Homework # 8 EE517 Fall 2000 Due October 24, 2000

In Woodson & Melcher start reading chapter 4.

Problem 1

Consider the inductor shown below with core cross sectional area Ac and window area Aw.

- Make a drawing of the inductor indicating the core cross sectional area A
_{c}. - Make a second drawing of the inductor indicating the window area A
_{w}. - Your are given that the inductor has N turns and that its maximum rms current is I
_{rms}. Also the maximum rms current density in the copper is J_{rms}. What is the minimum required wire cross sectional area a_{w}? - Your are given that the winding factor is K
_{w}. What is the minimum window area A_{w}? - The inductor is driven with a sine wave voltage with peak value V
_{p}and angular frequency w. What is the peak B field? - If the peak allowed B field is B
_{sat}, what is the minimum area of the core that is required in terms of the peak voltage V_{p}?

Problem 2

You are to design the actuator below. The actuator is to produce a force of 2 Newton at a plunger position x = 0.25cm. The plunger should be capable of moving at 10m/s with a voltage of 12V. The maximum B field at the given plunger position is 0.2Tesla, the maximum rms current density is 800A/cm^{2}, and the winding factor Kw = 0.35.

- Assuming the iron permeability is infinite, show that and .
- Write the force equation in terms of the B field and the current density by identifying these quantities in the above equations.
- Solve your force equation in part b) for the area product Ap = Ac Aw in terms of the B field, current density, winding factor, and plunger position. What is the required Ap?
- Unlike the inductor design, the designer cannot choose the air gap. In this case the required window area is constrained by the required B field. What is the required window area?
- Using the area product, what is the required core area?
- Choose the number of turns so the voltage due to the motion is equal to the applied voltage.
- Make a dimensioned sketch of the device.
- Which design requirement sizes the device and which design requirement determines the number of turns?
- What is the instantaneous power into the electrical terminal and out of the mechanical terminals at the design plunger position with the plunger moving at the design velocity? Neglect voltage droop across the inductance.

Problem 3

The fluxes linked by the rotor and stator windings of a 4 pole synchronous machine are

- What are the units of the mutual inductance M?
- Write down the expression for conservation of co-energy in terms of l
_{s}, i_{s}, l_{r}, i_{r}, Te, and q. You do not have to substitute in the above expressions for the fluxes. - Draw the path for computing the co-energy and write down the values of i
_{s}, i_{r}, q, di_{s}, di_{r}, and dq for each path segment. - Compute the machine's co-energy.
- What is the torque as a function of i
_{s}, i_{r}, and q? - Let i
_{s}= I_{a}cos(w_{e}t), q = w_{m}t + g and i_{r}= I_{f}. What is the voltage at the stator terminals and what is the relationship between w_{e}and w_{m}?