The purpose of this session is to further investigate single phase ac circuits with an emphasis on the equation of a sine wave and single pole low/high pass filters.
BTEC Outcomes Covered
- Unit 6 Pass 10
- Unit 6 Pass 11
In electrical terms a sine wave can be generated by rotating a conductor (a coil) within a magnetic field as shown by this graphic.
When the conductor is perpendicular (at right angles) to the magnetic field the generated voltage is at its strongest. When the coil is unaffected by the magnetic field then no voltage is generated. This link describes further how electricity is generated.
In the following video, the rotating vector (line), known as a phasor, shows the relationship between a rotation of 360 degrees (a full circle) and the formation of a sine wave.
Sine waveforms are the most efficient way of transmitting electrical power and electrical signals. You will encounter them everywhere when studying electronics so it is well worth knowing about them.
The equation of a sine wave
The following equation is used to describe a sine wave. I have annotated the important parts:
Suppose you wanted to know the exact voltage at a specified point in the future. The instantaneous voltage represents this value!!
This simply represents the peak voltage (the highest point) the waveform reaches for each positive cycle of the sine wave
ω pronounced Omega
This is a measure of how many cycles the sine wave will complete in 1 second. The cycles are measured in radians. There are 2π radians in every 360 degree rotation. Therefore 2πf (where f stands for frequency) will give the total amount of radians completed in 1 second
Stands for time, this is the time at which we will be calculating the instantaneous voltage. For example, say I wanted to know the instantaneous voltage 4 mS after t= 0? I would insert a value of 0.004 for t.
Ø pronounced phi
All sine waves are measured from t = 0. However sine waves can begin before or we say lead t = 0 or come after (lag) t = 0. The amount of lead or lag is measured in radians and added or subtracted from ωt to give an overall value within the brackets.
To find the instataneous voltage you follow this procedure
Within the brackets multiply ω by t (time)
If there is a phase shift then either add or subtract this value from the result
Take the resulting value within the brackets and convert it to degrees using:
(radian/2π) x 360
Use the sin function to convert the degrees to a ratio
Multiply the resulting ratio by Vmax
You have now found the instantaneous voltage!!
A sine wave is represented by the equation
Vinst = 150Sin(50πt =/- 0) <— There is no phase shift in this example
Find the instantaneous voltage after 5 mS have elapsed
Vinst = 150Sin(50π/2π x t) Divide ω/2π to find the frequency
Vinst = VmaxSin(25 x 0.005) Now multiply by the time
Vinst = 150Sin(0.125)
(0.125/6.283) x 360 = 57.17 degrees Now convert the bracket value to degrees
Sin 57.17 = 0.840 Find the ratio by using the sine function
Vinst = Vmax x 0.840 =126 Volts Multiply the ratio by Vmax
Find the instantaneous voltage after 10 mS have elapsed for the following sine wave:
Vinst = 240Sin(100π +/- 0)
Please show all of your working out
Low Pass Filter
Theory tells us that the bandwidth of any low pass filter can be found using the following equation:
Bandwidth Frequency = 1/(2πCR)
A Low Pass Filter
Where C is the value of the capacitor and R is the value of the resistor.
Find the bandwidth for a low pass filter with the following components:
R = 1000Ω
C = 10 nF (0.00000001 Farads)
Simulate the circuit using Multisim and prove that your calculations are correct. The following video shows you how to accomplish this.
Complete your report as usual. That concludes the work required for this week.