Reference no: EM13707096

Question 1:

Consider the geometry shown in the figure. The center is a metal sphere, radius a = 2 m and surface charge density P_sa = 2 C/m². The sphere is surrounded by a region of free space up to radius b = 5 M. A dielectric spherical shell of inner radius b, outer radius c = 6 in, and relative dielectric constant La = 4.00 is centered on the metal sphere. Outside the shell and extending to infinity is free space.

a) Use Gauss's Law to obtain an expression for the electric field in the free space region a ≤ r ≤ b.

b) Determine the electric field vector on the free space side of the dielectric at r = b. Use boundary conditions to determine the magnitude and direction of the electric field on the dielectric side at r = b.

c) Taking the potential at infinity to be V = 0 volts, determine the numerical value of the potential at r = a and r = 2a.

Consider the geometry below now. The inner spherical conductor is grounded (0 volts) and the outer spherical shell conductor is held at 10 V. In between is free space. Again use a = 2 m and b = 5 m.

d) Use Laplace's or Poisson's Equation to find an expression for the voltage between the plates.

e) Use the expression for the voltage to determine the Electric field between the plates.

Question 2:

A section of infinitely long and perfectly straight coaxial cable is shown on the left. The inner conductor has a radius a = 2 mm and carries a current of 1 A upwards (the positive z-direction). The inner conductor has relative permeability μ_r1 = 1.00. The outer conductor is a cylindrical shell with inner radius b = 5 mm and outer radius c = 6 mm. The outer shell carries an unknown current I₂ in the negative z-direction. The outer shell has relative permeability μ_r2 = 2.00. Assume the current is uniformly distributed over the cross-section of the conductors.

a) Determine the current density E for the inner conductor.

b) We require the field outside the cable (r > c) to be zero.

Use Ampere's Law to determine the current density J'₂ and current I₂ required to produce zero external magnetic field.

c) Use any technique to determine the expression for the magnetic field, H' between the conductors, a ≤ r ≤ b.

d) Determine the magnetic flux passing through a rectangular surface defined by the radial range a ≤ r ≤ b and axial range 0 ≤ z ≤ 1 m.

e) What is the value of the inductance per unit axial length (in the z-direction) for the coax cable?

f) Draw a graph of B' vs. r for the range 0 ≤ r ≤ 7 mm. Indicate the value of B' for r = 0, a, b, c.

Question 3:

The figure shows a magnetic circuit with a core region composed of three different materials. The top figure shows the 3-D perspective and depth while the lower figure provides the dimensions of each core segment. The material parameter values are: μ_r1 = 2, μ_r2 = 4, μ_r3 = 8, Ρ₁ = 1E - 7Ωm , p₂ = 2E - 7 Ωm, p₃ = 4E - 7 Ωm. The current into the coil is 1 A and the number of turns in the coil is N = 100.

a) Determine the value of the mmf.

b) Determine the reluctance for each core segment.

c) Draw the equivalent circuit representation using a resistor network connected to a battery.

d) Determine the magnetic flux in core segment 1 (the black one in the lower figure).

e) Suppose the current in the coil is adjusted such that the magnetic field in segment 3 of the core is made to be exactly 1 Tesla, and uniform across its cross-section (|B| = 1 Tesla). Determine the magnetic energy density in segment 3 and determine the total magnetic energy in segment 3.

f) Now suppose two small gaps of length L are introduced into the core region of segment 3. The current is again adjusted such that IBI = 1 Tesla in segment 3. Use the principle of virtual work to determine the magnitude of the force that tries to close the gaps.

Question 4:

a) Describe the two loss dominant loss mechanisms present in a real transformer. Indicate how they are minimized in transformer design. Identify at least one other non-ideal property of a transformer and discuss its origin.

b) A section of a power transmission network is shown in the Figure. There is a transformer on the generator side with turns ratio GN1:GN2, and a transformer on the load side with turns ratio LN₁:LN₂. Consider the transformers to be ideal.

i) Consider LN₁:LN₂ to be 999:1. Which transformer is step up and which is step down? Determine the voltage across the primary of the Load transformer, the current through the transmission resistance RT, and the value of RT.

ii) What would LN₁:LN₂ have to be in order to achieve maximum power transfer to the load?

c) A segment of a three phase transmission system is shown. The generators are balanced and supply 200 V AC (peak). The relative phase for each generator is A = 0°, B = 120° and C = 240° and the rotation sequence is ABC.

The generators are connected in a delta configuration. The load consists of three identical resistors connected in the star configuration. Each resistor is 1 kit

i) Determine the line-to-line voltage phasor V₁₂, indicating the phase relative to generator A.

ii) Determine the current phasor 1₁ in resistor R₁, indicating the phase relative to generator A.

Question 5

A linear motor is shown in the Figure. It consists of a movable bar of length L₁ = 1 m, in electrical contact with the metal rails. The bar is moving to the right at a constant velocity of 15 m/s. An external magnetic field of 1 Tesla, pointing into the page, is uniform over the structure.

a) Determine the emf induced across the bar

b) Determine the current magnitude and direction (up or down) of the current through the resistor.

c) Is the structure acting as a motor or a generator?

d) In the figure to the left, we see the movable bar has moved to the second rail pair with separation L₂ = 0.3 m. If the velocity of the bar and magnetic field were unchanged, is the structure acting as a motor or a generator? Justify your answer.

e) A device known as a commutator forms a key component in many D.C. motors. Describe the commutator and indicate its role for the D. C. Motor.

f) Explain the difference between synchronous motors and induction motors. You should discuss the structure of the motors, how shaft rotation is obtained, and how the rotation rate of the shaft is related to the AC electrical frequency. Use diagrams if necessary.

g) A DC motor is characterized with the following nameplate information: at no load V_in = 100 V, la = 5 A, torn = 2000 rpm, Ra = 0.5 Q. The equivalent electric circuit of the separately excited motor is shown in the figure along with measured parameters. All calculations and motor parameters stated are to be performed when steady state is achieved. A field voltage, V_f = 60 V and the field current, If = 0.5 A, are unchanged for all motor rotation rates and output power.

i. Determine the motor constant, armature loss, and rotational loss at no load.

ii. When the motor is loaded, its rotation rate is reduced to 1500 rpm. Assume rotational losses are proportional to the square of the rotation rate. Determine the following when loaded: input power, armature loss, developed power, rotational loss, output power, and efficiency.