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Electromechanics
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function of the coil current: N i 0 ax Ni = = R(x) 2lgap The force in the air gap is given by the expression fgap = (Ni 0 ax)2 ( 2lgap ) 2 dR(x) 0 a(N i)2 = = 2 2 2 dx 8lgap 0 ax 4lgap
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Thus, the force in the gap is proportional to the square of the current, and does not vary with plunger displacement Calculation of magnetomotive force To determine the required magnetomotive force, we observe that the magnetic force must overcome the mechanical (restoring) force generated by the spring Thus, fgap = kx = ka For the stated values, fgap = (10 N/m) (001 m) = 01 N, and Ni = 4lgap fgap = 0 a 4 0001 01 = 564 A t 4 10 7 001
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The required mmf can be most effectively realized by keeping the current value relatively low, and using a large number of turns
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Comments: The same mmf can be realized with an in nite number of combinations of
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current and number of turns; however, there are trade-offs involved If the current is very large (and the number of turns small), the required wire diameter will be very large Conversely, a small current will require a small wire diameter and a large number of turns A homework problem explores this trade-off
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EXAMPLE 1611 Transient Response of a Solenoid
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Problem
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Analyze the current response of the solenoid of Example 1610 to a step change in excitation voltage Plot the force and current as a function of time
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Solution
Known Quantities: Coil inductance and resistance; applied current Find: Current and force response as a function of time
turns V = 12 V Rcoil = 5
Schematics, Diagrams, Circuits, and Given Data: See Example 1610 N = 1000
Assumptions: The inductance of the solenoid is approximately constant, and is equal to
the midrange value (plunger displacement equal to a/2)
Analysis: From Example 1610, we have an expression for the reluctance of the solenoid:
Rgap (x) =
2lgap 0 ax
Using equation 1630, and assuming x = a/2, we calculate the inductance of the structure: L N2 N 2 0 a 2 106 4 10 7 10 4 = = = 314 mH Rgap |x=a/2 4lgap 4 10 3
16
Principles of Electromechanics
The equivalent solenoid circuit is shown in Figure 1642 When the switch is closed, the solenoid current rises exponentially with time constant = L/R = 63 ms As shown in 5, the response is of the form: V V 12 3 (1 e t/ ) = (1 e Rt/L ) = (1 e t/63 10 ) A R R 5 To determine how the magnetic force responds during the turn-on transient, we return to the expression for the force derived in Example 1610: i(t) = fgap (t) = 0 a(N i)2 4 10 7 10 2 106 2 = i (t) = i 2 (t) 4lgap 4 10 3 12 3 (1 e t/63 10 ) 5
The two curves are plotted in Figure 1642(b)
Solenoid Current and force response 4 Current, A 20 Force, N
t=0 V
Rcoil Lcoil
2 Current Force 0 0 001 002 003 Time, s (b) 004
0 005
Figure 1642 Solenoid equivalent electrical circuit and step response Comments: The assumption that the inductance is approximately constant is not quite
accurate The reluctance (and therefore the inductance) of the structure will change as the plunger moves into position However, allowing for the inductance to be a function of plunger displacement causes the problem to become nonlinear, and requires numerical solution of the differential equation (ie, the transient response results of 5 no longer apply) This issue is explored in a homework problem
Practical Facts About Solenoids Solenoids can be used to produce linear or rotary motion, either in the push or pull mode The most common solenoid types are listed below: 1 Single-action linear (push or pull) Linear stroke motion, with a restoring force (from a spring, for example) to return the solenoid to the neutral position 2 Double-acting linear Two solenoids back to back can act in either direction Restoring force is provided by another mechanism (eg, a spring)
3 Mechanical latching solenoid (bistable) An internal latching mechanism holds the solenoid in place against the load 4 Keep solenoid Fitted with a permanent magnet so that no power is needed to hold the load in the pulled-in position Plunger is released by applying a current pulse of opposite polarity to that required to pull in the plunger 5 Rotary solenoid Constructed to permit rotary travel Typical range is 25 to 95 Return action via mechanical means (eg, a spring) 6 Reversing rotary solenoid Rotary motion is from one end to the other; when the solenoid is energized again it reverses direction
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