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VECTOR SUBTRACTION OF TWO PHASORS
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⚡ This time, instead of adding both the horizontal and vertical components, we subtract them.

📘 If:

A=x+jy
B=w+jz
📌 Then phasor subtraction becomes:
A−B=(x−w)+j(y−z)

✅ This gives the new resultant phasor after subtraction.

⚡ THE 3-PHASE PHASOR DIAGRAMS

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📘 Previously we looked at single-phase AC waveforms, where one rotating coil generates one sinusoidal voltage.

⚡ But if three identical coils are placed at an electrical angle of 120° to each other on the same rotor shaft, a three-phase voltage supply is generated.
120∘
⚡ BALANCED THREE-PHASE SUPPLY

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📌 A balanced three-phase voltage supply consists of:

✔️ Three sinusoidal voltages
✔️ Equal magnitude
✔️ Same frequency
✔️ 120° phase difference between each phase

⚡ STANDARD PHASE COLORS

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🔴 Red
🟡 Yellow
🔵 Blue

📘 Normal phase sequence:

R→Y→B

⚡ THREE-PHASE PHASOR ROTATION

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⚡ Just like single-phase phasors, three-phase phasors rotate anti-clockwise around a central point at angular velocity:

ω rad/s

📌 All phase voltages are equal in magnitude, only their phase angles are different.

⚡ THREE-PHASE VOLTAGE EQUATIONS

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📘 If Red phase is taken as the reference:

VRN​=V∠0∘
VYN​=V∠−120∘
VBN​=V∠+120∘

📌 Yellow phase lags Red by 120°
📌 Blue phase leads Red by 120°

⚡ BALANCED SYSTEM RULE

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📘 In a balanced three-phase system, the phasor sum is always zero:

Va​+Vb​+Vc​=0

✅ This is one of the most important three-phase rules.

⚡ PHASOR DIAGRAMS SUMMARY

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✔️ Phasor diagrams are graphical representations of AC voltages and currents.
✔️ They are drawn as rotating vectors.
✔️ Reference phasor is drawn on the horizontal x-axis.
✔️ Only sinusoidal AC quantities can be represented.
✔️ All phasors must have the same frequency.
✔️ Leading phasors are ahead of reference.
✔️ Lagging phasors are behind reference.
✔️ Phasor length usually represents RMS value.
✔️ Different frequencies cannot be shown correctly on same diagram.
✔️ Two or more phasors can be added/subtracted into one resultant vector.
✔️ Horizontal side = Real part (x)
✔️ Vertical side = Imaginary part (y)
✔️ Hypotenuse = Resultant (r) vector
✔️ In balanced 3-phase systems each phasor is displaced by 120°.

⚡ NEXT LESSON

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📘 In the next tutorial about AC Theory, we will study Complex Numbers in:

✔️ Rectangular Form
✔️ Polar Form
✔️ Exponential Form

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✍️ Written by Sisira Senevirathna

PHASOR DIAGRAMS AND PHASOR ALGEBRA
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⚡ Phasor Diagrams are a visual way of representing the magnitude and directional relationship between two or more alternating quantities.

⚡ WHAT ARE PHASOR DIAGRAMS OF A WAVEFORM?

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📘 Phasor Diagrams present a graphical representation, plotted on a coordinate system, of the phase relationship between the voltages and currents within passive components or a whole circuit.

📌 Generally, phasors are defined relative to a reference phasor, which always points to the right along the horizontal x-axis.

🔹 Sinusoidal waveforms of the same frequency can have a phase difference between themselves, which represents the angular difference of the two sinusoidal waveforms.

🔹 Also, the terms lead, lag, in-phase, and out-of-phase are commonly used to indicate the relationship of one sinusoidal waveform to another.

📘 The generalised sinusoidal expression is:
A(t)=Am​sin(ωt±Φ)
📌 This represents the sinusoid in the time-domain form.

⚡ But when presented mathematically in this way, it can sometimes be difficult to visualize the angular or phasor difference between two or more sinusoidal waveforms.

✅ One way to overcome this problem is to represent the sinusoids graphically in the phasor-domain form using Phasor Diagrams.

⚡ THE ROTATION OF VECTORS

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🔹 Basically a rotating vector, also regarded as a Phase Vector, is a scaled line whose length represents an AC quantity that has both:

✔️ Magnitude (Peak Value)
✔️ Direction (Phase Angle)

📌 It is considered “frozen” at some point in time.

🔹 A vector has an arrow head at one end which signifies:

➡️ Maximum value of the quantity (Vmax or Imax)
➡️ Direction of rotation

🔹 Generally, vectors pivot at one fixed point known as the point of origin or initial point.

📍 Usually this is where the coordinate axes intersect:

(0,0)

🔹 One end of the vector is anchored there, while the arrowed end rotates freely.

⚡ DIRECTION OF ROTATION

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🔄 Anti-clockwise rotation = Positive rotation
🔄 Clockwise rotation = Negative rotation

📘 The angular velocity is:

⚡ DIFFERENCE BETWEEN VECTOR AND PHASOR

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🔹 Although both terms are used for rotating quantities:

✔️ Vector magnitude = Peak value of the sinusoid
✔️ Phasor magnitude = RMS value of the sinusoid

📌 In both cases, the phase angle, direction, and angular velocity remain the same.

⚡ WHAT DOES A PHASOR DIAGRAM LOOK LIKE?

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📘 The phase of an alternating quantity at any instant in time can be represented by phasor diagrams.

📌 Thus, phasor diagrams can be thought of as representing functions of time.

🔹 A complete sine wave can be constructed by a single vector rotating anti-clockwise at angular velocity ω.

🔹 Then a phasor is a quantity that has both:

✔️ Magnitude
✔️ Direction

⚡ PHASOR ALGEBRA

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📌 Vectors obey the parallelogram law of addition and subtraction, so they can be added graphically to produce a vector sum.

📘 Phasors can also be represented mathematically in:

✔️ Rectangular Form
a+jb

✔️ Polar Form
✔️ Exponential Form

📌 Phasor notation defines the effective RMS voltage and RMS current magnitudes.

⚡ SUMMARY

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✅ Phasor diagrams simplify AC waveform analysis.
✅ Show phase angle clearly.
✅ Useful for voltage/current comparison.
✅ Important for AC circuit calculations.

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✍️ Written by Sisira Senevirathna

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 analyzing the behavior 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 zero values occurring along the horizontal baseline at:

➡️ 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 is represented by the Greek letter Φ (phi) or 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 0° to 360° or 0 to 2π radians depending on 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. 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 developed previously must be modified to take account of the phase angle of the waveform. This new general expression becomes.

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✍️ Written by Sisira Senevirathna