6.302: Magnetic Levitation Design Project

Tony Hyun Kim

May 7, 2009

1 Introductory Overview

We have assembled and compensated a feedback-based magnetic levitation (“maglev”)device. Figure 1 shows a cartoon schematic of the mechanical layout, alongside anactual photograph of our system.

Figure 1: (Left) A cartoon schematic of the magnetic levitator. (Right) A photographof the actual apparatus. The magnet is inside the levitated box, which also containssome counterweights.

The position of the levitated object (containing a small neodymium magnet) is deter-mined by the Hall-effect sensor mounted at the bottom of the solenoid. The sensoroutput is processed by the accompanying circuit, which in turn drives the solenoid withthe goal of stabilizing the position of the levitated object. A block diagram of thisfeedback network is shown in Figure 2.

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Figure 2: Block diagram representation of the magnetic levitation device. The voltageinput was utilized only in the characterization of the device.

We now point out some unique features of our device:

1. The solenoid (and its chassis) and the drive electronics are machined as two sepa-rate parts, which are connected by a standard DB-9 connector. This arrangementminimizes clutter.

2. The drive electronics is contained in an attractive case, which has a DB-9 port forcommunication with the solenoid/sensor, a BNC input port for the +15V powersource, and an LED power indicator.

3. The drive electronics has a DPDT switch that swaps the orientation of currentflow through the solenoid. This feature was very handy in the initial constructionof the maglev.

In other words, we have spent some time preparing a nice packaging for the maglev.My product will be submitted to the 6.302 “most artistic” competition, where I take“artistic” to mean a clean, “industrial look-and-appeal”. Figure 3 shows photos of themaglev electronic box.

2 Theory of magnetic levitation

We present a brief summary of magnetic levitator theory, which guided our compensatordesign strategy. The discussion follows that found in the 6.302 course notes[5], inparticular pp. 76-79.

The levitated object is acted on by two forces: gravity, and the magnetic force from thesolenoid. The gravitational force is important in determining the large-signal operatingpoint of the levitation, but is not relevant for the small-signal analysis.

For the small-signal analysis, let xb represent the levitation position, and im be thecurrent through the solenoid. The small-signal dynamics is captured by the following

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Figure 3: Front and rear panels of the maglev electronic box. Features includeLED power indicator, BNC +15V power input, DB-9 connector to communicate tosolenoid/sensor, solenoid polarity DPDT switch.

differential equation:mxb = kXxb − kIim (1)

where m is the mass of the object, and kX and kI are constants. This differentialequation can be easily transformed into a transfer function:

Xb(s)

Im(s)= − kI

ms2 − kX

. (2)

We conclude: The (open-loop) maglev is a two-pole system with poles at s = ±√

kX

m.

3 Characterization of the basic system

The electronic signal chain is as follows:

• The SD495 Hall-effect sensor, mounted at the bottom of the solenoid, measuresthe position of the levitated object.

• In the basic system, the sensor output is connected directly to the input of theMIC502 fan-management chip.

• The fan-management chip produces a pulse-width modulated (PWM) drive signalto the LMD18201 H-bridge chip. The PWM scheme adjusts the average currentin the solenoid (i.e. im in Eq. 1).

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The schematic of the basic system is shown in Figure 4. We have found surprisinglygood performance under visual inspection, even with this bare-bones system. (Themass of the levitated object was cruicial, however.)

Figure 4: Schematic of the bare-bones maglev system.

In order to quantitatively characterize the performance, we added a control input port,which was added to the sensor reading before it was fed back to the fan-managementchip. The step-response is shown in Fig. 5. We have filtered the sensor output with apassive RC filter with τ = RC = 0.1 ms.

Figure 5: The blue 50mV step is the input step to the system. The top, orangetrace shows the corresponding sensor response. Note the persistent ripple in the stepresponse.

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In particular, note the 1kHz, persistent ripple. We will design a compensator circuitwith the intent of removing this oscillation.

4 Compensation

Our compensation strategy is based on root-locus diagrams of the basic two-pole system.Figure 6a shows the root-locus of the basic two-pole system. For any gain, the closed-loop poles are located on the jω-axis, which indicates perpetual oscillations. Thisis consistent with the observation of long-lived oscillations in the basic-system stepresponse.

Figure 6: (a) Root locus diagram of the basic two-pole system. (b) Root locus diagramof the compensated system. The system poles can be pulled into the left-half plane.This root locus diagram is only qualitative, since the trajectory is heavily dependenton the actual location of the open-loop poles and zeroes.

Therefore, our compensation strategy is to include a zero and a pole, far to the leftof the system poles. (The pole is added because I didn’t know of a convenient circuitto add only a zero.) As seen in the root locus diagram of Fig. 6b, the system polescan be “pulled” into the left-half plane (LHP), which corresponds to damping of theoscillations.

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Note that the system poles can be pulled further into the LHP by placing the compen-sator pole further to the left. This was our main design consideration.

We have utilized the compensation circuit as in Figure 7. This circuit fragment, which

Figure 7: The compensator circuit (a lead network). The opamp (non-inverting am-plifer) stage “undoes” the attenuation due to the initial passive circuit stage. We usedR1 = 100kΩ, R2 = 1kΩ and C = 0.1µF .

incidentally is a standard lead-network, has the transfer function:

Vout

Vin

=sR1C + 1

s(R1||R2)C + 1≈ sR1C + 1

sR2C + 1(3)

where the last equality is valid under the approximation R2 << R1. By choosingR1 = 100kΩ and R2 = 1kΩ, we place the pole a hundred times further out along thenegative axis in the s-plane.

In the actual implementation, we have also included a variable gain stage (non-invertingopamp amplifier) in order to scan the system root locus (as in Fig. 6b).

The compensated system was characterized by a step response, shown in Figure 8.Note that the undesirable ripple has been completely eliminated. The downside ofthis compensation scheme, however, is that the speed of the circuit has been greatlyreduced. The new time-constant is roughly 40 times larger than the original system.

5 Conclusion

We have assembled and compensated a magnetic levitation system. The compensationwas designed in order to remove the long-lived high frequency ripples in the basicsystem, and was derived from a theoretical root-locus analysis.

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Figure 8: The step response of the compensated system. The blue trace represents thestep input, and the orange trace is the sensor output.

Most importantly, the system was manufactured with an attractive appearance in mind;namely, a sleek, industrial design.

References

[1] Honeywell. SS490 Series Datasheet. (Hall-effect sensor)

[2] Microchip Technology Inc. MCP601/1R/2/3/4 Datasheet, 2007. (Single-supply opamp.)

[3] Micrel, Inc.. MIC502 Datasheet, March 2003. (Fan Management IC)

[4] National Semiconductor Corporation. LMD18201 3A, 55V H-Bridge, 2004. (H-Bridge IC)

[5] Lundberg, Kent. Feedback Systems for Analog Circuit Design, 6.302 Spring 2009Notes.

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