higher input impedance than a TR stator. This ensures that the CT draws negligible circulating current from the CX, preventing voltage drops and thermal loading along the interconnect lines.
3. De-energized Rotor Output: The CT rotor winding is NEVER connected to the AC reference power supply. Instead, its rotor terminals (R1 and R2) serve purely as an electrical signal generator, delivering an AC error voltage into the high-input-impedance terminals of a servo amplifier.

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The $90^\circ$ Electrical Null & Error Generation

In a control synchro loop, the stator coils of the CT are energized by the stator voltages transmitted from the CX. These currents establish an alternating magnetic flux in the CT core whose spatial angle matches the shaft position of the command transmitter ($\theta_{CX}$):

The Mathematical Error Formulation

The voltage induced in the CT rotor winding ($V_{error}$) depends on the physical orientation of the rotor coil relative to this stator magnetic field:

$$V_{error}(t) = V_{max} \sin(\theta_{CX} - \theta_{CT}) \sin(\omega t + \phi)$$

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Why is it Called the $90^\circ$ Electrical Null?

• When the CT rotor shaft is mechanically aligned with the CX command shaft ($\theta_{CT} = \theta_{CX}$), the physical turns of the CT rotor coil are oriented exactly perpendicular ($90^\circ$) to the direction of the stator magnetic flux lines.
• Under Faraday's Law of Induction, induced voltage is proportional to the rate of change of magnetic flux linking the coil: $e = -N (d\Phi/dt)$. When the coil is perpendicular to the flux lines, the net flux cutting through the winding turns is zero. Hence, the induced voltage across terminals R1 and R2 is zero ($V_{error} = 0$). This zero-signal position is defined as the $90^\circ$ electrical null.

Off-Null Error Dynamics & Phase Sensitivity

When the pilot or flight control computer commands a new position, $\theta_{CX}$ changes, altering the angle of the CT stator field. Because the CT rotor is momentarily stationary, misalignment exists ($\Delta \theta = \theta_{CX} - \theta_{CT} \neq 0$):

• Voltage Magnitude: An AC voltage appears across CT terminals R1 and R2. For angular errors up to $\pm 15^\circ$, $\sin \Delta \theta \approx \Delta \theta$ (in radians), making the error voltage magnitude strictly linear with angular error ($V_{error} \propto \Delta \theta$).
• Phase Polarity: The phase of $V_{error}$ relative to the $400\text{ Hz}$ reference bus indicates the direction of misalignment:
  • If $\theta_{CX} > \theta_{CT}$ (clockwise error), $V_{error}$ is in-phase ($0^\circ$) with the reference.
  • If $\theta_{CX} < \theta_{CT}$ (counter-clockwise error), $V_{error}$ is $180^\circ$ out-of-phase with the reference.
• Servo Nulling Action: The phase-sensitive servo amplifier senses this magnitude and phase, supplying drive power of the correct polarity to the servomotor. The motor drives both the heavy flight surface load and the CT rotor shaft via a mechanical feedback gear train until the CT rotor re-establishes the $90^\circ$ perpendicular relationship. At that point, $V_{error} \to 0$, the motor stops, and the system locks in rigid alignment.

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Resolvers: Physical Architecture & Coordinate Conversion

A resolver is a specialized, high-precision electromagnetic rotary sensor designed for analog trigonometric computation and angular position measurement.

Construction: Orthogonal $90^\circ$ Windings

Unlike synchros, which utilize $120^\circ$ wye-connected stator windings, a resolver uses two-phase orthogonal windings:

• Stator: Contains two separate windings physically and electrically positioned at $90^\circ$ to each other in space (labeled Stator 1 / Stator 2, or $S_{1-3}$ and $S_{2-4}$).
• Rotor: Contains two separate windings also physically and electrically positioned at $90^\circ$ to each other in space (labeled Rotor 1 / Rotor 2, or $R_{1-3}$ and $R_{2-4}$), terminated via four slip rings.

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Mathematical Formulation

When a single rotor winding (e.g., Rotor 1) is energized with reference voltage $v_{ref}(t) = V_{in} \sin(\omega t)$, the alternating magnetic field couples into the two orthogonal stator windings in proportion to the sine and cosine of the rotor shaft angle ($\theta$):

$$V_{S1-S3} = K \cdot V_{in} \sin(\theta) \sin(\omega t)$$ $$V_{S2-S4} = K \cdot V_{in} \cos(\theta) \sin(\omega t)$$

where $K$ is the transformation ratio.

Primary Avionics Applications

1. Radar Coordinate Transformation (Polar to Cartesian): Airborne weather radar and search radar detect targets in polar coordinates: slant range ($R$) and antenna azimuth angle ($\theta$). Cockpit plan position indicator (PPI) displays and digital sweep scan converters require Cartesian coordinates ($X$ and $Y$ raster coordinates): $$X = R \cos(\theta)$$ $$Y = R \sin(\theta)$$ By applying a linear range sweep ramp voltage ($V_{range} \propto R$) to the rotor of an antenna-driven resolver, the two orthogonal stator windings directly output the instantaneous $X$ and $Y$ sweep voltages, driving the display deflection plates or digitizers.
2. Inertial Reference Units (IRU / INS): Resolvers mounted on the gimbal axes of stabilized gyro platforms perform real-time Euler angle transformations, resolving aircraft roll, pitch, and yaw rates from the aircraft body frame into the local-vertical geographic navigation frame (North, East, Down).

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Inductive & Capacitive Transmitters in Aircraft Sensing

Beyond rotary angle transmission, avionics instrumentation utilizes variable-reluctance electromagnetic and capacitive sensors for linear and pressure parameter sensing. The Part-66 syllabus names these two families explicitly: inductance transmitters, which vary mutual inductance or magnetic reluctance (the E-and-I bar transmitter below, together with the LVDT and RVDT of Section 7.2), and capacitance transmitters, which vary dielectric capacitance:

1. E and I Bar Differential Transformers (Inductance Transmitters)

The E-and-I bar transmitter is an inductive sensor widely used in cabin altimeters, engine pressure ratio (EPR) indicators, and manifold pressure transducers:

• Construction: A stationary laminated magnetic core shaped like the letter "E" carries three coils. The center leg carries a primary winding energized by $400\text{ Hz}$ AC. The two outer legs carry identical secondary sensing coils connected in series-opposing (differential connection). A movable laminated armature shaped like the letter "I" is pivoted above the pole faces, mechanically connected to a capsule (such as a Bourdon tube, evacuated aneroid bellows, or differential diaphragm).
• Operation: When the I-bar is centered (pressure neutral), the magnetic reluctance of the two outer magnetic loops is identical. Equal voltages are induced in both outer secondary coils. Because they are connected series-opposing, their voltages cancel, yielding a net output of $0\text{ V}$ (null).
• When pressure deflects the capsule, the I-bar tilts. This narrows the air gap on one outer leg (reducing reluctance and increasing secondary voltage) while widening the air gap on the opposite leg (increasing reluctance and lowering secondary voltage). The net output is an AC error voltage whose amplitude is proportional to capsule displacement (pressure magnitude) and whose phase ($0^\circ$ or $180^\circ$) indicates deflection direction.

2. Capacitance Transmitters: Fuel Quantity Probes

Aircraft fuel quantity indication systems (FQIS) require accurate mass measurement across extreme flight attitudes and temperatures ($-55^\circ\text{C}$ to $+50^\circ\text{C}$):

• Construction: A capacitance probe consists of two concentric, open-ended aluminum tubes mounted vertically inside the aircraft fuel tank, acting as an electrical capacitor.
• Dielectric Physics: The capacitance of concentric cylindrical plates is governed by: $$C = \frac{2\pi \epsilon_0 \epsilon_r L}{\ln(r_o / r_i)}$$ where $\epsilon_r$ is the relative permittivity (dielectric constant) of the medium between the plates. Dry air has a dielectric constant of $\epsilon_{air} \approx 1.0$, whereas Jet-A/Jet-A1 aviation kerosene has a dielectric constant of $\epsilon_{fuel} \approx 2.1$.
• Linear Mass Sensing: As fuel fills the tank, it rises between the concentric tubes, replacing air with kerosene. This increases total capacitance linearly. Crucially, as fuel temperature changes, fuel volume expands or contracts (density changes), but its dielectric constant ($\epsilon_r$) shifts in direct proportion to density. Consequently, a capacitance FQIS measures true fuel mass (kilograms or pounds) rather than volume—vital because aircraft engine thrust, takeoff weight, and range calculations depend strictly on the mass of fuel available for combustion.

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Worked Engineering Calculations

Calculation 1: Control Transformer (CT) Error Voltage & Phase

A $400\text{ Hz}$ control synchro loop has a Control Transformer sensitivity rating of $K_{CT} = 1.05\text{ V AC RMS per degree}$ of angular misalignment. The flight computer commands a rudder trim change such that the CX rotor moves to $\theta_{CX} = +42.5^\circ$, while the CT follow-up shaft is at $\theta_{CT} = +38.0^\circ$.

Step 1: Calculate the angular misalignment error ($\Delta \theta$)

$$\Delta \theta = \theta_{CX} - \theta_{CT} = 42.5^\circ - 38.0^\circ = +4.5^\circ$$

Step 2: Compute the CT output error voltage ($V_{error}$)

Using the small-angle linear sensitivity:

$$V_{error} = K_{CT} \times \Delta \theta = 1.05\text{ V/deg} \times 4.5^\circ = 4.725\text{ V AC RMS}$$

Step 3: Determine phase angle

Because $\theta_{CX} > \theta_{CT}$ (positive error), $V_{error}$ is in-phase ($0^\circ$ phase shift) relative to the $400\text{ Hz}$ reference line.

Step 4: Evaluate opposite deflection

If the rudder trim commanded $\theta_{CX} = +33.5^\circ$ while $\theta_{CT} = +38.0^\circ$:

$$\Delta \theta = 33.5^\circ - 38.0^\circ = -4.5^\circ$$

$V_{error} = 4.725\text{ V AC RMS}$, with a $180^\circ$ phase shift relative to the reference line. Servomechanism Result: The servo amplifier detects the $180^\circ$ phase reversal, reversing the motor drive current to rotate the trim actuator counter-clockwise back to null.

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Calculation 2: Resolver Radar Coordinate Conversion

An airborne weather radar antenna is scanning at an azimuth angle of $\theta = 60.0^\circ$. The transmitter produces an instantaneous range sweep ramp voltage of $V_{in} = 12.0\text{ V}$ across the resolver Rotor 1 winding ($R_{1-3}$), representing an airborne weather return at a slant range of $R = 40.0\text{ nautical miles}$. The resolver transformation ratio is $K = 1.00$.

Calculate the Cartesian display sweep voltages across the two orthogonal stator windings ($V_X$ across $S_{2-4}$ and $V_Y$ across $S_{1-3}$):

Step 1: Compute $X$-sweep (Cosine) Stator Voltage ($V_{S2-S4}$)

$$V_X = K \cdot V_{in} \cos(\theta) = 1.00 \times 12.0\text{ V} \times \cos(60.0^\circ) = 12.0 \times 0.5000 = +6.00\text{ V}$$

Step 2: Compute $Y$-sweep (Sine) Stator Voltage ($V_{S1-S3}$)

$$V_Y = K \cdot V_{in} \sin(\theta) = 1.00 \times 12.0\text{ V} \times \sin(60.0^\circ) = 12.0 \times 0.8660 = +10.392\text{ V}$$

Step 3: Verify vector magnitude

$$V_{resultant} = \sqrt{V_X^2 + V_Y^2} = \sqrt{(6.00)^2 + (10.392)^2} = \sqrt{36.00 + 107.99} = \sqrt{143.99} \approx 12.00\text{ V}$$

Display Electronics Result: The $X$-deflection plates receive $+6.00\text{ V}$ and the $Y$-deflection plates receive $+10.39\text{ V}$, deflecting the electron beam or digital raster to the exact screen position representing $40\text{ NM}$ at $60^\circ$ azimuth.