We can work on Analytical & Computational Methods

You work as a Test Engineer for a global manufacturer of electrical and mechanical
components and systems. Your Line Manager is responsible for delegating to you and your
colleagues the testing of theory, principles, and hypotheses from several worldwide
company divisions. She has asked you to undertake a series of such evaluations.

Task 1

Note: You need to have your calculator in radians (RAD) mode for this task (since the angles are
given in radians – i.e., π is featured).

a) A current waveform in a robotic arm may be described by…

is=13cos�2πft−π
4� [𝐴𝐴]

           where frequency, f = 1Hz and t represents time.

Make time (t) the subject of this formula and hence determine a point in time when the
current waveform has a magnitude of +10A.

b) The instantaneous value of a power signal may be described by;

12∠�3π
8 � [W]

Find the magnitude of the vertical and horizontal components of this signal.
Note that the symbol ∠ indicates ‘angle’.

Task 2

a) A resistor, R, is connected in series with an inductor, L. An a.c. current, 𝑖𝑖, flows through this
RL combination, causing a voltage (𝑉𝑉𝑅𝑅) of 30V to be developed across the resistor, and a
voltage (𝑉𝑉𝐿𝐿) of 40V to be developed across the inductor.

Assuming that 𝑉𝑉𝑅𝑅 is in-phase with 𝑖𝑖, and 𝑉𝑉𝐿𝐿 leads 𝑉𝑉𝑅𝑅 by 90𝑜𝑜, draw a vector diagram for this
arrangement and then calculate the magnitude of the resultant voltage across the whole
RL combination.

b) A current-carrying filament is subjected to a strong magnetic field within an experimental
chamber. It is required to find the force on the filament. The current may be modelled in
three-dimensional space as:

�
�𝐼=2𝑖𝑖+3𝑗𝑗−4𝑘𝑘
and the magnetic field as:
�
�𝐵=3𝑖𝑖−2𝑗𝑗+6𝑘𝑘

Find the Cross Product of these two vectors to ascertain the characteristics of the force on
the filament (i.e., find 𝐼𝐼×𝐵𝐵).

Sketch this Cross Product (or use software to do so).

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Page 3 of 4 Issue 1 – 2024/25

Task 3

PART 1
The two signals below are sensed by a signal processor;
�
�𝑣1=40sin(4𝑡𝑡)
�
�𝑣2=𝐴𝐴cos(4𝑡𝑡)
The signal processor adds the signals to form a third signal, which must be described as a distinct
signal in the following form;
�
�𝑣𝑜𝑜=50sin(4𝑡𝑡+𝛼𝛼)
Use a compound angle identity to determine the value of A (the amplitude of 𝑣𝑣2).
Ensure that you have your calculator in Radians (RAD) mode when determining your answer.

PART 2
The third harmonic of a sound wave is given by;
4cos(3𝜃𝜃)−6sin(3𝜃𝜃)
Express this sound wave in the form;
�
�𝑅sin(3𝜃𝜃+𝛽𝛽)

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Sample Answer

 

 

 

Task 1

a) Current Waveform

Given:

  • is = 13cos(2πft - π/4) [A]
  • f = 1 Hz
  • is = 10 A

We need to find t when is = 10 A.

  1. Substitute the given values:
    • 10 = 13cos(2π(1)t - π/4)
  2. Divide by 13:
    • 10/13 = cos(2πt - π/4)
  3. Find the inverse cosine:
    • arccos(10/13) = 2πt - π/4
  4. Calculate arccos(10/13):
    • arccos(10/13) ≈ 0.6926 rad

Full Answer Section

 

 

 

 

 

  1. Solve for 2πt:
    • 0.6926 = 2πt - π/4
    • 2πt = 0.6926 + π/4
    • 2πt ≈ 0.6926 + 0.7854
    • 2πt ≈ 1.478
  2. Solve for t:
    • t = 1.478 / (2π)
    • t ≈ 0.2353 seconds

b) Instantaneous Power Signal

Given:

  • 12∠(3π/8) [W]

We need to find the horizontal and vertical components.

  1. Horizontal component (real part):
    • 12 * cos(3π/8)
    • 3π/8 ≈ 1.1781 rad
    • cos(3π/8) ≈ 0.3827
    • 12 * 0.3827 ≈ 4.5924 W
  2. Vertical component (imaginary part):
    • 12 * sin(3π/8)
    • sin(3π/8) ≈ 0.9239
    • 12 * 0.9239 ≈ 11.0868 W

Task 2

a) Resistor and Inductor Voltages

Given:

  • VR = 30 V (in-phase with i)
  • VL = 40 V (leads VR by 90°)
  1. Vector Diagram:
    • Draw a horizontal vector representing VR with a length of 30 units.
    • Draw a vertical vector upward from the end of VR representing VL with a length of 40 units.
    • The resultant voltage V is the hypotenuse of the right triangle formed by VR and VL.
  2. Calculate the resultant voltage:
    • V = √(VR^2 + VL^2)
    • V = √(30^2 + 40^2)
    • V = √(900 + 1600)
    • V = √2500
    • V = 50 V

b) Cross Product of Vectors

Given:

  • I = 2i + 3j - 4k
  • B = 3i - 2j + 6k

We need to find I × B.

  1. Calculate the cross product:

    • I × B = (3 * 6 - (-4) * (-2))i - (2 * 6 - (-4) * 3)j + (2 * (-2) - 3 * 3)k
    • I × B = (18 - 8)i - (12 + 12)j + (-4 - 9)k
    • I × B = 10i - 24j - 13k

  2. Sketch (or use software):

    • The force vector I × B has components (10, -24, -13) in the x, y, and z directions, respectively.
    • This represents a vector pointing in the positive x-direction, negative y-direction, and negative z-direction.

Task 3

PART 1

Given:

  • v1 = 40sin(4t)
  • v2 = Acos(4t)
  • vo = 50sin(4t + α)
  1. Use the compound angle identity:
    • vo = 50sin(4t + α) = 50(sin(4t)cos(α) + cos(4t)sin(α))
  2. Compare with v1 + v2:
    • v1 + v2 = 40sin(4t) + Acos(4t)
  3. Equate coefficients:
    • 50cos(α) = 40
    • 50sin(α) = A
  4. Solve for cos(α) and sin(α):
    • cos(α) = 40/50 = 0.8
    • sin(α) = A/50
  5. Use the identity sin^2(α) + cos^2(α) = 1:
    • (A/50)^2 + (0.8)^2 = 1
    • A^2/2500 + 0.64 = 1
    • A^2/2500 = 0.36
    • A^2 = 900
    • A = 30

PART 2

Given:

  • 4cos(3θ) - 6sin(3θ)

We need to express it in the form Rcos(3θ + α).

  1. Use the identity Rcos(3θ + α) = R(cos(3θ)cos(α) - sin(3θ)sin(α)).
  2. Equate coefficients:
    • Rcos(α) = 4
    • Rsin(α) = 6
  3. Find R:
    • R^2 = (Rcos(α))^2 + (Rsin(α))^2 = 4^2 + 6^2 = 16 + 36 = 52
    • R = √52 ≈ 7.21
  4. Find α:
    • tan(α) = (Rsin(α)) / (Rcos(α)) = 6/4 = 1.5
    • α = arctan(1.5) ≈ 0.9828 rad
  5. Express the wave:
    • 7.21cos(3θ + 0.9828)

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