In the field of electrical power systems, the terms GMD (Geometric Mean Distance) and GMR (Geometric Mean Radius) are crucial for calculating inductance and capacitance of transmission lines. These parameters play a vital role in power system analysis, especially for high voltage transmission lines.
This article explains the meaning, formulas, and differences between GMD and GMR, along with their importance in power transmission. If you're preparing for exams like SSC JE, RRB JE, GATE, or UPPSC AE, this is a must-know concept.
What is GMR (Geometric Mean Radius)?
GMR, also known as Self GMD, is the equivalent radius of a conductor that accounts for the internal flux linkage due to the current flowing in the conductor itself.
🔹 GMR Definition:
GMR is the self-geometric mean distance of a conductor, which is used to calculate the inductance of a single conductor or a group of conductors.
🔹 GMR Formula:
For a single solid conductor:
Where:
r = actual radius of the conductor
r' = equivalent GMR radius
For bundled conductors or multiple conductors, GMR is calculated using the geometric mean of all mutual and self-distances.
What is GMD (Geometric Mean Distance)?
GMD, also known as Mutual GMD, is the average distance between two or more conductors. It is used to calculate the inductive reactance and capacitance between conductors in a transmission line.
🔹 GMD Definition:
GMD is the average of all the distances between the conductors of different phases.
🔹 GMD Formula:
For a 2-conductor system:
For a 3-phase system with conductors spaced unequally:
Where are the distances between the conductors.
Importance of GMR and GMD in Power Systems
Both GMR and GMD are essential in transmission line calculations:
| Parameter | Used For | Impact |
|---|---|---|
| GMR | Inductance calculation | Higher GMR → Lower inductance |
| GMD | Mutual inductance & capacitance | Higher GMD → Lower interference |
Inductance of a Single-Phase Line Using GMD and GMR
Where:r' = GMR of the conductor
Capacitance of a Transmission Line
Where:
- ε = Permittivity of the medium
- D = Distance between conductors (GMD)
- r= Radius of the conductor
Difference Between GMD and GMR
| Feature | GMD | GMR |
|---|---|---|
| Full Form | Geometric Mean Distance | Geometric Mean Radius |
| Also Known As | Mutual GMD | Self GMD |
| Type of Distance | Between conductors | Within a conductor |
| Use | For inductance and capacitance | For inductance only |
| Value | Always greater than GMR | Smaller value |
FAQs on GMD and GMR
❓ What is the unit of GMD and GMR?
Both are measured in meters (m).
❓ Why is GMR multiplied by 0.7788?
The factor 0.7788 comes from the mathematical integration of flux inside the conductor, accounting for the internal inductance.
❓ How does increasing GMD affect transmission lines?
Increasing GMD reduces inductive coupling and improves voltage regulation.
MCQs on GMD and GMR for Electrical Engineering Exams (SSC JE, RRB JE, GATE, UPPSC AE)
Question 1:
Which method is used to evaluate the inductance of a transmission line?
A) Proximity effect
B) Transposition of lines
C) Skin effect
D) Self GMD
Answer: D) Self GMD
Explanation: Self GMD (Geometric Mean Radius) is used to evaluate the inductance of a transmission line.
Question:
What is the Geometric Mean Radius (GMR) of a conductor with three strands, each of radius 'r', placed in the form of a triangle?
Options:
A) r × r × r
B) 3 × e^(-0.25) × r × 2r × 2r
C) 3 × e^(-0.5) × r × 2r × 2r
D) 3 × e^(-1/3) × r × 2r × 2r
Answer:
B) 3 × e^(-0.25) × r × 2r × 2r
Explanation:
When three strands are placed in a triangular shape, the GMR is calculated using their mutual distances and radius. The factor e^(-0.25) comes from the flux linkage calculations for such an arrangement.
Question 3:
By bundling conductors in a transmission line, what effect is observed?
A) Increase in inductance
B) Decrease in capacitance
C) Increase in GMR
D) Increase in resistance
Answer: C) Increase in GMR
Explanation: Bundling conductors increases the Geometric Mean Radius (GMR), which in turn reduces the inductance and increases the capacitance of the transmission line.
Question 4:
In a three-phase system with conductors spaced unequally, how is the Geometric Mean Distance (GMD) calculated?
A) Arithmetic mean of all distances
B) Harmonic mean of all distances
C) Geometric mean of all distances
D) Root mean square of all distances
Answer: C) Geometric mean of all distances
Explanation: GMD is the geometric mean of the distances between conductors in different phases.
Question 5:
Why is the factor 0.7788 used in calculating GMR for a solid conductor?
A) To account for the skin effect
B) To consider the proximity effect
C) To account for internal flux linkage
D) To adjust for temperature variations
Answer: C) To account for internal flux linkage
Explanation: The factor 0.7788 arises from the integration of magnetic field inside the conductor to calculate internal inductance.
Question 6:
What is the unit of measurement for both GMD and GMR?
A) Centimeters
B) Meters
C) Millimeters
D) Kilometers
Answer: B) Meters
Explanation: Both GMD and GMR are measured in meters (m).
Question 7:
How does increasing the Geometric Mean Distance (GMD) between conductors affect a transmission line?
A) Increases inductive coupling
B) Decreases inductive coupling
C) Increases voltage drop
D) Decreases line resistance
Answer: B) Decreases inductive coupling
Explanation: Increasing GMD reduces inductive coupling between conductors, improving voltage regulation and reducing interference.
Question 8:
Which of the following statements is true regarding GMD and GMR?
A) GMD is always less than GMR
B) GMD is always equal to GMR
C) GMD is always greater than GMR
D) GMD and GMR are unrelated
Answer: C) GMD is always greater than GMR
Explanation: In a transmission line, the GMD (distance between conductors) is always greater than the GMR (self-distance within a conductor).