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Comparator Techniques in Protective Relays: Amplitude & Phase

Learn academic principles of amplitude and phase comparators in protective relays, featuring duality principles, Mho relays, and numerical implementation.

#protective-relays#electrical-engineering#power-systems#mho-relay#distance-protection#amplitude-comparator#phase-comparator#static-relays
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Comparator Techniques in Protective Relays

Concepts, Working Principles, Amplitude & Phase Comparator Methods

Amplitude Comparators   |   Phase Comparators   |   Relay Implementation

Department of Electrical Engineering   |   Power Systems & Protection

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Table of Contents

1
Introduction to Protective Relays
2
Role of Comparators in Relaying
3
Types of Comparators – Overview
4
Amplitude Comparator – Concept & Theory
5
Amplitude Comparator – Methods & Characteristics
6
Phase Comparator – Concept & Theory
7
Phase Comparator – Methods & Characteristics
8
Duality Between Amplitude & Phase Comparators
9
Mho Relay Implementation
10
Applications in Distance Protection

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Section 1

Introduction to Protective Relays

"A protective relay is an electrical device designed to detect abnormal conditions in a power system and initiate corrective action by tripping circuit breakers."

•
First line of defense in power system protection
•
Detects faults: short circuits, overloads, ground faults
•
Initiates tripping of circuit breakers to isolate faults
•
Must be: Fast, Selective, Sensitive, Reliable
System Block Diagram
Source
CT/PT
Relay
Circuit Breaker
Trip Signal
Load
2
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Section 1

Basic Relay Operating Principles

Relays operate based on measured electrical quantities:
Voltage (V), Current (I), Impedance (Z), Power (P)
Based fundamentally on an Operating Torque vs Restraining Torque principle.
General trip condition logic:
Operating quantity > Restraining quantity
Trip Condition
|S₁| > |S₂|
OR
∠(S₁ - S₂) > threshold
S₁ Operating Signal
S₂ Restraining Signal
OPERATING BOUNDARY Re Im S₁ S₂
Comparators are the core decision-making elements inside protective relays.
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Section 2

Role of Comparators in Protective Relaying

• A comparator is a device that compares two electrical quantities
• In relays, comparators compare: magnitudes (amplitude) or phase angles (phase)
• Output: Binary decision — Operate (Trip) or Restrain
• Replaces electromagnetic torque balance in digital/static relays
• Two input signals: S₁ (operating) and S₂ (restraining)
Input S₁
Input S₂
COMPARATOR
Trip Signal

Amplitude Comparator

compares |S₁| vs |S₂|

Phase Comparator

compares ∠(S₁ vs S₂)

The comparator defines the operating characteristic (circle, straight line, lens) of the relay on the R-X diagram.

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Section 3

Types of Comparators – Overview

AMPLITUDE COMPARATOR

  • Compares magnitudes of two signals
  • Operates when |S₁| ≥ |S₂|
  • Produces circular/offset characteristics
  • Examples: Overcurrent relay, Impedance relay
  • Also called: Modulus comparator

PHASE COMPARATOR

  • Compares phase angles of two signals
  • Operates when -90° ≤ ∠(S₁,S₂) ≤ +90°
  • Produces MHO / lens / straight line characteristics
  • Examples: Mho relay, Directional relay
  • Also called: Cosine / Sine comparator

Any amplitude comparator can be converted to a phase comparator and vice versa — DUALITY PRINCIPLE

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Section 4

Amplitude Comparator – Concept & Theory

An amplitude comparator compares the magnitudes of two input signals S₁ and S₂ and produces a trip output when the magnitude of S₁ exceeds that of S₂.
Operating Criterion
→ OPERATE if:  |S₁| ≥ |S₂|
→ RESTRAIN if: |S₁| < |S₂|
Where:
S₁ = Operating signal (function of fault current/voltage)
S₂ = Restraining signal (reference/threshold)
Key Characteristics
  • • Linear input-output relationship
  • • Can handle AC sinusoidal signals
  • • Implementation: rectifier bridge comparators, Hall-effect, digital logic
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Section 4

Amplitude Comparator – Mathematical Formulation

General form of input signals:

S₁ = aV + bI Operating
S₂ = cV + dI Restraining

Where:
V = voltage phasor
I = current phasor
a, b, c, d = complex weights

Operating condition in terms of impedance Z = V/I:

|aV + bI| ≥ |cV + dI|
Divide by |I| →
|aZ + b| ≥ |cZ + d|

This represents a CIRCLE on the R-X plane:

Center Zc = (d·ā - b·c̄) / (|a|² - |c|²)

Radius r = |a·d - b·c| / | |a|² - |c|² |

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Section 5

Amplitude Comparator – Methods of Implementation

Rectifier Bridge Method

Uses rectifier bridges to convert AC signals to DC. Compares DC levels. Simple and robust. Common in electromechanical and early static relays.

Peak Detection / Sampling Method

Samples instantaneous values of S₁ and S₂ at regular intervals. Digital processor compares sampled magnitudes. Used in numerical/digital relays.

RMS Comparison Method

Computes RMS values of S₁ and S₂ over one cycle. Compares RMS: If RMS(S₁) ≥ RMS(S₂) → trip. Most accurate for sinusoidal signals.

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Section 5

Amplitude Comparator – Characteristics on R-X Plane

The operating characteristic of an amplitude comparator is a CIRCLE on the R-X impedance plane.

Three Key Cases:

Case 1 |a| > |c| → Circle encloses origin → Impedance relay
Case 2 |a| = |c| → Straight line (infinite rad) → Reactance relay
Case 3 |a| < |c| → Circle excludes origin → Offset imp. relay

Key relay characteristics produced:

  • Plain impedance relay: Circle centered at origin
  • Offset mho relay: Circle offset from origin
  • Reactance relay: Horizontal straight line

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R jX O
Impedance Relay
Reactance Relay
Offset Mho
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Section 5

Amplitude Comparator – Applications in Relays

01

Overcurrent Relay

Compares fault current |Ifault| with pickup setting |Iset|.
Trips when |Ifault| ≥ |Iset|.
Equation S1 = I, S2 = Iset
02

Impedance Relay

Compares |V| with |I·Zset|.
Trips when |V| ≤ |I·Zset|.
Equation S1 = I·Zset, S2 = V
03

Differential Relay

Compares differential current |Id| with restraining current |Ir|.
Trips when |Id| ≥ k·|Ir|.
Equation S1 = Id, S2 = k·Ir
04

Voltage Relay

Compares terminal voltage |V| with reference |Vref|.
Trips on under/over voltage.
Equation S1 = Vref, S2 = V (for undervoltage)
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Section 6

Phase Comparator – Concept & Theory

A phase comparator compares the phase angle between two input signals S₁ and S₂. It produces a trip output when the phase difference falls within a defined range.

Operate if:
−90° ≤ ∠(S₁/S₂) ≤ +90°
i.e., cos∠(S₁, S₂) ≥ 0
Equivalent: Re(S₁ · S₂*) ≥ 0
Where S₂* = complex conjugate of S₂
🔷
Cosine Comparator: operates when cos(θ) ≥ 0 (θ = phase difference)
🔷
Sine Comparator: operates when sin(θ) ≥ 0

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SECTION 6

Phase Comparator – Mathematical Formulation

Input signals

S1 = aV + bI
S2 = cV + dI

Phase comparator operating condition

-90° ≤ arg(S1 / S2) ≤ 90°
Equivalently: Re(S1 · S̄2) ≥ 0

Expanding in terms of Z = V/I

Re[(aZ + b)(c̄Z̄ + d̄)] ≥ 0
This describes a CIRCLE passing through the origin on the R-X plane when appropriate coefficients are chosen.

The Mho Relay Condition

S1 = V, S2 = I · Zr (where Zr is relay setting)
Operating condition: -90° ≤ ∠(V / I · Zr) ≤ 90°
→ Circle passing through origin with diameter = Zr
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Section 7

Phase Comparator – Methods of Implementation

01

METHOD 1 – Coincidence Timing Method

Measures the time during which both S₁ and S₂ have the same polarity. If the coincidence time > T/2 (half cycle), the relay operates. Simple and widely used in static relays.

Coincidence time > T/2 → Operate

02

METHOD 2 – Product (Multiplication) Method

Computes the product S₁ × S₂. If the time-average of the product is positive: ⟨S₁·S₂⟩ > 0 → Trip. Equivalent to checking cos(θ) ≥ 0. Common in analog multiplier circuits.

⟨S₁·S₂⟩ = |S₁||S₂|cosθ ≥ 0 → Operate

03

METHOD 3 – Digital Phase Angle Measurement

Microprocessor measures the zero-crossing instants of S₁ and S₂. Calculates phase difference θ digitally. Trips if |θ| ≤ 90°. Used in modern numerical relays.

Discrete Fourier Transform (DFT) extracts phase angles

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Section 7

Phase Comparator – Characteristics on R-X Plane

The exact characteristic shape depends on the geometric mapping choices for S₁ and S₂ on the complex impedance plane:

Case 1 — Mho Relay

S₁ = V, S₂ = IZr

  • Circle passing through origin, diameter = Zr
  • Self-polarized Mho characteristic

Case 2 — Offset Mho

S₁ = V + IZr, S₂ = V - IZr

  • Circle offset from origin framing standard reach

Case 3 — Lens Characteristic

Intersecting Mho Circles

  • Lens-shaped operating zone
  • More selective, avoids load encroachment

Case 4 — Straight Line

S₁ = I·Zr, S₂ = V

  • Straight line through origin
  • Pure directional element operation

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Section 8

Duality Between Amplitude & Phase Comparators

DUALITY PRINCIPLE: Any amplitude comparator with inputs (S₁, S₂) is equivalent to a phase comparator with inputs (S₁+S₂, S₁-S₂), and vice versa.

Amplitude → Phase Conversion

Amplitude comparator: |S₁| ≥ |S₂|
Define: A = S₁ + S₂, B = S₁ - S₂
→ Equivalent phase comparator:-90° ≤ ∠(A,B) ≤ 90°
Proof note: |S₁|² ≥ |S₂|² ⇔ Re(S₁·S₂*) ≥ 0
(when |S₁|=|S₂|, on boundary)

Phase → Amplitude Conversion

Phase comparator: -90° ≤ ∠(S₁,S₂) ≤ 90°
Define: P = S₁ + S₂, Q = S₁ - S₂
→ Equivalent amplitude comparator:|P| ≥ |Q|
Amplitude Inputs Phase Inputs
(S₁, S₂) (S₁+S₂, S₁-S₂)
Impedance relay Mho relay

NOTE: This duality means the same physical relay hardware can be configured as either type.

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Section 9

Mho Relay – Phase Comparator Implementation

The Mho relay is the classic implementation of a phase comparator.

Relay Inputs from CT and VT

  • Voltage: V (from VT at relay location)
  • Current: I (from CT at relay location)

Comparator Signals

S₁ = V
(operating signal)
S₂ = I × Zr
(polarizing/reference signal, Zr = relay setting impedance)

Operating Condition

-90° ≤ ∠(V, I·Zr) ≤ +90°
→ Equivalent: -90° ≤ (∠Z - ∠Zr) ≤ +90°
→ The measured impedance Z lies within the Mho circle

The Mho Circle

  • Passes through the origin (R-X plane)
  • Diameter = |Zr| at angle ∠Zr
  • Self-polarized (uses its own voltage)
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Section 9

Directional Relay – Phase Comparator Application

A directional relay determines the direction of fault (forward or reverse).

Comparator signals:
S₁ = I (fault current)
S₂ = V_pol (polarizing voltage)
Usually 90° shifted or memory voltage.
Two polarization methods:
• Voltage polarized: Uses line voltage rotated by 90°
• Current polarized: Uses neutral or ground current
Operating Condition:
-90° ≤ ∠(I, V_pol) ≤ +90°
→ FORWARD fault → Trip
→ Outside range = REVERSE → Restrain
Maximum Torque Angle (MTA) / Max Sensitivity:
• The relay is most sensitive when ∠(S₁, S₂) = 0°
• MTA is adjusted to match line impedance angle (typically 60°–75°)
Phase Comparator Phasor Diagram
Forward Zone (Trip) Reverse Zone (Restrain) Zero Torque Line (±90° Boundary) Operating Range {-90°,+90°} V_pol (Ref / MTA) I_F (Forward Fault) I_R (Reverse Fault)
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Section 9

Distance Protection Using Comparator Techniques

Distance relays protect transmission lines by continuously measuring impedance Z = V/I. A fault immediately creates a dangerously low impedance value that actively falls inside the predefined relay characteristic.

Zone 1 80–85% of line Z  →  Instantaneous trip
Zone 2 120% of line Z  →  Time-delayed (0.3–0.5s)
Zone 3 220% of line Z  →  Backup (1.0–1.5s)

Comparator Roles in Distance Relay

  • Zone 1: S₁ = V, S₂ = 0.85·I·ZL (Phase/Mho)
  • Zone 2: S₁ = V, S₂ = 1.2·I·ZL
  • Each protection zone relies on an independent comparator.
R jX O Line Characteristic Z_L Zone 3 Zone 2 Zone 1 Z_F (Fault) Load Region

Fig: R-X Plane demonstrating concentric Mho characteristic circles passing through the origin.

! Relay trips if measured impedance Z falls inside ANY active zone circle.

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Section 10

Comparator Realization in Static Relays

Static Amplitude Comparator

Uses operational amplifiers (op-amps) and rectifier circuits.

S₁ → [Rectifier] → |S₁| ─┐ ├→ [Diff. Amp] → [Level Detector] → Trip S₂ → [Rectifier] → |S₂| ─┘

Key components

•
Full-wave bridge rectifiers
•
Differential op-amp
•
Schmitt trigger for clean switching
•
Output relay or thyristor

Static Phase Comparator (Coincidence Type)

Detects overlapping positive half-cycles of S₁ and S₂.

S₁ → [Zero-Crossing Detector] ─┐ ├→ [AND Gate] → [Timer] → Trip S₂ → [Zero-Crossing Detector] ─┘

Key components

•
Zero-crossing detectors
•
AND logic gate
•
Monostable timer (measures coincidence period)
•
Output: Trip if coincidence > T/2

"Static relays replaced electromagnetic relays in 1970s–1980s, offering faster response and improved accuracy."

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Section 10

Comparator Implementation in Numerical (Digital) Relays

CT / VT
➔
Analog Filter
➔
A/D Converter
➔
DSP / Microprocessor
➔
Comparator Algorithm
➔
Trip Logic

Amplitude Comparator in Numerical Relay

  • Computes phasor magnitudes using DFT (Discrete Fourier Transform)
  • |S₁|² = (Re_S₁)² + (Im_S₁)²
  • Compare: if |S₁| ≥ |S₂| → set Trip flag
  • Updated every half-cycle or per-sample

Phase Comparator in Numerical Relay

  • Computes phasors S₁ and S₂ via DFT
  • Phase angle:
    θ = arg(S₁) − arg(S₂) = arctan(Im/Re)
  • Or: uses dot product:
    Re(S₁·S₂*) ≥ 0 → Trip
  • Updated every sample interval (e.g., every 1ms)
i
Numerical relays sample at 1–4 kHz, process signals digitally, and implement multiple relay functions (overcurrent, distance, differential) simultaneously in software.
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Summary

Amplitude vs Phase Comparators – Comparative Analysis

Feature Amplitude Comparator Phase Comparator
Operating Criterion |S₁| ≥ |S₂| -90° ≤ ∠(S₁,S₂) ≤ +90°
R-X Characteristic Circle (any position) Circle through origin / Lens
Typical Relay Type Impedance, Overcurrent, Differential Mho, Directional, Distance
Implementation Rectifier bridge, RMS comparison Coincidence, Product, Phase angle
Effect of Arc Resistance Less affected More affected (circle through origin)
Load Encroachment Higher risk Lower risk (Mho is more directional)
Polarization Needed Not required Required for directional characteristic
Duality Can be converted to phase comparator Can be converted to amplitude comparator
Use in Modern Relays Yes (digital implementation) Yes (preferred for distance protection)

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Section 10

Practical Considerations & Limitations

Design Considerations

  • • CT/VT accuracy directly affects comparator inputs — must use class PS/X CTs
  • • Signal filtering needed to remove harmonics before comparison
  • • DC offset in fault current can cause errors — mitigated by Fourier filtering
  • • For phase comparators: polarization source must remain stable during faults
  • • Memory voltage polarization used when VT voltage collapses during close-in faults
  • • Temperature drift in analog circuits must be compensated

Limitations

  • • Amplitude comparators: susceptible to arc resistance (inflates impedance)
  • • Phase comparators (Mho): reduced reach for high resistance faults
  • • Load encroachment can cause incorrect operation during heavy load
  • • Mutual coupling on parallel lines can cause errors in phase measurement
  • • Instrument transformer errors introduce measurement inaccuracies
  • • Saturation of CTs during heavy faults can corrupt comparator inputs

Modern numerical relays use adaptive algorithms and multiple comparators in parallel zones to overcome these limitations.

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ADVANCED TOPICS

Advanced Topics: Quadrilateral Characteristic & Adaptive Relaying

Quadrilateral Characteristic

  • ■ Combines four separate comparator elements (boundaries)
  • ■ Boundaries configuration:
    • •Left boundary: Reactance line (horizontal)
    • •Right boundary: Resistance limiter
    • •Top boundary: Reactance limiter
    • •Bottom boundary: Directional element
  • ■ Created using FOUR phase comparators in AND logic
  • ■ Advantage: Handles high resistance faults better than Mho
  • ■ Used in: Line protection, especially for short/medium lines

Boundary Equations

Reactance: Im(V·Ī·e-jφ) ≤ Im(IZr·Ī·e-jφ)
Resistance: Re(V/I) ≤ RF_limit

Adaptive Relaying

  • ■ Relay settings are dynamically adjusted based on:
    • • System topology changes
    • • Load conditions
    • • Real-time network data from EMS/SCADA
  • ■ Comparator thresholds (Zr, angle) updated in real-time
  • ■ Benefits: Improved selectivity, prevents load encroachment
  • ■ Implementation in numerical relays via adaptive algorithms
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Conclusion

Summary & Conclusion

Comparator Fundamentals

Comparators are the decision core of protective relays, comparing either amplitudes or phase angles of two derived signals to produce trip/restrain outputs.

Amplitude Comparators

Operate when |S₁| ≥ |S₂|. Produce circular characteristics on R-X plane. Implemented via rectifier bridges or digital RMS comparison. Used in: Overcurrent, Impedance, Differential relays.

Phase Comparators

Operate when phase difference is within ±90°. Produce Mho/lens characteristics. Implemented via coincidence timing or digital DFT. Used in: Mho relay, Directional relay, Distance protection.

Duality Principle

Any amplitude comparator ↔ phase comparator conversion is possible. Inputs transform as: (S₁,S₂) ↔ (S₁+S₂, S₁-S₂). This unifies relay theory.

Modern numerical relays implement both comparator types digitally using DFT-based phasor estimation, offering flexibility, precision, and multiple protection functions in one device.

References: Phadke & Thorp – Computer Relaying | Blackburn – Protective Relaying | Anderson – Power System Protection

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Comparator Techniques in Protective Relays: Amplitude & Phase

Learn academic principles of amplitude and phase comparators in protective relays, featuring duality principles, Mho relays, and numerical implementation.

Comparator Techniques in Protective Relays

Concepts, Working Principles, Amplitude & Phase Comparator Methods

Amplitude Comparators &nbsp;&nbsp;|&nbsp;&nbsp; Phase Comparators &nbsp;&nbsp;|&nbsp;&nbsp; Relay Implementation

Department of Electrical Engineering &nbsp;&nbsp;|&nbsp;&nbsp; Power Systems & Protection

Table of Contents

Introduction to Protective Relays

Role of Comparators in Relaying

Types of Comparators – Overview

Amplitude Comparator – Concept & Theory

Amplitude Comparator – Methods & Characteristics

Phase Comparator – Concept & Theory

Phase Comparator – Methods & Characteristics

Duality Between Amplitude & Phase Comparators

Mho Relay Implementation

Applications in Distance Protection

Department of Electrical Engineering &nbsp;&nbsp;|&nbsp;&nbsp; Power Systems & Protection

Section 1

Introduction to Protective Relays

A protective relay is an electrical device designed to detect abnormal conditions in a power system and initiate corrective action by tripping circuit breakers.

First line of defense in power system protection

Detects faults: short circuits, overloads, ground faults

Initiates tripping of circuit breakers to isolate faults

Must be:

Fast, Selective, Sensitive, Reliable

System Block Diagram

Source

CT/PT

Relay

Circuit Breaker

Load

Trip Signal

2

Section 1

Basic Relay Operating Principles

Relays operate based on measured electrical quantities:<br><span style="color: #0a1f4b; font-weight: 700; display: block; margin-top: 8px;">Voltage (V), Current (I), Impedance (Z), Power (P)</span>

Based fundamentally on an <span style="color: #0a1f4b; font-weight: 700;">Operating Torque vs Restraining Torque</span> principle.

General trip condition logic:<br><span style="color: #0a1f4b; font-weight: 700; display: block; margin-top: 8px;">Operating quantity > Restraining quantity</span>

Trip Condition

Comparators are the core decision-making elements inside protective relays.

Section 2

Role of Comparators in Protective Relaying

A comparator is a device that compares two electrical quantities

In relays, comparators compare: magnitudes (amplitude) or phase angles (phase)

Output: Binary decision — Operate (Trip) or Restrain

Replaces electromagnetic torque balance in digital/static relays

Two input signals: S₁ (operating) and S₂ (restraining)

Input S₁

Input S₂

COMPARATOR

Trip Signal

Amplitude Comparator

compares |S₁| vs |S₂|

Phase Comparator

compares ∠(S₁ vs S₂)

The comparator defines the operating characteristic (circle, straight line, lens) of the relay on the R-X diagram.

Section 3

Types of Comparators – Overview

AMPLITUDE COMPARATOR

Compares magnitudes of two signals

Operates when |S₁| ≥ |S₂|

Produces circular/offset characteristics

<strong style="color: #c8962b;">Examples:</strong> Overcurrent relay, Impedance relay

<strong style="color: #c8962b;">Also called:</strong> Modulus comparator

PHASE COMPARATOR

Compares phase angles of two signals

Operates when -90° ≤ ∠(S₁,S₂) ≤ +90°

Produces MHO / lens / straight line characteristics

<strong style="font-weight: 800; color: #0a1f4b;">Examples:</strong> Mho relay, Directional relay

<strong style="font-weight: 800; color: #0a1f4b;">Also called:</strong> Cosine / Sine comparator

Any amplitude comparator can be converted to a phase comparator and vice versa — <strong style="color:#c8962b; font-weight: 800;">DUALITY PRINCIPLE</strong>

Section 4

Amplitude Comparator – Concept & Theory

An amplitude comparator compares the magnitudes of two input signals S₁ and S₂ and produces a trip output when the magnitude of S₁ exceeds that of S₂.

Operating signal (function of fault current/voltage)

Restraining signal (reference/threshold)

Section 4

Amplitude Comparator – Mathematical Formulation

Section 5

Amplitude Comparator – Methods of Implementation

Rectifier Bridge Method

Uses rectifier bridges to convert AC signals to DC. Compares DC levels. Simple and robust. Common in electromechanical and early static relays.

Peak Detection / Sampling Method

Samples instantaneous values of S₁ and S₂ at regular intervals. Digital processor compares sampled magnitudes. Used in numerical/digital relays.

RMS Comparison Method

Computes RMS values of S₁ and S₂ over one cycle. Compares RMS: If RMS(S₁) ≥ RMS(S₂) → trip. Most accurate for sinusoidal signals.

Section 5

Amplitude Comparator – Characteristics on R-X Plane

Key relay characteristics produced:

Department of Electrical Engineering &nbsp;&nbsp;|&nbsp;&nbsp; Power Systems & Protection

Section 5

Amplitude Comparator – Applications in Relays

01

Overcurrent Relay

Compares fault current |I<sub>fault</sub>| with pickup setting |I<sub>set</sub>|.<br/>Trips when |I<sub>fault</sub>| &ge; |I<sub>set</sub>|.

S<sub>1</sub> = I, S<sub>2</sub> = I<sub>set</sub>

02

Impedance Relay

Compares |V| with |I&middot;Z<sub>set</sub>|.<br/>Trips when |V| &le; |I&middot;Z<sub>set</sub>|.

S<sub>1</sub> = I&middot;Z<sub>set</sub>, S<sub>2</sub> = V

03

Differential Relay

Compares differential current |I<sub>d</sub>| with restraining current |I<sub>r</sub>|.<br/>Trips when |I<sub>d</sub>| &ge; k&middot;|I<sub>r</sub>|.

S<sub>1</sub> = I<sub>d</sub>, S<sub>2</sub> = k&middot;I<sub>r</sub>

04

Voltage Relay

Compares terminal voltage |V| with reference |V<sub>ref</sub>|.<br/>Trips on under/over voltage.

S<sub>1</sub> = V<sub>ref</sub>, S<sub>2</sub> = V <span style="font-family: 'Segoe UI', Roboto, sans-serif; font-size: 18px; font-weight: 400; color: #666; font-style: normal; margin-left: 10px;">(for undervoltage)</span>

Section 6

Phase Comparator – Concept & Theory

A phase comparator compares the phase angle between two input signals S₁ and S₂. It produces a trip output when the phase difference falls within a defined range.

Operate if:

−90° ≤ ∠(S₁/S₂) ≤ +90°

i.e., cos∠(S₁, S₂) ≥ 0

Equivalent: Re(S₁ · S₂*) ≥ 0

Where S₂* = complex conjugate of S₂

Cosine Comparator:

operates when cos(θ) ≥ 0 (θ = phase difference)

Sine Comparator:

operates when sin(θ) ≥ 0

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Phase Comparator – Mathematical Formulation

SECTION 6

Department of Electrical Engineering &nbsp;&nbsp;|&nbsp;&nbsp; Power Systems & Protection

<i>S<sub>1</sub> = aV + bI</i>

<i>S<sub>2</sub> = cV + dI</i>

-90&deg; &le; arg(<i>S<sub>1</sub> / S<sub>2</sub></i>) &le; 90&deg;

Re(<i>S<sub>1</sub> &middot; S̄<sub>2</sub></i>) &ge; 0

Re[(<i>aZ + b</i>)(<i>c̄Z̄ + d̄</i>)] &ge; 0

This describes a CIRCLE passing through the origin on the R-X plane when appropriate coefficients are chosen.

<i>S<sub>1</sub> = V, S<sub>2</sub> = I &middot; Z<sub>r</sub></i>

-90&deg; &le; &ang;(<i>V / I &middot; Z<sub>r</sub></i>) &le; 90&deg;

Circle passing through origin with diameter = <i>Z<sub>r</sub></i>

Section 7

Phase Comparator – Methods of Implementation

METHOD 1 – Coincidence Timing Method

Measures the time during which both S₁ and S₂ have the same polarity. If the coincidence time > T/2 (half cycle), the relay operates. Simple and widely used in static relays.

Coincidence time > T/2 → Operate

METHOD 2 – Product (Multiplication) Method

Computes the product S₁ × S₂. If the time-average of the product is positive: ⟨S₁·S₂⟩ > 0 → Trip. Equivalent to checking cos(θ) ≥ 0. Common in analog multiplier circuits.

⟨S₁·S₂⟩ = |S₁||S₂|cosθ ≥ 0 → Operate

METHOD 3 – Digital Phase Angle Measurement

Microprocessor measures the zero-crossing instants of S₁ and S₂. Calculates phase difference θ digitally. Trips if |θ| ≤ 90°. Used in modern numerical relays.

Discrete Fourier Transform (DFT) extracts phase angles

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Section 7

Phase Comparator – Characteristics on R-X Plane

The exact characteristic shape depends on the geometric mapping choices for S₁ and S₂ on the complex impedance plane:

Case 1 — Mho Relay

S₁ = V, S₂ = IZr

<ul style="margin:0; padding-left: 20px; color: #555;"><li style="margin-bottom: 5px;">Circle passing through origin, diameter = Zr</li><li>Self-polarized Mho characteristic</li></ul>

Case 2 — Offset Mho

S₁ = V + IZr, S₂ = V - IZr

<ul style="margin:0; padding-left: 20px; color: #555;"><li>Circle offset from origin framing standard reach</li></ul>

Case 3 — Lens Characteristic

Intersecting Mho Circles

<ul style="margin:0; padding-left: 20px; color: #555;"><li style="margin-bottom: 5px;">Lens-shaped operating zone</li><li>More selective, avoids load encroachment</li></ul>

Case 4 — Straight Line

S₁ = I·Zr, S₂ = V

<ul style="margin:0; padding-left: 20px; color: #555;"><li style="margin-bottom: 5px;">Straight line through origin</li><li>Pure directional element operation</li></ul>

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Section 8

Duality Between Amplitude & Phase Comparators

<strong style="font-weight: 800; letter-spacing: 0.5px;">DUALITY PRINCIPLE:</strong> Any amplitude comparator with inputs <span style="font-family: 'Georgia', serif; font-weight: 700;">(S₁, S₂)</span> is equivalent to a phase comparator with inputs <span style="font-family: 'Georgia', serif; font-weight: 700;">(S₁+S₂, S₁-S₂)</span>, and vice versa.

Amplitude → Phase Conversion

<div style="font-size: 23px; color: #444; line-height: 1.6;"> <div style="display: flex; align-items: baseline; margin-bottom: 16px;"> <div style="width: 8px; height: 8px; background: #c8962b; border-radius: 50%; margin-right: 15px; transform: translateY(-3px); flex-shrink: 0;"></div> <span>Amplitude comparator: <strong style="color: #0a1f4b; font-family: 'Georgia', serif; margin-left: 8px; font-size: 26px;">|S₁| ≥ |S₂|</strong></span> </div> <div style="display: flex; align-items: baseline; margin-bottom: 22px;"> <div style="width: 8px; height: 8px; background: #c8962b; border-radius: 50%; margin-right: 15px; transform: translateY(-3px); flex-shrink: 0;"></div> <span>Define: <strong style="color: #0a1f4b; font-family: 'Georgia', serif; margin-left: 8px; font-size: 26px;">A = S₁ + S₂, B = S₁ - S₂</strong></span> </div> <div style="display: flex; align-items: center; margin-bottom: 22px; padding: 14px 20px; background: rgba(200, 150, 43, 0.1); border-left: 5px solid #c8962b; border-radius: 0 8px 8px 0;"> <span style="font-size: 24px; color: #c8962b; margin-right: 15px; font-weight: bold;">&rarr;</span> <span>Equivalent phase comparator:<strong style="color: #0a1f4b; font-family: 'Georgia', serif; margin-left: 12px; font-size: 26px;">-90° ≤ ∠(A,B) ≤ 90°</strong></span> </div> <div style="font-size: 20px; color: #555; font-style: italic; background: rgba(10, 31, 75, 0.04); padding: 12px 20px; border-radius: 6px;"> <strong style="color: #0a1f4b;">Proof note:</strong> |S₁|² ≥ |S₂|² &iff; Re(S₁&middot;S₂*) ≥ 0 <br> <span style="font-size: 18px; color:#777;">(when |S₁|=|S₂|, on boundary)</span> </div> </div>

Phase → Amplitude Conversion

<div style="font-size: 23px; color: #444; line-height: 1.6;"> <div style="display: flex; align-items: baseline; margin-bottom: 16px;"> <div style="width: 8px; height: 8px; background: #c8962b; border-radius: 50%; margin-right: 15px; transform: translateY(-3px); flex-shrink: 0;"></div> <span>Phase comparator: <strong style="color: #0a1f4b; font-family: 'Georgia', serif; margin-left: 8px; font-size: 26px;">-90° ≤ ∠(S₁,S₂) ≤ 90°</strong></span> </div> <div style="display: flex; align-items: baseline; margin-bottom: 22px;"> <div style="width: 8px; height: 8px; background: #c8962b; border-radius: 50%; margin-right: 15px; transform: translateY(-3px); flex-shrink: 0;"></div> <span>Define: <strong style="color: #0a1f4b; font-family: 'Georgia', serif; margin-left: 8px; font-size: 26px;">P = S₁ + S₂, Q = S₁ - S₂</strong></span> </div> <div style="display: flex; align-items: center; margin-bottom: 22px; padding: 14px 20px; background: rgba(200, 150, 43, 0.1); border-left: 5px solid #c8962b; border-radius: 0 8px 8px 0;"> <span style="font-size: 24px; color: #c8962b; margin-right: 15px; font-weight: bold;">&rarr;</span> <span>Equivalent amplitude comparator:<strong style="color: #0a1f4b; font-family: 'Georgia', serif; margin-left: 12px; font-size: 26px;">|P| ≥ |Q|</strong></span> </div> </div>

This duality means the same physical relay hardware can be configured as either type.

Section 9

Mho Relay – Phase Comparator Implementation

The Mho relay is the classic implementation of a phase comparator.

Relay Inputs from CT and VT

Voltage: V (from VT at relay location)

Current: I (from CT at relay location)

Comparator Signals

S₁ = V

(operating signal)

S₂ = I × Zr

(polarizing/reference signal, Zr = relay setting impedance)

Operating Condition

-90° ≤ ∠(V, I·Zr) ≤ +90°

→ Equivalent: -90° ≤ (∠Z - ∠Zr) ≤ +90°

→ The measured impedance Z lies within the Mho circle

The Mho Circle

Passes through the origin (R-X plane)

Diameter = |Zr| at angle ∠Zr

Self-polarized (uses its own voltage)

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Section 9

Directional Relay – Phase Comparator Application

Department of Electrical Engineering &nbsp;&nbsp;|&nbsp;&nbsp; Power Systems & Protection

Section 9

Distance Protection Using Comparator Techniques

Distance relays protect transmission lines by continuously measuring impedance <b style="color: #0a1f4b; font-size: 27px;">Z = V/I</b>. A fault immediately creates a dangerously low impedance value that actively falls <b style="color: #c8962b;">inside the predefined relay characteristic</b>.

<svg width="680" height="680" viewBox="0 0 700 700" xmlns="http://www.w3.org/2000/svg" style="border-radius:12px; font-family:'Segoe UI', sans-serif;"> <defs> <marker id="arrowHead" markerWidth="6" markerHeight="6" refX="5" refY="3" orient="auto"> <path d="M0,0 L0,6 L6,3 z" fill="#333" /> </marker> </defs> <!-- Faint Grid for visual context --> <g stroke="rgba(0,0,0,0.03)" stroke-width="1.5"> <line x1="200" y1="120" x2="680" y2="120"/> <line x1="200" y1="220" x2="680" y2="220"/> <line x1="200" y1="320" x2="680" y2="320"/> <line x1="200" y1="420" x2="680" y2="420"/> <line x1="300" y1="50" x2="300" y2="520"/> <line x1="400" y1="50" x2="400" y2="520"/> <line x1="500" y1="50" x2="500" y2="520"/> <line x1="600" y1="50" x2="600" y2="520"/> </g> <!-- X-axis (R) --> <line x1="80" y1="520" x2="650" y2="520" stroke="#333" stroke-width="2.5" marker-end="url(#arrowHead)" /> <text x="640" y="555" fill="#222" font-size="22" font-weight="bold" font-style="italic">R</text> <!-- Y-axis (jX) --> <line x1="200" y1="620" x2="200" y2="50" stroke="#333" stroke-width="2.5" marker-end="url(#arrowHead)" /> <text x="160" y="65" fill="#222" font-size="22" font-weight="bold" font-style="italic">jX</text> <!-- Origin point --> <text x="175" y="545" fill="#222" font-size="22" font-weight="bold">O</text> <!-- Line Impedance Characteristic (Angle = 60 deg) --> <line x1="200" y1="520" x2="490" y2="17.72" stroke="#444" stroke-width="3" stroke-dasharray="6,4" /> <text x="390" y="165" fill="#444" font-size="20" font-weight="700" transform="rotate(-60 390 165)">Line Characteristic Z_L</text> <!-- Center and Circle Formulas: Angle=60, Origin=(200, 520) Cx = 200 + R*0.5 Cy = 520 - R*0.86602 --> <!-- Zone 3 Circle (Radius 265) --> <circle cx="332.5" cy="290.5" r="265" fill="rgba(136,136,136,0.02)" stroke="#888888" stroke-width="3" stroke-dasharray="14,10" /> <text x="605" y="295" fill="#888888" font-size="22" font-weight="bold">Zone 3</text> <!-- Zone 2 Circle (Radius 150) --> <circle cx="275" cy="390.1" r="150" fill="rgba(200,150,43,0.03)" stroke="#c8962b" stroke-width="4.5" stroke-dasharray="12,8" /> <text x="440" y="445" fill="#c8962b" font-size="22" font-weight="bold">Zone 2</text> <!-- Zone 1 Circle (Radius 100) --> <circle cx="250" cy="433.4" r="100" fill="rgba(10,31,75,0.08)" stroke="#0a1f4b" stroke-width="4" /> <text x="325" y="510" fill="#0a1f4b" font-size="22" font-weight="bold">Zone 1</text> <!-- Fault Point Z_F (Inside Zone 1, dist 120 along line -> X=260, Y=416) --> <circle cx="260" cy="416" r="8" fill="#e53935" stroke="#fff" stroke-width="2.5" /> <text x="275" y="415" fill="#e53935" font-size="22" font-weight="bold">Z_F (Fault)</text> <!-- Marker for Origin to Z_F --> <line x1="200" y1="520" x2="260" y2="416" stroke="#e53935" stroke-width="4" /> <!-- Load Region (Bottom Right Quadrant representation) --> <path d="M 400 450 Q 550 420 620 450 Q 640 500 500 500 Q 400 480 400 450 Z" fill="rgba(10,31,75,0.05)" stroke="#0a1f4b" stroke-width="2" stroke-dasharray="6,6" /> <text x="460" y="475" fill="#0a1f4b" font-size="20" font-weight="bold">Load Region</text> </svg>

Section 10

Comparator Realization in Static Relays

Static Amplitude Comparator

Uses operational amplifiers (op-amps) and rectifier circuits.

S₁ → [Rectifier] → |S₁| ─┐ ├→ [Diff. Amp] → [Level Detector] → Trip S₂ → [Rectifier] → |S₂| ─┘

<div style="display: flex; flex-direction: column; gap: 14px;"><div style="display: flex; align-items: flex-start;"><span style="color: #c8962b; margin-right: 15px; font-weight: bold; font-size: 26px; line-height: 22px;">•</span><div>Full-wave bridge rectifiers</div></div><div style="display: flex; align-items: flex-start;"><span style="color: #c8962b; margin-right: 15px; font-weight: bold; font-size: 26px; line-height: 22px;">•</span><div>Differential op-amp</div></div><div style="display: flex; align-items: flex-start;"><span style="color: #c8962b; margin-right: 15px; font-weight: bold; font-size: 26px; line-height: 22px;">•</span><div>Schmitt trigger for clean switching</div></div><div style="display: flex; align-items: flex-start;"><span style="color: #c8962b; margin-right: 15px; font-weight: bold; font-size: 26px; line-height: 22px;">•</span><div>Output relay or thyristor</div></div></div>

Static Phase Comparator (Coincidence Type)

Detects overlapping positive half-cycles of S₁ and S₂.

S₁ → [Zero-Crossing Detector] ─┐ ├→ [AND Gate] → [Timer] → Trip S₂ → [Zero-Crossing Detector] ─┘

<div style="display: flex; flex-direction: column; gap: 14px;"><div style="display: flex; align-items: flex-start;"><span style="color: #c8962b; margin-right: 15px; font-weight: bold; font-size: 26px; line-height: 22px;">•</span><div>Zero-crossing detectors</div></div><div style="display: flex; align-items: flex-start;"><span style="color: #c8962b; margin-right: 15px; font-weight: bold; font-size: 26px; line-height: 22px;">•</span><div>AND logic gate</div></div><div style="display: flex; align-items: flex-start;"><span style="color: #c8962b; margin-right: 15px; font-weight: bold; font-size: 26px; line-height: 22px;">•</span><div>Monostable timer (measures coincidence period)</div></div><div style="display: flex; align-items: flex-start;"><span style="color: #c8962b; margin-right: 15px; font-weight: bold; font-size: 26px; line-height: 22px;">•</span><div>Output: Trip if coincidence > T/2</div></div></div>

Static relays replaced electromagnetic relays in 1970s–1980s, offering faster response and improved accuracy.

Section 10

Comparator Implementation in Numerical (Digital) Relays

CT / VT

Analog Filter

A/D Converter

DSP / Microprocessor

Comparator Algorithm

Trip Logic

Amplitude Comparator in Numerical Relay

Computes phasor magnitudes using <b style="color: #0a1f4b;">DFT</b> (Discrete Fourier Transform)

<span style="color: #0a1f4b; font-size: 28px; font-family: monospace;"><b>|S&#8321;|&#178;</b> = (Re_S&#8321;)&#178; + (Im_S&#8321;)&#178;</span>

Compare: if <span style="color: #0a1f4b; font-size: 28px; font-family: monospace;"><b>|S&#8321;| &ge; |S&#8322;|</b></span> &rarr; set Trip flag

Updated every half-cycle or per-sample

Phase Comparator in Numerical Relay

Computes phasors <b style="color: #0a1f4b;">S&#8321;</b> and <b style="color: #0a1f4b;">S&#8322;</b> via DFT

Phase angle:<br/><span style="color: #0a1f4b; font-size: 28px; font-family: monospace;"><b>&theta;</b> = arg(S&#8321;) &minus; arg(S&#8322;) = arctan(Im/Re)</span>

Or: uses dot product:<br/><span style="color: #0a1f4b; font-size: 28px; font-family: monospace;"><b>Re(S&#8321;&middot;S&#8322;*) &ge; 0</b></span> &rarr; Trip

Updated every sample interval (e.g., every 1ms)

Numerical relays sample at 1&ndash;4 kHz, process signals digitally, and implement multiple relay functions (overcurrent, distance, differential) simultaneously in software.

Summary

Amplitude vs Phase Comparators – Comparative Analysis

Feature

Amplitude Comparator

Phase Comparator

Operating Criterion

|S₁| ≥ |S₂|

-90° ≤ ∠(S₁,S₂) ≤ +90°

R-X Characteristic

Circle (any position)

Circle through origin / Lens

Typical Relay Type

Impedance, Overcurrent, Differential

Mho, Directional, Distance

Implementation

Rectifier bridge, RMS comparison

Coincidence, Product, Phase angle

Effect of Arc Resistance

Less affected

More affected (circle through origin)

Load Encroachment

Higher risk

Lower risk (Mho is more directional)

Polarization Needed

Not required

Required for directional characteristic

Duality

Can be converted to phase comparator

Can be converted to amplitude comparator

Use in Modern Relays

Yes (digital implementation)

Yes (preferred for distance protection)

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Section 10

Practical Considerations & Limitations

Design Considerations

CT/VT accuracy directly affects comparator inputs — must use class PS/X CTs

Signal filtering needed to remove harmonics before comparison

DC offset in fault current can cause errors — mitigated by Fourier filtering

For phase comparators: polarization source must remain stable during faults

Memory voltage polarization used when VT voltage collapses during close-in faults

Temperature drift in analog circuits must be compensated

Limitations

Amplitude comparators: susceptible to arc resistance (inflates impedance)

Phase comparators (Mho): reduced reach for high resistance faults

Load encroachment can cause incorrect operation during heavy load

Mutual coupling on parallel lines can cause errors in phase measurement

Instrument transformer errors introduce measurement inaccuracies

Saturation of CTs during heavy faults can corrupt comparator inputs

Modern numerical relays use adaptive algorithms and multiple comparators in parallel zones to overcome these limitations.

ADVANCED TOPICS

Advanced Topics: Quadrilateral Characteristic & Adaptive Relaying

Quadrilateral Characteristic

Adaptive Relaying

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Conclusion

Summary & Conclusion

Comparator Fundamentals

Comparators are the decision core of protective relays, comparing either amplitudes or phase angles of two derived signals to produce trip/restrain outputs.

Amplitude Comparators

Operate when |S₁| ≥ |S₂|. Produce circular characteristics on R-X plane. Implemented via rectifier bridges or digital RMS comparison. Used in: Overcurrent, Impedance, Differential relays.

Phase Comparators

Operate when phase difference is within ±90°. Produce Mho/lens characteristics. Implemented via coincidence timing or digital DFT. Used in: Mho relay, Directional relay, Distance protection.

Duality Principle

Any amplitude comparator ↔ phase comparator conversion is possible. Inputs transform as: (S₁,S₂) ↔ (S₁+S₂, S₁-S₂). This unifies relay theory.

Modern numerical relays implement both comparator types digitally using DFT-based phasor estimation, offering flexibility, precision, and multiple protection functions in one device.

References: Phadke & Thorp – Computer Relaying | Blackburn – Protective Relaying | Anderson – Power System Protection

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  • electrical-engineering
  • power-systems
  • mho-relay
  • distance-protection
  • amplitude-comparator
  • phase-comparator
  • static-relays