In this post we comprehensively discuss sinusoidal waveforms and learn how to calculate the involved specifications and parameters.
What are Sinusoidal Waves?
Basically, sinusoidal waveforms are periodic waveforms whose shape can be plotted using the sine or cosine function of trigonometry. We commonly refer to electrical circuits supplied by sinusoidal waveforms as “AC” voltages and current sources, in which the polarity alternates after each cycle.
When we have an electric current flowing through a wire or conductor, a circular magnetic field is created around the wire.
The strength of this magnetic field is related to the value of the current passing. If we move or rotate a single wire conductor, within a stationary magnetic field, an “EMF” (Electro-Motive Force) is induced within the conductor due to the movement through the magnetic flux.
Relationship Between Electricity and Magnetism
From the above we clearly find a relationship between Electricity and Magnetism. This relationship was discovered by Michael Faraday, giving us the principle of “Electromagnetic Induction.” It is this basic principle that electrical machines and generators use to generate a Sinusoidal Waveform for our mains supply.
AC Generator Working Principle
When a conductor is positioned parallel to the magnetic field, there is no induction of electromagnetic force. However, if the conductor is at a right angle to the magnetic field maximum flux is cut resulting in the highest induced EMF.
The strength of this EMF is dependent on the angle between the conductor and magnetic flux, as well as the intensity of the magnetic field.
By using Faraday’s law of electromagnetic induction, alternating current (AC) generators are able to convert mechanical energy into a sinusoidal waveform. This is achieved, by a rotating rectangular loop of wire within a fixed magnetic field generated by permanent magnets.
The wire loop traverses the magnetic force lines formed between the north and south poles at various angles as the coil spins counterclockwise around the central axis, which is at right angles to the magnetic field.
The wire loop’s rotational angle determines the amount of induced electromagnetic field (EMF) present in the loop at any one moment.
Electrons in a wire begin travelling in a single direction across the loop as it rotates. The electrons in the wire loop quickly alter and flow in the reverse direction as soon as it has rotated through the 180° mark and crosses the magnetic lines of force in that direction.
The polarity of the induced voltage is then determined by the direction of the electron’s travel.
Therefore, it is evident that one full sinusoidal waveform is formed for every rotation of the coil, when the loop or coil physically rotates one full revolution, or 360°.
Carbon brushes and slip rings are employed to make electrical connections to the coil as it spins inside the magnetic field. These connections are created in order to efficiently conduct the electrical current that is generated in the coil.
Factors that Determine the Strength of EMF
The following three parameters influence the quantity of electromagnetic field (EMF) induced into a coil that cuts the magnetic lines of force.
- Speed: The coil’s rotational speed within the magnetic field.
- Strength: The magnetic field’s strength.
- Length: the total length of the conductor or coil that is exposed to the magnetic field.
Relationship Between Rotational Speed and Frequency
We are aware that a supply’s frequency, which is expressed in Hertz, is the number of times a cycle occurs in a second.
A constant speed of rotation of the coil will result in a uniform amount of cycles per second, or a fixed frequency, as each full rotation of the coil within the magnetic field consisting of the north and south poles produces one cycle of induced emf.
Thus, the frequency will begin to increase in tandem with the coil’s increased rotational speed. As a result, frequency and rotational speed are proportional, which can be expressed as ƒ ∝ N, where Ν = r.p.m.
Additionally, our uncomplicated single coil generator mentioned earlier only possesses two poles – one magnetic north and one magnetic south – resulting in just a solitary pair of poles.
Should we introduce more magnetic poles to the aforementioned generator, increasing the total count to four (comprising two magnetic north poles and two magnetic south poles), each revolution of the coil will yield two cycles for the same rotational speed.
Consequently, the frequency is directly proportional to the quantity of pairs of magnetic poles, denoted as ( ƒ ∝ P ), where P signifies the count of “pairs of poles.”
Consequently, based on these aforementioned observations, we can assert that the frequency produced by an alternating current (AC) generator is:
ƒ ∝ N and ƒ ∝ P
ƒ = N x P (cycles per minute)
Since we measure frequency in Hertz:
Frequency, (ƒ) = (N x P) / 60 Hz
Where: Ν denotes the speed of rotation in r.p.m. P represents the number of “pairs of poles” and the figure 60 transforms the result into seconds.
Instantaneous Value of Sinusoidal Waveform
The electromotive force (EMF) induced in the winding at any point relies on the pace or speed at which the winding intersects the lines of magnetic flux amid the poles.
This dependency is determined by the angle of rotation, Theta ( θ ), of the generating mechanism. Due to the fact that an alternating current (AC) waveform perpetually modifies its magnitude, the waveform at any particular moment will be different from its previous moments magnitude.
For example, the magnitude at 1 millisecond might be different from the magnitude at 1.2 milliseconds, and so forth.
These sizes are commonly denoted as the Immediate Values or Vi. As a result, the immediate size of the waveform and its orientation will shift based on the placement of the winding within the magnetic field.
The instantaneous magnitudes of a sinusoidal waveform can be expressed using the formula
Instantaneous magnitude = Peak value x sin θ, and this is universally expressed by the equation.
Vi = Vmax x sin θ
In the above formula Vmax represents the maximum voltage induced in the coil and θ = ωt, denotes the coil’s rotational angle with respect to time.
The instantaneous quantities at different positions throughout the waveform may be determined by applying the above formula if we know the waveform’s highest or peak value. One may create a sinusoidal waveform shape by charting these numbers on graph paper.
To simplify we will plot the sinusoidal waveform’s instantaneous values at each 45-degree rotation, which will give us eight plotting points. Once more, for simplicity’s sake, we’ll use a maximum voltage of 100V (VMAX).
A more precise sinusoidal waveform might be created by graphing the instantaneous values at smaller time periods, which means every 30° (12 points) or 10° (36 points), for example.
How to Plot and Draw a Sinusoidal Waveform
One complete waveform is formed, when the wire loop or coil rotates 360°, or one complete revolution. The coordinates on the sinusoidal waveform are acquired by projecting across from the different places of rotation between 0 and 360° to the ordinate of the waveform which relates to the angle, θ.
The sinusoidal waveform figure indicates that the generated EMF is zero whenever θ equals 0°, 180°, or 360° since the coil traverses the fewest possible lines of flux at those degrees. However, the produced EMF reaches its greatest magnitude when θ equals 90° and 270° since the maximum amount of flux is reduced at those times.
As a result, a sinusoidal waveform contains two peaks: one at 90° and the other at 270°. The EMF values for positions B, D, F, and H are determined by the formula e = Vmax.sinθ.
Since the waveform structure created by our basic single loop generator happens to be sinusoidal in nature, it is called a sine wave. Also, since it is founded on the trigonometric sine function x(t) = Amax.sinθ, this kind of waveform is known as a sine wave.
Understanding Radians
The unit of measurement employed on a waveform’s horizontal axis can range from either time, degrees, or radians while working with sine waves in the time domain, especially for current-related sine waves.
Instead of using degrees for determining an angle across a horizontal axis, the Radian is more favorably used in electrical engineering. As an illustration, ω = 500 rad/s or 100 rad/s.
In mathematics, the radius, or rad, is the quadrant of a circle where the length of the circle’s radius (r) equals the distance subtended on its circumference. A circle has to have 2π radians surrounding its 360°, simply because its circumference is equal to 2π x radius.
Relationship between Radians and Degrees
A sinusoidal waveform incorporating radians as the unit of measurement gives 2π radians for a complete 360° cycle. Subsequently, 1π radians, or simply π (pi), must correspond to half of a sinusoidal waveform. Considering the value of pi, (π) equals 3.142, the following formula represents the connection between degrees and radians for a sinusoidal waveform:
Radians = (π / 180°) x degrees
Degrees = (180° / π) x Radians
If we implement the above equations across various points on the waveform, we get:
- 30o → Radians = π / 180o (30o) = π / 6 rad.
- 90o → Radians = π / 180o (90o) = π / 2 rad.
- 5π / 4 rad → Degrees = 180o / π (5π / 4) = 225o
- 3π / 2 rad → Degrees = 180o / π (3π / 2) = 270o
Sinusoidal Waveform and its Angular Velocity
A generator’s rotational velocity about its central axis establishes the sinusoidal waveform’s frequency.
The waveform is defined by an angular frequency, ω, in radians per second, which is in addition to its frequency, which is expressed as ̒ Hz, or cycles per second.
Therefore, the sinusoidal waveforms’ angular velocity is expressed by the formula.
ω = 2πf = rad/sec
In countries wheer the mains AC frequency is 50 Hz, the above equation can be solved as:
ω = 2πf = 2π x 50 = 314.2 radians / second
In countries where the mains AC frequency is 60 hz, the above equation becomes:
ω = 2πf = 2π x 60 = 377 radians / second
We now understand that the generator’s rotational velocity, or ω, which is additionally referred to as its angular velocity, controls the frequency of the sinusoidal waveform. However, we should also be aware that the sinusoidal waveform’s periodic time, or T, is equal to the amount of time needed to carry out one full rotation.
Since frequency is inversely proportional to its time period, ƒ = 1/T, we may replace the analogous periodic time variable in the previous equation with the frequency quantity, which gives us the formula:
ω = 2π / T (Radians / Second)
According to the above formula, the sinusoidal waveform’s angular velocity must be bigger for a smaller sinusoidal waveform’s periodic time. Similar to this, the angular velocity increases with increasing frequency in the equation above.
Solving Sinusoidal Waveform Problems (Example#1)
Consider a sinusoidal waveform expressed as Vm = 170 sin(377t) volts. Now, after 7 milliseconds have passed, determine the waveform’s RMS voltage, frequency, and instantaneous voltage value (Vi).
As we already know, a sinusoidal waveform may be expressed generally as follows:
Vt = sine(ωt)
Since for our sinusoidal waveform above we have Vm = 170 sin(377t), this implies that the waveform’s peak voltage value would be 170 volts.
So, we calculate the RMS of this waveform using the following formula:
V(RMS) = 0.707 x peak value = 0.707 x 170 = 120.19 volts
In the problem we also know that angular velocity (ω) is 377 rad/s, which means 2πƒ = 377. Using this data, we can calculate the frequency of the waveform as shown below:
Frequency (f) = 377 / 2π = 60 Hz
Next, to calculate the instantaneous Voltage Vi, we use the following formula:
Vi = Vm sine(ωt)
Vi = 170 sine (377 x 0.006) = 170 sine (2.262) volts peak.
2.262 radians x 57.3o = 129.6o
Vi = 170 sine (129.6) = 170 x 0.771
Vin = 131.07 peak voltage
Since the angular velocity is expressed in radians (rads) at time t = 6mS. If desired, we could translate this into a corresponding angle expressed in degrees and apply that number in place of the original calculation for the instantaneous voltage value. As a result, the instantaneous voltage value’s angle in degrees is as follows:
Degrees = (180o / π) x radians
= 180o / π x 2.262 = 57.3o x 2.262 = 129.6o
References: Sine wave – Wikipedia
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