← Back to the TL-SAG App

⚡ TL-SAG Physics Sandbox

Understand the physics, mathematics, and field application of sag measurement methods

📷 Looking for the flagship Photo Catenary method? Trace the conductor in a span photograph and the app fits a catenary — tension comes directly from the curve shape (T = w·C). Its geometry model and accuracy physics are explained inside the app itself, under "How the photo calibration works" in the Photo Sag Tracker panel. The three classical methods below remain the best cross-checks for it.
Interactive Simulation

Visual Sag Board Sighting

Manually clamp visual boards on both towers to sight the bottom curve tangent.

Board A Board B GRAVITY HORIZONTAL DATUM PLANE L (Horizontal Span w.r.t. Gravity) Target Sag (d) Target Sag (d)
🔴 Conductor Out of Alignment

Adjust the tension slider. The lowest point of the conductor must touch the yellow dashed line of sight between the two target boards clamped to the towers.

The Physics

Target Sighting Method

This is a mechanical, direct visual sagging method that dates back to early line construction, but remains widely used due to its simplicity.

How it Works:

  • Step 1: Calculate Target Sag: Engineers compute the required sag for the given temperature and span, using sag-tension tables.
  • Step 2: Place Sighting Boards: Clamps or target boards are placed on both towers at a vertical distance exactly equal to the target sag **below** the conductor support point.
  • Step 3: Direct Sighting: A lineman stands on Tower A, placing their eye in line with Board A, and sights across to Board B. This visual axis forms a dashed sight line tangent to the target catenary curve.
  • Step 4: Pull to Tangency: The ground crew tensions the conductor. As it rises, the bottom of the loop climbs. When it is perfectly aligned with the sight line, tension is clamped off.

Quick Target Sag Calculator

Required Pull Tension
19.45 kN
T = (w * L^2) / (8 * D) T = (15.912 * 300^2) / (8 * 9.2) T = 1,432,080 / 73.6 = 19,457.6 N
Interactive Simulation

Return Wave Timing

Kick the wire, time the reflection travels with a stopwatch, and extract tension.

GRAVITY HORIZONTAL DATUM PLANE L (Horizontal Span w.r.t. Gravity) d (Mid-span Sag w.r.t. Gravity)
Stopwatch 0.00s
Returns 0 / 3
Wave Timing Calculation Output

Click "Kick Cable" to initiate a wave. The time it takes for 3 return trips will allow us to calculate both the sag and the tension.

The Physics

Return Wave Method

The Return Wave Method relies on the physics of **transverse wave propagation** through a tensioned string. It is highly accurate and requires zero optical survey instruments.

The Physics Equations:

  • Wave Velocity: A physical pulse travels along a cable at a velocity ($v$) that depends entirely on its tension ($T$) and mass per unit length ($m$):
    v = sqrt( Tension / mass )
  • Loop Distance: A round trip from one tower to the other and back covers a distance of exactly `2 * Span` ($2 \cdot L$). For $N$ complete return trips, the total travel distance is `2 * N * L`.
  • Stopwatch Relation: By measuring the total time ($t$) for $N$ returns:
    v = (2 * N * L) / t
  • Direct Sag Extract: By substituting the velocity equation into the standard level-span sag formula, we get the elegant direct stopwatch relationship:
    Sag = (g * t^2) / (32 * N^2)
    Where `g = 9.81 m/s²`.

Wave Stopwatch Calculator

Calculated Sag & Tension
SAG:
8.51 m
TENSION:
18.29 kN
Sag = (9.81 * t^2) / (32 * N^2) Sag = (9.81 * 15.8^2) / (32 * 3^2) = 2448.97 / 288 = 8.50 m Tension = (mass * g * L^2) / (8 * Sag) T = (1.622 * 9.81 * 280^2) / (8 * 8.50) = 18,290 N = 18.29 kN
Interactive Simulation

Three-Point Coordinate Sighting

Shoot a laser to both tower support tips and any arbitrary point on the wire.

A (0, yA) B (L, yB) P (xp, yp) D(xp) GRAVITY HORIZONTAL DATUM PLANE L = 300.0 m xp = 100.0 m h = 35.00 m
Survey calculations at point P
xp Pos:
300.0 m
Chord y:
100.0 m
Chord-Sag D(xp):
8.50 m
The Physics

Three-Point Survey Method

By capturing three 3D coordinate points along a single span, a total station survey instrument can mathematically model the conductor's parabolic catenary curve in space, automatically determining both the sag and internal horizontal tension.

⚠️ GEOMETRICAL REFERENCE CLARIFICATION:
In all transmission line sag-tension analysis, the Span Length (L) and the Measurement Position (xp) are strictly the horizontal projections w.r.t. gravity (measured along a horizontal datum plane, e.g., using a Total Station or GPS map projection), NOT the sloped linear slant distance between tower hooks. The Elevation Difference (h) and Sag Offsets (D) are strictly the vertical projections parallel to the vector of gravity (absolute elevation sea level differences).

The Mathematical Derivation:

  • 1. Reference the Chord Line: Connect support A at `(0, yA)` and support B at `(L, yB)` with a straight line. At any position `xp`, the height of this chord is:
    y_chord(xp) = yA + (h / L) * xp
  • 2. Measure Sag Offset: The vertical sag offset `D(xp)` is the difference between the chord line and the measured wire height `yp`:
    D(xp) = y_chord(xp) - yp
  • 3. Solve the Quadratic: Because the conductor fits the parabolic curve equation `y = a*x^2 + b*x + c` with `a = w / (2*T)`, fitting the points yields the universal tension equation:
    Tension (T) = (w * xp * (L - xp)) / (2 * D(xp))
  • Universal Result: Regardless of elevation difference `h` or tower slopes, measuring the offset `D` at any distance `xp` solves the horizontal tension `T` instantly.

Three-Point Surveyor Calculator

Calculated Horizontal Tension
30.02 kN
T = (w * xp * (L - xp)) / (2 * D(xp)) T = (15.912 * 100 * (300 - 100)) / (2 * 5.3) T = (15.912 * 100 * 200) / 10.6 T = 318,240 / 10.6 = 30,022.6 N = 30.02 kN