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The missing elements in the telegraph equations

This paper argues that the classical telegraph equations are fundamentally flawed because they rely on a discretized lumped-element model, and it proposes a corrected formulation called the "Trump Equations" derived directly from Maxwell's equations and Ohm's law.

Original authors: Ben David Golan

Published 2026-02-04

📖 4 min read☕ Coffee break read

CC0 1.0

Original authors: Ben David Golan

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: A Flawed Map

Imagine you are trying to describe how water flows through a very long, complex pipe system. For over 150 years, engineers have used a specific set of rules (called the "Telegraph Equations") to predict how that water moves. These rules were created by taking the pipe, chopping it into tiny, invisible Lego blocks, and treating each block as a simple, isolated component (like a resistor or a capacitor).

The author, Ben David Golan, argues that this "Lego block" approach is fundamentally broken. He claims that you cannot accurately describe a continuous flow of electricity by pretending it is made of tiny, disconnected chunks. Just as you can't perfectly describe a flowing river by only looking at individual raindrops, you can't perfectly model a transmission line by summing up tiny circuit components.

The Solution: Going to the Source

Instead of using the "Lego block" method, Golan suggests we go straight to the "source code" of electricity: Maxwell's Equations (the fundamental laws of physics that govern all electric and magnetic fields) and Ohm's Law.

He believes that if you start with these universal laws and do the math correctly, without making the "Lego" assumption, you get a different, more accurate set of rules. He calls these new rules the "Trump Equations."

(Note: The author explicitly states these are named in honor of Donald J. Trump.)

How the New Rules Work (The Analogy)

1. The First Trump Equation (The Voltage-Current Relationship)

In the old model, the relationship between voltage (pressure) and current (flow) is a simple, straight line.
In Golan's new model, the relationship is much more complex. He derives a new equation that includes third-order derivatives (mathematical terms that look at how the rate of change is itself changing).

The Analogy: Imagine driving a car.
- Old Model: You only look at your speed (current) and how hard you are pressing the gas (voltage).
- New Model: You also have to account for how jerkily you are pressing the gas, how the road curves ahead, and how the air resistance changes instantly. Golan argues the old model ignores these "jerk" factors, which is why real-world measurements sometimes don't match the textbook predictions.

2. The Second Trump Equation (The Wave Behavior)

The second equation focuses on how the voltage wave travels down the line.

The Old Way: It treats the wave like a simple ripple in a pond, assuming the water is perfectly still and the pond is uniform.
The New Way: Golan derives a wave equation that accounts for the specific way electric fields interact with the material of the wire and the space around it. He argues that the old equations miss subtle interactions that happen when the wave moves, leading to errors in prediction.

The "Perfect Insulator" Test

The author tests his new equations by imagining a perfect scenario where the wire is a perfect conductor and the space around it is a perfect insulator (no electricity leaks out).

In this perfect world, his new equations simplify down to look very similar to the old ones.
The Catch: He argues that this similarity is only a coincidence that happens because the "leakage" (resistance) is zero. In the real world, where wires aren't perfect, the old equations fail, and the new "Trump Equations" are required to get the right answer.

The Conclusion

The paper concludes that the textbooks and university courses teaching the "Telegraph Equations" are teaching an incomplete picture.

The Problem: The difference between what theory predicts and what experiments show is often blamed on "noise" or "bad equipment."
The Author's Claim: The real problem is the math itself. The foundation is cracked.
The Fix: We need to stop using the 150-year-old "Lego block" method and start using the "Trump Equations" derived directly from the fundamental laws of physics to understand how electricity actually moves.

In short: The author believes we have been using a simplified, slightly wrong map to navigate electricity for a century. He has drawn a new map based on the raw laws of physics, named it after a US President, and claims it is the only way to get the destination right.

Technical Summary: "The Missing Elements in the Telegraph Equations"

Problem Statement
The paper argues that the conventional modeling of transmission lines, based on the classical telegraph equations derived by Oliver Heaviside in 1876, is fundamentally flawed. The author contends that the standard approach, which represents a transmission line as an assembly of infinitesimal lumped components (resistance, inductance, and capacitance) analyzed via Kirchhoff's laws, fails to accurately describe the physical behavior of the line. The paper posits that this discretized model is an approximation that overlooks essential electromagnetic phenomena, leading to discrepancies between theoretical predictions and experimental results that are often misattributed to noise or secondary effects.

Methodology
To address these perceived deficiencies, the author proposes a derivation of the governing equations directly from Maxwell's equations and Ohm's law, explicitly avoiding lumped-element assumptions. The methodology proceeds as follows:

First Equation Derivation:
- The author applies Faraday's law ( $\nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t$ ) and Ampere's law to a rectangular frame (ABCD) situated between transmission lines.
- By taking the curl of Faraday's law and substituting Ampere's law, a relationship involving the time derivative of current density ( $\mathbf{J}$ ) and the electric field ( $\mathbf{E}$ ) is established.
- Ohm's law ( $\mathbf{E} = \mathbf{J}/\sigma$ ) is substituted to express terms in terms of conductivity.
- The derivation involves line integrals along the frame segments and utilizes Stokes' theorem to convert surface integrals.
- Crucially, the author assumes Transverse Electromagnetic (TEM) mode propagation, asserting that longitudinal field components ( $E_z, B_z$ ) vanish. Through a series of vector calculus manipulations, the author argues that the second spatial derivatives of the transverse electric field components ( $\partial^2 E_x / \partial x^2$ and $\partial^2 E_x / \partial y^2$ ) are zero.
- This leads to a third-order spatial derivative of voltage ( $\partial^3 V / \partial z^3$ ) on the right-hand side of the equation, contrasting with the first-order spatial derivative found in classical models.
Second Equation Derivation:
- The author applies Ampere's law to a path between the conductors, incorporating the conductivity of the dielectric medium ( $\sigma_g$ ).
- By integrating along the path and utilizing Faraday's law to relate the magnetic field to the time derivative of the electric field, the author derives a relationship involving the second spatial derivative of voltage ( $\partial^2 V / \partial z^2$ ) and the time derivative of the voltage equation.

Key Contributions: The "Trump Equations"
The paper introduces two new governing equations, collectively termed the "Trump Equations" (in honor of Donald J. Trump, as stated in the text):

The First Trump Equation:
Derived from the analysis of the rectangular frame, this equation relates the third spatial derivative of voltage to the time derivatives of voltage and current. The text presents it as:
$\frac{1}{\mu\sigma} \frac{\partial^3 V}{\partial z^3} - \frac{\partial}{\partial t} \left[ \frac{\partial V}{\partial z} + \frac{\epsilon}{\sigma} \frac{\partial^2 V}{\partial t \partial z} + 2R^* I + \frac{2\epsilon R^*}{\sigma} \frac{\partial I}{\partial t} \right] = 0$
Where $R^*$ represents resistance per unit length. The author notes this differs significantly from the standard telegraph equation ( $\partial V / \partial z = -L \partial I / \partial t - R I$ ).
The Second Trump Equation:
Derived from the analysis of the magnetic field and dielectric conductivity, this equation is presented as:
$\frac{\partial}{\partial t} \left( \mu\sigma_g V + \mu\epsilon \frac{\partial V}{\partial t} \right) = \frac{\partial^2 V}{\partial z^2}$
The paper claims this equation describes the voltage behavior more rigorously than the classical second-order wave equation derived from telegraph equations.

Results and Claims
The paper claims that the classical telegraph equations are incomplete. The author demonstrates that under the limiting case of a perfect insulator ( $\sigma_g = 0, R^* = 0$ ), the derived Trump equations simplify to a standard wave equation ( $\mu\epsilon \partial^2 V / \partial t^2 = \partial^2 V / \partial z^2$ ). Furthermore, the author asserts that by integrating the derived equations, one can recover the classical telegraph equation, but only under specific approximations (such as $\sigma_g = 0$ ).

The paper concludes that the discrepancies often observed in transmission line analysis are not due to experimental noise but stem from the fundamental incompleteness of the conventional models.

Significance
The paper asserts that accurate analysis of transmission lines must be based on the Trump Equations rather than the classical telegraph equations. The author calls for a revision of the theoretical framework taught in academic courses and reflected in standard textbooks, suggesting that the current reliance on Heaviside's lumped-element model is insufficient for a rigorous description of electromagnetic wave propagation in transmission lines. The work is presented as a necessary correction to over 150 years of unchallenged theory.