--- title: "Method of images for line capacitance" subject: "Power System Analysis I" chapter: 4 key: "semester_4/power_system_analysis_i:method-of-images" source: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/method-of-images site: BCT Engineering --- # Method of images for line capacitance Power System Analysis I · Chapter 4: Line Parameter Calculations - Asked in: 2077 Chaitra, 6 marks - Video: [41. Image Method for Capacitance Calculation](https://www.youtube.com/watch?v=MRdbBoB9ODM) — Bright Engineers - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/power_system_analysis_i/method-of-images - Whole chapter: https://www.bctengineer.com/md/semester_4/power_system_analysis_i/chapter-4 ## Question What is the method of images? How can it be used to take into account the presence of ground in calculating the capacitance of a line? ## Short answer Method of images replaces the earth’s conducting plane with imaginary conductors carrying equal and opposite charges, placed symmetrically below ground. Capacitance is calculated from potentials due to real and image charges. This accounts for the ground effect, which slightly increases line capacitance compared to neglecting earth. ## Answer The **method of images** is a technique for finding the electric field (and hence capacitance) of charged conductors above a conducting plane. The earth is treated as a perfectly conducting, zero-potential plane. Its effect is replaced by imaginary "image" conductors placed as far below the ground as the real conductors are above it, carrying equal and opposite charges. #### Basis - The earth's surface is an equipotential (zero potential) surface. - A charge $+q$ at height $h$ and a charge $-q$ at depth $h$ below the surface together produce zero potential everywhere on the plane midway between them. - So, for the region above ground, the field of "conductor + earth" is the same as the field of "conductor + image conductor" with the earth removed. ```text +q o a +q o b (real conductors) | h | ==================================== earth (V = 0) | h | -q o a' -q o b' (images) ``` #### Use in capacitance calculation 1. Replace the earth by image conductors: for each conductor of charge $q$ at height $h$, add an image of charge $-q$ at depth $h$. 2. Write the voltage of each conductor due to all real and image charges, using $V = \dfrac{q}{2\pi\varepsilon_0}\ln\dfrac{D_2}{D_1}$ for each charge. 3. For a single-phase line (conductors a, b, spacing $D$, height $h$), this gives $$ V_{ab} = \frac{q}{\pi\varepsilon_0}\ln\frac{2hD}{r\sqrt{4h^2+D^2}} $$ $$ C_{ab} = \frac{\pi\varepsilon_0}{\ln\left(\dfrac{D}{r\sqrt{1+D^2/4h^2}}\right)}\ \text{F/m} $$ 4. For a transposed three-phase line, average over the transposition cycle: $$ C_n = \frac{2\pi\varepsilon_0}{\ln\dfrac{D_{eq}}{r} - \ln\dfrac{\sqrt[3]{H_{12}H_{23}H_{31}}}{\sqrt[3]{H_1H_2H_3}}}\ \text{F/m} $$ where $H_1, H_2, H_3$ are the distances from each conductor to its own image and $H_{12}$ etc. are the distances to the images of the other conductors. #### Effect The extra term subtracted in the denominator is positive, so the denominator becomes smaller and the ground **increases** the line capacitance slightly. The effect is small when the conductors are high above ground compared with their spacing, so it is usually neglected except for low lines or accurate studies. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the method of images, where is the image of a conductor of charge +q at height h placed? A. At height h above the conductor with charge +q B. At depth h below the ground with charge -q C. At depth 2h below the ground with charge -q D. At depth h below the ground with charge +q Answer: B. The image lies as far below the ground as the real conductor is above, with equal and opposite charge, so the ground plane is at zero potential. (Key point: Image conductor) 2. Why can the earth be replaced by image conductors? A. Earth is a perfect insulator B. Earth is a zero-potential equipotential surface, and +q and -q together give zero potential on the plane midway C. Earth has infinite capacitance with no field D. Images cancel all the field above ground Answer: B. The real and image charges produce zero potential on the mid-plane, matching the earth surface, so the field above ground is unchanged. (Key point: Basis: earth equipotential) 3. What is the effect of the ground on line capacitance as found by the method of images? A. It decreases the capacitance slightly B. It has no effect at all C. It increases the capacitance slightly D. It makes the capacitance zero Answer: C. The extra term subtracted in the denominator is positive, so the denominator shrinks and capacitance increases slightly. (Key point: Effect on capacitance) 4. When can the ground effect on capacitance usually be neglected? A. For low lines with close spacing B. When conductors are high above ground compared with their spacing C. When the line is underground D. When the conductor radius is large Answer: B. For high conductors the correction term is small, so it is neglected except for low lines or accurate studies. 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