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ElectroLearn Hub, kurunegala , maho, Maho (2026)

06/04/2026

INSTANTANEOUS WAVEFORM EQUATION
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⚡ Instantaneous Waveform Equation

📌 Where:
🔹 Amax – is the maximum amplitude of the waveform.
🔹 ωt – is the angular frequency of the waveform in radian/sec (could also be: 2πƒt)
🔹 Φ (phi) – is the phase angle in degrees or radians that the waveform has shifted either left or right from the reference point.

⬅️ If the positive slope of the sinusoidal waveform passes through the horizontal axis “before” t = 0 then the waveform has shifted to the left. So Φ > 0. Thus the phase angle will be positive in nature, +Φ giving a “leading phase angle”. In other words it appears earlier in time than 0° producing an anticlockwise rotation of the vector.

➡️ Likewise, if the positive slope of the sinusoidal waveform passes through the horizontal x-axis some time “after” t = 0 then the waveform has shifted to the right. So Φ < 0. Thus the phase angle will be negative in nature -Φ producing a “lagging phase angle” as it appears later in time after 0° producing a clockwise rotation of the vector. Both cases are shown below.

🔄 Phase Relationship of a Sinusoidal Waveform

📌 Firstly, let’s consider that two instantaneous quantities such as a voltage, v and a current, i have the same frequency ƒ in Hertz and both start at t = 0. As the frequency of the two quantities is the same their angular velocity, ω must also be the same. So at any instant in time we can say that the phase of voltage, v will be the same as the phase of the current, i.

📊 Then the angle of rotation within a particular time period will always be the same. The difference between the two quantities of v and i will therefore be zero. That is: Φ = 0.

✅ As the frequency of the voltage, v and the current, i are the same they must both reach their maximum positive, negative and zero values during one complete cycle at the same time (although their amplitudes may be different). Then the two alternating quantities, v and i are said to be “in-phase”.

📉 Two Sinusoidal Waveforms – “in-phase”

⚡ Now let us consider that the voltage, v and the current, i have a phase difference between themselves of 30°. Therefore, (Φ = 30° or π/6 radians).

📌 As both alternating quantities rotate at the same speed, they will have the same frequency. Therefore this phase difference will remain constant for all instants in time. Thus the phase shift of 30° between the two waveforms is represented by phi, Φ as shown below.

📐 30° Phase Difference of a Sinusoidal Waveform

📊 The voltage waveform above starts at zero (0°) along the horizontal reference axis. However, at that same instant of time the current waveform is still negative in value and does not cross this reference axis until 30° later.

📍 Then we can see that there exists a difference between the phases of the two waveforms as the current waveform crosses the horizontal reference axis reaching its maximum peak and zero values 30° after the voltage waveform.

❓ What are Leading and Lagging Waveforms?

⚠️ Since the two sinusoidal waveforms are no longer “in-phase”, they must therefore be “out-of-phase” with each other by an amount determined by phi, Φ and in our simple example this is: 30°. So we can say that the two waveforms are now 30° “out-of-phase” with each other.

⏱️ Then the current waveform can also be said to be “lagging” behind the voltage waveform by the phase angle, Φ of 30°. Thus the two waveforms have a Lagging Phase Shift and the expression for both the voltage and current above is given as:

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✍️ Sisira Senevirathna

06/04/2026

PHASE DIFFERENCE AND PHASE SHIFT
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⚡ Phase Difference is used to describe the difference in degrees or radians between two or more alternating quantities when they reach their maximum or zero values

🔹 What is the Phase Difference Between Two Waveforms

📌 Phase Difference, also known as Phase Shift or Phase Delay, defines the difference in time between two sinusoidal waveforms of the same frequency. The phase difference of two waveforms indicates how much one leads or lags behind the other.

⚙️ Phasors are an effective way of analysing the behaviour of elements within an AC circuit when the circuit frequencies are the same. The result of adding together two phasors depends on their relative phase, whether they are “in-phase” or “out-of-phase” due to some phase difference.

📊 Characteristics of a Sinusoidal Waveform

📌 A Sinusoidal Waveform is an alternating quantity that can be presented graphically in the time domain along a horizontal axis using the trigonometric functions of sine or cosine. Expressed as:
👉 A(t) = Amax*sin(ωt).

📍 As a time-varying quantity, sinusoidal waveforms have a positive maximum value at time π/2 (90°) and a negative maximum value at time 3π/2 (270°) with its zero values occurring along the horizontal baseline at the: 0, π and 2π points.

🔄 Horizontal Shifting of an AC Waveform

📌 However, not all sinusoidal waveforms of the same frequency will pass exactly through the zero axis point at the same time. For example, when comparing a voltage waveform to that of a current waveform.

➡️ Thus, compared to one reference waveform, some waveforms may be “shifted” to the right of 0° by some value represented by ƒ(ωt – t₀), while others may be shifted to the left of 0° by some value represented by ƒ(ωt + t₀). That is, the waveform moves along the zero axis without changing its shape.

⚡ This difference produces an angular shifting of the sinusoidal waveforms creating what is known as a Phase Difference between them. Any sine wave that does not pass through zero at t = 0 will generally have a “phase shift” in degrees or radians of some amount.

📐 How to Measure Phase Difference in Waveforms?

📌 The difference or phase shift of a Sine Wave is the angle, in degrees or radians that a waveform has shifted, left or right, from a certain reference point along the horizontal zero axis compared to another. In other words, it is the lateral difference between two or more waveforms along a common axis of the same frequency.

🔣 The primary symbol for electrical phase difference (phase angle) is represented by the Greek capital letter Φ (phi), or by its lowercase φ (phi). Both symbols represent the same angle and therefore phase shift.

📊 Then the difference between phases (Φ) of an alternating waveform can vary from between zero degrees (or radians) to its maximum time period, T of the waveform during one complete cycle. This phase shift can be anywhere along the horizontal axis between, Φ = 0 to 2π (radians) or Φ = 0 to 360° depending upon the angular units used.

⏱️ Phase difference can also be expressed as a time shift of τ (tau) in seconds representing a fraction of the time period, T for example, +10mS or –50uS. But generally it is more common to express the difference between two sinusoidal waveforms as an angular measurement.

📌 So the equation for the instantaneous value of a sinusoidal voltage or current waveform we developed in the previous sinusoidal waveform tutorial will need to be modified to take account of the phase angle of the waveform. This new general expression becomes.

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✍️ Sisira Senevirathna

01/04/2026

⚡️📘 SINUSOIDAL WAVEFORMS SUMMARY
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📌 Note that the angular velocity at time t = 6ms is given in radians (rads). We could, if so wished, convert this into an equivalent angle in degrees and use this value instead to calculate the instantaneous voltage value.

📐 The angle in degrees of the instantaneous voltage value is therefore given as:

θ=ωt=377×0.006
📊 Giving us the same rotational angle of 129.6°

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📚 Sinusoidal Waveforms Tutorial Summary

🌊 We have seen here in this tutorial that a Sinusoidal Waveform (or sine wave) is a fundamental waveform of periodic oscillations over time that can be described by the trigonometric sine or cosine functions.

🏠 The sinusoidal waveforms which supply electric power to our homes and workplaces are produced by a rotating machine which creates a waveform which continuously oscillates between a maximum or peak positive and a negative value.

📈 Thus, sinusoidal waveforms can be defined by their:
⚡️ Amplitude (maximum value)
⏱️ Frequency (cycles per second)

📊 With the generalised format used for analysing and calculating their various values, given as follows:

v(t)=Vax sin(ωt)

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🔌 Note that in electronics, low-power sinusoidal waveforms can be created using LC or RC Oscillators which use positive feedback circuits to produce a continuous and smooth sine wave output at a specific frequency.

⏭️ In the next tutorial about Phase Difference we will look at the relationship between two sinusoidal waveforms that are of the same frequency but pass through the horizontal zero axis at different time intervals.

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✍️ Written by: Sisira Senevirathna

01/04/2026

⚡️📊 ANGULAR VELOCITY OF SINUSOIDAL WAVEFORMS
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🌍 In Europe and the United Kingdom, the angular velocity of the mains utility supply is given as:

ω=2π×50=314 rad/s

🌎 In North America and the USA, as their mains supply frequency is 60Hz, it is calculated as:

ω=2π×60=377 rad/s

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⚙️ So hopefully we now know that the velocity at which the generator rotates around its central axis determines the frequency of the sinusoidal waveform it produces, and which can also be called its angular velocity (ω).

⏱️ But we should by now also know that the time required to complete one full revolution is equal to the periodic time (T) of the sinusoidal waveform.

📊 As mentioned above, the frequency of an AC waveform is inversely proportional to its time period:
📈 The above equation states that for a smaller periodic time of the sinusoidal waveform, the greater must be the angular velocity of the waveform. Likewise, the higher the frequency, the higher the angular velocity.

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🧮 Sinusoidal Waveforms Worked Example

📌 A sinusoidal waveform is defined as:

v(t)=169.8sin(377t)

📊 From this:
⚡️ Maximum voltage (Vmax) = 169.8 V

📉 The waveform’s RMS voltage is calculated as:

Vrms= Vmax
2

👉 Therefore:
⚡️ Vrms = 169.8 / √2 ≈ 120 V

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📡 Frequency Calculation

From the given equation: ω = 377

👉 Therefore:
📊 f = 377 / (2π) ≈ 60 Hz

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⏱️ Instantaneous Voltage at t = 6ms

📌 t = 6 ms = 0.006 s

vi=169.8sin(377×0.006)

👉 Therefore:
📊 vi ≈ 169.8 × sin(2.262)
📉 vi ≈ ~130.7 V
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✍️ Written by: Sisira Senevirathna

01/04/2026

⚡️📐 ANGULAR DEFINITION OF A RADIAN
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📊 Thus, using radians as the unit of measurement for a sinusoidal waveform would give 2π radians for one full cycle of 360°. Then half a sinusoidal waveform must be equal to 1π radians or just π (pi).

📐 Then knowing that pi (π) is equal to 3.142 (22/7), the relationship between degrees and radians for a given sinusoidal waveform is therefore calculated as:

2
𝜋
radians
=
360
∘
2π radians=360
∘

📊 Relationship between Degrees and Radians

🔄 Applying these two equations to various points along the waveform gives us the conversion between degrees and radians for the more common equivalents used in sinusoidal AC analysis.

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⚙️ The velocity at which the generator rotates around its central axis determines the frequency of the sinusoidal waveform.

⏱️ As the frequency of the waveform is given as ƒ (Hz) or cycles per second, the waveform also has angular frequency, ω (Greek letter Omega), in radians per second.

📈 Then the angular velocity of sinusoidal waveforms is given as:🔥 Want more simple electronics lessons like this?
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✍️ Written by: Sisira Senevirathna

01/04/2026

⚡️📉 SINUSOIDAL WAVEFORM CONSTRUCTION
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📊 The points on the sinusoidal waveform are obtained by projecting across from the various positions of rotation between 0° and 360° to the ordinate of the waveform that corresponds to the angle, θ. Thus when the wire loop or coil has completed one full revolution, or 360°, one complete sinusoidal waveform is produced as shown.

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📈 How to Find Instantaneous Voltage in a Sine Wave

⚡️ From the plot of the sinusoidal waveform we can see that when Theta (θ) is equal to 0°, 180° or 360°, the generated EMF will be zero. This is because as we discussed above, the rotating coil cuts the minimum amount of lines of flux since the coils conductors are rotating parallel to them.

📊 However, when Theta (θ) is equal to 90° and 270° the generated EMF is at its maximum value as the maximum amount of lines of flux are cut horizontally to them.

🔄 This results in a sinusoidal waveform having a positive peak at 90° and a negative peak at 270°. Positions B, D, F and H generates an EMF voltage value corresponding to the formula:

𝑣𝑖= 𝑉𝑚𝑎𝑥 sin ( 𝜃 )
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🌊 Then the waveform shape produced by our simple single loop generator is commonly referred to as a Sine Wave as it is said to be sinusoidal in its shape.

📐 As we have seen, this type of waveform is called a sinusoidal wave because it is based on the trigonometric sine function used in mathematics. That is:

𝑥(𝑡) = 𝐴𝑚𝑎𝑥 sin (𝜃)

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⏱️ When dealing with sinusoidal waveforms in the time domain and especially related to alternating currents, the unit of measurement used along the horizontal axis to plot an AC waveform can be either:

🕒 Time (μs, ms, seconds)
📐 Degrees (90°, 180°, 360°)
🔄 Radians (π/2, π, 2π), etc.

⚡️ In electrical engineering it is generally more common to use the Radian as the angular measurement of the angle along the horizontal axis rather than degrees.

📊 For example: ω = 100 rad/s, or 500 rad/s

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📐 What are Radians?

📏 The Radian (rad) is defined mathematically as a quadrant of a circle where the distance subtended on the circumference of the circle is equal to the length of the radius (r) of the same circle.

🔵 Since the circumference of a circle is equal to 2π times its radius (r), there must therefore be 2π radians around the whole 360° of the circle.

📊 In other words, the radian is a unit of angular measurement and the length of one radian (r) will fit 6.284 (2π) times around the whole circumference of a circle.

📐 Thus one radian equals:

360 1 radian = ━━ ≈ 57.3∘
2𝜋
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🔄 Thus the radian in a sinusoidal waveform represents the angular measurement of the waveform cycle along the horizontal axis.

📉 Where one full cycle of the waveform equals 2π radians (or 360°). The use of radians in Electrical Engineering is very common so it is important to remember this relationship.

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✍️ Written by: Sisira Senevirathna

01/04/2026

⚡️📊 FREQUENCY & INSTANTANEOUS VALUES OF SINUSOIDAL WAVEFORM
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🔄 Then from these two facts we can say that the frequency output from an AC generator is:

📡 Frequency Relationship
Where: Ν is the speed of rotation in revs per minute (rpm). P is the number of “pairs of poles” and 60 converts it into seconds.

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📈 Sinusoidal Waveforms Instantaneous Voltage

⚡️ The EMF induced in the coil at any instant of time depends upon the rate or speed at which the rotating coil cuts the lines of magnetic flux between the two poles. This EMF is therefore dependant upon the angle of rotation, Theta (θ) of the generating device.

🔄 Therefore, because the EMF induced in the rotating coil is constantly changing its value or amplitude, the AC waveform produced at any instant in time will have a different value from its next instant in time (t), or angle (θ).

⏱️ For example, the EMF value at 1.0 millisecond will be different to the EMF value measured at 1.2 milliseconds and so on. These values are known generally as Instantaneous Values.

📊 That is, an instantaneous value represents the exact and changing value of the sinusoidal waveform as it alternates above and below its horizontal zero-axis.

📉 Then the instantaneous value of the waveform, either voltage (vi) or current (ii), depends on the angular position of the coil within the magnetic field as shown below.

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🧮 The instantaneous values of a sinusoidal waveform is given as:

𝑣𝑖 = 𝑉𝑚𝑎𝑥 SIN(𝜃)

📌 That is, it is calculated by multiplying the maximum (peak) value by the sine of the angle.

👉 Where:
⚡️ Vmax is the maximum voltage induced in the coil
🔄 θ = ωt is the rotational angle of the coil with respect to time

📊 Then for a voltage waveform: vi = VMAX sin(θ)
📊 And for a current waveform: ii = IMAX sin(θ)

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📉 If we know the maximum or peak value of the waveform, by using the formula above the instantaneous values at various points along the waveform can be calculated. By plotting these values out onto graph paper, a sinusoidal waveform shape can be constructed.

📍 In order to keep things simple we will plot the instantaneous values for the sinusoidal waveform at every 45° of rotation giving us 8 points to plot.

⚡️ Again, to keep it simple we will assume a maximum voltage, VMAX value of 100V.

📈 Plotting the instantaneous values at shorter intervals, for example at every 30° (12 points) or 10° (36 points), would result in a more accurate plotting of the sinusoidal waveform construction.

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✍️ Written by: Sisira Senevirathna

01/04/2026

⚡️🔄 BASIC AC GENERATOR & SINUSOIDAL EMF
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🧲 A simple generator consists of a pair of permanent magnets producing a fixed magnetic field between a north and a south pole. Inside this magnetic field is a single rectangular loop of wire that can be rotated around a fixed axis allowing it to cut the magnetic lines of flux at various angles as shown below.

⚙️ Basic Single Coil AC Generator
🔄 How a Sinusoidal EMF is Generated

🔁 As this single coil rotates anticlockwise around a central axis which is perpendicular to the magnetic field, the wire loop cuts the magnetic lines of force set up between the north and south poles at different angles as it rotates.

📊 Thus the amount of EMF induced in the wire loop at any instant of time is therefore proportional to the rotational angle of the wire loop as the rate of change of magnetic flux changes.

➡️ As this wire loop rotates from 0°, electrons in the wire flow in one direction around the loop. Now when the wire loop has rotated past the 180° point, it moves across the magnetic lines of force in the opposite direction.

🔄 As the coil moves through the flux the electrons in the wire loop change and flow in the opposite direction. Then the direction of the electron movement determines the polarity of the induced voltage.

🌊 So we can see that when the loop or coil physically rotates one complete revolution, or 360°, one full sinusoidal waveform is produced with one cycle of the waveform being produced for each full revolution of the coil.

🔌 Rotating the coil within the magnetic field means that the electrical connections made to the coil are by means of carbon brushes and slip-rings. These are used to transfer the electrical current induced in the coil to an external circuit.

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📌 What is the Induced EMF Proportional To?

⚡️ The amount of EMF induced into a coil cutting the magnetic lines of force is determined by the following three factors:

🚀 Speed – the speed at which the coil rotates inside the magnetic field.
🧲 Strength – the strength of the magnetic field.
📏 Length – the length of the coil or conductor passing through the magnetic field.

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📡 The Frequency of a Sinusoidal Waveform

⏱️ We know that the frequency of a supply is the number of times a waveform repeats itself per unit of time, usually in one second, and that frequency is measured in Hertz.

🔄 As one cycle of induced EMF is produced each full revolution of the coil through a magnetic field comprising of a north and south pole as shown above, if the coil rotates at a constant speed a constant number of cycles will be produced per second giving a constant frequency.

⚡️ So by increasing the speed of rotation of the coil the frequency will also be increased. Therefore, frequency is proportional to the speed of rotation, ( ƒ ∝ Ν ). Where Ν = rpm.

🧲 Also, our simple single coil generator above only has two poles, one north and one south pole, giving just one pair of poles. If we add more magnetic poles to increase the amount of magnetism inside the generator above so that it now has four poles in total, two north and two south, then for each revolution of the coil two cycles will be produced for the same rotational speed.

📊 Therefore, frequency is proportional to the number of pairs of magnetic poles, ( ƒ ∝ P ) of the generator. Where P = the number of “Pairs of Poles”. That is, two individual poles would be one pair of poles. So for a 4-pole generator, P = 2.

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✍️ Written by: Sisira Senevirathna

01/04/2026

⚡️🧲 ELECTROMAGNETIC INDUCTION
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⚡️ In the Electromagnetic Induction tutorial we said that when a single wire conductor moves through a permanent magnetic field thereby cutting its lines of flux, an EMF is induced in it.

🚫 However, when the rotating conductor moves in parallel with the magnetic field as in the case of points A and B, no lines of flux are cut or crossed. Therefore, no EMF is induced into the conductor.

📈 When the conductor moves at right angles to the magnetic field as in the case of points C and D, the maximum amount of magnetic lines of flux are cut between the north and south poles producing the maximum amount of induced EMF.

🔄 Therefore, when the conductor moves in an anticlockwise direction through the magnetic field at different angles rotating between points A and C (0° and 90°), the amount of induced EMF will lie somewhere between this zero and its maximum value.

📊 Then the amount of EMF induced within a conductor will depend on the rotational angle between the conductor and the corresponding magnetic flux, as well as the strength of the magnetic field in which it rotates.

🏭 An electrical generator uses the principle of Faraday’s electromagnetic induction to convert a mechanical energy such as rotation, into electrical energy, producing a Sinusoidal Waveform as it rotates.

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✍️ Written by: Sisira Senevirathna

01/04/2026

📡🔌 SINUSOIDAL WAVEFORMS
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⚡️ Sinusoidal Waveforms, or sine waves, are continuous, periodic alternating signals which can be defined mathematically by the sine or cosine function representing an AC voltage or current

🔄 The Periodic Sinusoidal Waveform
Sinusoidal Waveforms are continuous periodic waveforms whose smooth shape can be plotted using the trigonometric sine or cosine function to represent an AC voltage or current.

🔌 An electrical supply whose polarity changes every cycle between a maximum positive and negative value are commonly known as “AC” (Alternating Current) voltages and current sources.

⚙️ How are Sinusoidal Waveforms Produced

🧲 Sinusoidal waveforms are produced by electromagnetic induction when a coil of wire rotates at a constant speed within a uniform magnetic field. However, the reverse is also true.

🔋 When an electric current flows through a straight wire or conductor, a circular magnetic field is created around the wire whose strength is related to the strength of the current value.

🔄 If this single wire conductor is moved or rotated within a stationary magnetic field, an “EMF” (Electro-Motive Force) is induced within the conductor due to the movement of the conductor through the magnetic flux.

⚡️ From this fundamental effect we can see that a relationship exists between Electricity and Magnetism giving us, as Michael Faraday discovered the effect of “Electromagnetic Induction”.

🏭 It is this basic principle that electrical machines and generators use to generate a sinusoidal waveform supply.

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✍️ Written by: Sisira Senevirathna

24/03/2026

📘 Next Lesson 👉 Sinusoidal Waveforms

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✍️ Written by Sisira Senevirathna

24/03/2026

AC WAVEFORM EXAMPLE WORKED NO2
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⚡ A sinusoidal alternating current of 6 amps is flowing through a resistance of 40Ω. Calculate the average voltage and the peak voltage of the supply.

📊 The waveforms R.M.S. Voltage value is calculated as:

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🔄 The use and calculation of Average, RMS, Form Factor and Crest Factor can also be use with any type of periodic waveform including Triangular, Square, Sawtoothed or any other irregular or complex voltage/current waveform shape.

📐 The conversion between various sinusoidal values can sometimes be confusing so the following table gives a convenient way of converting one sine wave value to another.

🚀 In the next tutorial about Sinusoidal Waveforms we will look at the principal of generating a sinusoidal AC waveform (a sinusoid) along with its angular velocity representation.

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✍️ Written by Sisira Senevirathna

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