INDUSTRIAL ELECTRONICS 2 EENG27000 Problem Sheet 1 PB1-1 ISSUE 2, REVISION E

UNIVERSITY OF BRISTOL

Department of Electrical and Electronic Engineering

INDUSTRIAL ELECTRONICS 2: Electro-Mechanical Energy Conversion

Problem Sheet 1: Magnetic Circuits, Stored Energy and Ferro-magnetic Actuators

1. Figure 1 shows an iron core with two excitation coils. The mean path of the magnetic flux is shown in this figure by the dashed lines.

Figure 1: Double Looped Core

The relative permeability,r , of the core is 1500. Dimensions shown are in centimetres and the depth

of the core is 2cm. Given this information, determine:

(i) The equivalent magnetic circuit including the reluctances of paths bafe, bcde, be (excluding air gap) and of the air-gap

(ii) The air-gap flux and flux-density

(iii) The air-gap magnetic field intensity

Answers: (i) ,

,

,

(ii) ,

(iii)

2. A d.c. choke inductor has an inductance given by the expression:

(PB1-1)

Where x is the length, in metres, of the air-gap in the inductor core and is equal to 2mm. Calculate:

(i) The energy stored in the inductor when carrying a d.c. current of 3A.

(ii) The force across the faces of the air-gap at the same level of d.c. current.

Answers: (i) 81mJ, (ii) 33.8N

PB1

INDUSTRIAL ELECTRONICS 2 EENG27000 Problem Sheet 1 PB1-2 ISSUE 2, REVISION E

3. Figure 2 represents the magnetic circuit for a primitive relay device. The coil has 2000 turns and the mean core path is .

Figure 2: Primitve Relay

When both of the air-gap lengths are 1.5mm, a flux density of 0.8T is required to actuate the relay. The core is made from steel which has a relative permeability of 1248.3.

(i) Find the total required mmf and hence the current in the coil.

(ii) If the air-gap is zero, find the current in the coil for the same flux density (0.8T) in the core

Answers: (i) , , (ii)

4. Figure 3 shows a C-core whose relative permeability r = 3000 and whose cross-sectional area is equal to that of the air-gap and where Ac = Ag = 1500mm2, find:

Figure 3: C-Core

(i) Flux and flux density in the air-gap

(ii) Inductance of the coil.

Answers: (i) , (ii)

INDUSTRIAL ELECTRONICS 2 EENG27000 Problem Sheet 1 PB1-3 ISSUE 2, REVISION E

5. The toroidal (circular cross section) core shown in Figure 4 is made from cast steel.

Figure 4: Toroidal Core

(i) Calculate the coil current required to produce a core flux density of 1.2T at the mean radius of the toroid. (Assume relative permeability of cast steel of 955)

(ii) What is the core flux, in Webers assuming a uniform flux density in the core?

(iii) IF a 2mm wide air-gap were made in the toroid along the line A-A, determine the new coil current required to maintain core flux density of 1.2T.

Answers: (i) 2.51A, (ii) 1.51 x 10-3 Wb, (iii) 12.07A

6. Figure PB1-5 shows the configuration of a rudimentary form of relay in which the only significant reluctance is the air-gap of length, x, and cross-sectional area, A.

Figure 5: Pivoted Ferromagnetic Relay

(i) Derive an expression for the coil inductance and hence show the magnitude of the force on the hinged armature is given by:

(PB1-2)

(ii) For a particular relay having N = 1000 turns, A = 120 mm2 and x = 5mm when the armature is in the open position and x = 3mm when the armature is in the closed position. If the spring exerts a force of 3N, calculate the current required to close the relay, and the current at which it will re-open.

(iii) Why are the two currents not equal and why is this effect desirable in the relay operation.

Answers: (i)

, (ii) ,

INDUSTRIAL ELECTRONICS 2 EENG27000 Problem Sheet 1 PB1-4 ISSUE 2, REVISION E

7. The system shown in Figure 3 has an air-gap with a length of 2mm and an area of 1500mm2. Neglecting the core reluctance and fringing and leakage fluxes, determine:

(i) The force of attraction between both sides of the air-gap.

(ii) The energy stored in the air-gap.

Answers: (i) 37.7N , (ii) 75.4mJ

8. A ferro-magnetic actuator is used in a lifting device and is shown in Figure 6.

Figure 6: Electro-magnetic Lifting Device

The coil has 2500 turns and the flux density in the air-gap is 1.25T. Assuming that the core used in the device does not saturate and that the air-gap length, g =10mm, determine:

(i) The coil current

(ii) The energy stored in the magnetic field

(iii) The force exerted upon the load (sheet of metal) and hence, assuming that this force holds the load in the air, the mass of the load.

Now, assuming an air-gap, g = 5mm, determine:

(iv) The current required to lift the load

Answers: (i) 7.96A (ii) 19.89J (iii) 1991.8N, 203kg (iv) 3.98A

9. In a linear motion actuator based on a coil, the relationship is given by:

(PB1-3)

For 0 < x < 1m, where I is the current passing through the coil used in the linear actuator. Determine the force on the moving part of the actuator when it is at a distance of x = 0.6m.

Answer:

INDUSTRIAL ELECTRONICS 2 EENG27000 Problem Sheet 1 PB1-5 ISSUE 2, REVISION E

10. The relationship for a solenoid actuator can be approximated by the following relationship:

(PB1-4)

Where g represents the air-gap present in the solenoid actuator. The solenoid actuators initial position is such the that air-gap length, g, is equal to 100mm.

If the coil in the actuator has a resistance of 12 Ohms and is supplied from a voltage source whose output voltage = 24V determine:

(i) Stored energy in the magnetic field and co-energy of the system

The voltage in the coil remains constant and the air-gap in the actuator after 0.5s is 5cm. Determine:

(ii) Stored energy in the magnetic field and co-energy of the system

Hence, determine:

(iii) Mechanical work done on the solenoid by the magnetic field

(iv) Energy transferred from the electrical system.

Answers: (i) 11.31J and 22.62J (ii) 22.64J and 45.24J (iii) 22.62J (iv) 33.92J