Analytical & Computational Methods and Calculation

Task 1: 

A current waveform may be described by… 

Where, 

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

Solution: 

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

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

Solution: 

Task 2: 

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

Assuming that ? R is in-phase with ? i , and ? L leads ? R by , draw a vector diagram for this arrangement and then calculate the magnitude of the resultant voltage across the whole RL combination. 

Solution: 

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: 

and the magnetic field as: 

Find the Cross Product of these two vectors to ascertain the characteristics of the force on the filament (i.e. find ?? × ?? ). Sketch this Cross Product (or use software to do so). 

Solution: 

Task 3: 

Part 1 

The two signals below are sensed by a signal processor; 

The signal processor adds the signals to form a third signal, which must be described as a distinct signal in the following form; 

Use a compound angle identity to determine the value of A (the amplitude of v 2 ). Ensure that you have your calculator in Radians (RAD) mode when determining your answer. 

Use graphical software to plot/model the inputs and output of the signal processor. How do you think graphical methods of sine wave combination compare with analytical methods? 

Solution: 

The amplitude of V 0 is the resultant of the amplitudes of V 1 and V 2 . 

Analytical methods are more concise and accurate than graphical methods. 

Part 2 

The third harmonic of a sound wave is given by; 

Express this sound wave in the form; 

Solution: 

Here, we have; 

Alternatively, we can first convert 146.31 degrees to radians. 

Therefore, 

Or, 

U2 A3 

Task 1 

Differentiate each of the following voltage functions with respect to time and hence determine the ‘rate of change’ for each of the functions when time (t) is 5 seconds. 

1.  

Therefore, 

When t = 5 

1.  

Therefore, 

When t = 5 

1.  

When t = 5 

1.  

When t = 5 

1.  

When t = 5 

1.  

When t = 5 

Task 2: 

Part 1 

For the circuit shown below, assuming zero charge on the capacitor at t = 0, the current flowing may be quantified as… 

Integrate this equation with respect to t and hence find the charge stored in the capacitor one second after the switch is closed. 

Solution: 

Part 2 

The current through a 10 mH inductor (L) has a relationship with time (t) as follows; 

Determine the inductor current when the time is 0.9 seconds. 

Solution: 

Therefore; 

Task 3: 

Part 1 

Determine the area under the following exponential growth curve between 2 seconds and 4 seconds; 

Part 2 

Determine the area under the following exponential decay curve between 2 seconds and 4 seconds; 

Task 4 

Part 1 

Locate the co-ordinates of the turning point for the following function and determine whether it is a maxima or minima; 

At the turning point; 

Therefore; 

But; 

Therefore, 

The coordinates of the turning point is: 

Or 

Maxima or Minima? 

Therefore, the turning point is a minima. 

Part 2 

Find the maxima and minima values for the function; 

At the turning point; 

Therefore; 

But, 

The coordinates of the turning point are: 

Maxima or Minima? 

Therefore, the turning point is minima.