Relay Protection Academy Module 03 of 25
03
Module 03 Foundational

Fundamental
Theory

⏱ ~2 hours 📖 IEC / IEEE 📑 15 slides

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§ 3.1

Why Mathematical Analysis?

Key concept: Phasor A sinusoid is represented as , where V is the RMS magnitude. Time derivatives become multiplication by .
  • §3.2: Vector algebra: four equivalent representations
  • §3.3: The 'a' operator: the 120° rotation building block of symmetrical components
  • §3.4: Circuit quantities: impedance, power, three-phase systems
  • §3.5: Circuit theorems: KVL/KCL, Thévenin, star/delta reduction
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Illustration prompt

Side-by-side illustration: left panel shows a sinusoidal AC waveform v(t) on a time axis; right panel shows the same quantity as a phasor arrow V at angle φ on a complex plane (real and imaginary axes). An equals sign connects the two representations. Dark background, clean technical style, blue accent color.

§ 3.2

Vector Algebra: Complex Quantities

Rectangular
Real part a, imaginary part b. Best for addition and subtraction.
Polar
Magnitude and angle. Best for multiplication and division.
Exponential (Euler)
. Used in derivations and signal analysis.
Trigonometric
Equivalent to exponential form. Makes real and imaginary components explicit.
Conversion Polar → Rect:  |  Rect → Polar:
§ 3.2

Arithmetic on Complex Quantities

Add / Subtract: use rectangular

Multiply: use polar

Divide: use polar

Conjugate

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Illustration prompt

Complex plane diagram showing phasor addition: two phasor arrows Z1 and Z2 drawn tip-to-tail, with the resultant Z1+Z2 completing the triangle. Separately shows phasor multiplication as angle addition (Z1 at 30°, Z2 at 45°, product Z1·Z2 at 75°). Dark background, labeled arrows in blue and green, clean technical style.

§ 3.3

The 'a' Operator

Definition
Multiplying any phasor by a rotates it 120° anticlockwise without changing its magnitude.
ExpressionPolarRectangularMeaning
Identity
Rotate 120° CCW
Rotate 240° CCW
Full rotation: returns to 1
Balanced phasors sum to zero
Purely imaginary; used in sequence networks
Convention The IEC convention uses a = 1∠120°. Some older IEEE texts define a = 1∠−120°. Confirm before mixing references.
§ 3.4

Circuit Quantities: Impedance

RMS (effective) values: what every relay nameplate uses Relay nameplates, settings, and test values are all stated in RMS. The RMS value delivers the same heating as an equal DC quantity:
Phasors represent RMS magnitudes, not peak values. When you read on a relay, that is 5 A rms; peak is .
Resistance R
Real part. Dissipates energy. Voltage in phase with current.
Inductive
Positive X. Voltage leads current 90°. Dominant in power system plant.
Capacitive
Negative X. Current leads voltage 90°. Important for long EHV lines.
Admittance
. Used in nodal (Y-bus) analysis.
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Illustration prompt

Impedance triangle diagram on a complex plane: horizontal axis = R (resistance), vertical axis = X (reactance). The hypotenuse is labeled |Z| with angle θ. Separate small diagrams show a resistor symbol (in-phase V and I), inductor symbol (V leads I by 90°), and capacitor symbol (I leads V by 90°). Dark background, IEC schematic style.

§ 3.4

Power in AC Circuits

Active power P [W]
Real part of S. Actual energy consumed. Positive for loads.
Reactive power Q [VAr]
Imaginary part of S. Positive Q = inductive; negative Q = capacitive (IEC convention).
Apparent power |S| [VA]
Determines equipment rating. Power factor .
Sign convention IEC 60375: Q positive for inductive (lagging current). Some older IEEE references use the opposite sign for Q; verify before mixing formulae.
§ 3.4

Three-Phase Systems

Under balanced conditions the three phases are equal in magnitude and displaced 120° using the a operator:

Star connection

Delta connection

Three-phase power

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Illustration prompt

Phasor diagram showing three balanced three-phase voltage phasors Va, Vb, Vc at 120° spacing on a complex plane. Each phasor is a different color (blue, green, yellow). Separately shows a star-connected winding diagram and a delta-connected winding diagram with labeled line and phase voltages. Dark background, clean technical style.

§ 3.5

Circuit Laws

KVL: Kirchhoff's Voltage Law Phasor sum of voltages around any closed loop = 0
KCL: Kirchhoff's Current Law Phasor sum of currents entering any node = 0

Series and Parallel

Sign conventions: double-suffix vs diagrammatic Double-suffix notation: voltage means the potential of node A with respect to node B; the subscript order defines polarity. Diagrammatic notation: an arrow on the diagram indicates the assumed positive direction; a negative result simply means the actual direction is reversed. IEC practice favours double-suffix notation. Mixing the two conventions in the same calculation is the most common source of sign errors in fault studies.
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Illustration prompt

Two IEC-style circuit diagrams side by side: left shows KVL with voltage source and three impedances in a loop, arrows showing voltage polarities summing to zero; right shows KCL at a node with four branch currents labeled I1–I4 with arrows, summing to zero. Dark background, clean schematic style, blue accent.

§ 3.5

Superposition & Thévenin's Theorem

Superposition In a linear network, the response at any point is the sum of responses from each source acting alone. Used to overlay the pre-fault network with the fault network.
Thévenin's Theorem Any two-terminal linear network reduces to one voltage source in series with one impedance :

In fault analysis is the pre-fault voltage and determines fault current magnitude.

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Illustration prompt

Two-step diagram showing Thévenin reduction: left panel shows a complex network with multiple sources and impedances connected to two terminals A and B; right panel shows the equivalent Thévenin circuit with a single voltage source Vth in series with impedance Zth at the same two terminals. A "simplifies to" arrow between them. IEC schematic style, dark background.

§ 3.5

Star ↔ Delta Conversion

Delta → Star (Δ→Y)

Star → Delta (Y→Δ)

Balanced special case
Reduction workflow Convert all deltas to star → combine series impedances → combine parallel impedances → apply Thévenin at the terminal of interest.
Mutual coupling: parallel lines on common towers When two transmission lines run on the same tower structure, their magnetic fields interact. Each line has a self-impedance and a mutual impedance . During network reduction, this mutual coupling must be modelled explicitly. The usual approach is to common the two neutral nodes and insert the mutual impedance as a separate branch, yielding modified series impedances and with in the common branch. Ignoring mutual coupling produces significant errors in the zero-sequence network, because zero-sequence currents flow in the same direction in both conductors and the coupling is strongest at that sequence. Distance relays calibrated without mutual coupling will overreach or underreach their intended protection zone.
§ 3.6

Per-Unit & Percentage Impedance

Every impedance in a power system has a physical value in ohms, but ohmic values change whenever the voltage base changes through a transformer. Per-unit (p.u.) notation eliminates this nuisance by expressing each quantity as a fraction of a chosen base.

Base quantities & per-unit definition Choose any two of {MVAb, kVb}; the rest follow:
Percentage impedance is simply .
Base-change formula Equipment data sheets quote Z% on the equipment's own rating. When combining items on a common system base, convert:
Why per-unit? In a correctly chosen per-unit system, transformer turns ratios disappear from the network equations: all impedances referred to either winding side give the same per-unit value. Fault currents and relay settings can then be stated as fractions of rated current, making them independent of voltage level and directly comparable across the network. Base voltages are chosen to follow transformer turns ratios, not nominal nameplate voltages.
Worked Example

Per-Unit Base Conversion: G1 & T1

Given: Generator G1 (66.6 MVA, 11 kV, X = 26%). Transformer T1 (75 MVA, 11/145 kV, X = 12.5%). System base: 100 MVA, 132 kV (HV side).

Conversion formula
Generator G1 (LV side)
kV bases match (11 kV = 11 kV), so voltage factor = 1.
Transformer T1 (HV side)
Nameplate HV = 145 kV; system base = 132 kV. The mismatch matters.
Why 145 kV, not 132 kV? The transformer nameplate voltage (145 kV) is its actual open-circuit turns ratio voltage, not the nominal system voltage (132 kV). Using 132 kV here gives a voltage factor of 1.0 and understates the transformer impedance by about 21%. This error would cause distance relays to underreach or overreach their intended protection zones. Always use the transformer's rated HV winding voltage in the base-change formula.
Interactive Tool

Phasor Calculator

Module 03

Knowledge Check