A current waveform in a robotic arm

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)