The law of squares is a theorem concerning transmission lines. It states that the current injected into the line by a step in voltage reaches a maximum at a time proportional to the square of the distance down the line. The theorem is due to William Thomson, the future Lord Kelvin. The law was of crucial importance to the first submarine telegraph cables.
The law For a step increase in the voltage applied to a transmission line, the law of squares can be stated as follows,
t max = 1 2 R C x 2 {\displaystyle t_{\text{max}}={1 \over 2}RCx^{2}}
where,
t max {\displaystyle t_{\text{max}}} is the time at which the current on the line reaches a maximum
R {\displaystyle R} is the resistance per metre of the line
C {\displaystyle C} is the capacitance per metre of the line
x {\displaystyle x} is the distance in metres from the input of the line. The law of squares is not just limited to step functions. It also applies to an impulse response or a rectangular function which are more relevant to telegraphy. However, the multiplicative factor is different in these cases. For an impulse it is 1/6 rather than 1/2 and for rectangular pulses it is something in between depending on their length.
History The law of squares was proposed by William Thomson (later to become Lord Kelvin) in 1854 at Glasgow University. He had some input from George Gabriel Stokes. Thomson and Stokes were interested in investigating the feasibility of the proposed transatlantic telegraph cable. Thomson built his result by analogy with the heat transfer theory of Joseph Fourier (the transmission of an electrical step down a line is analogous to suddenly applying a fixed temperature at one end of a metal bar). He found that the equation governing the instantaneous voltage on the line, v ( x , t ) {\displaystyle v(x,t)} is given by,
∂ 2 v ∂ x 2 = R C ∂ v ∂ t . {\displaystyle {\frac {\partial ^{2}v}{\partial x^{2}}}=RC{\frac {\partial v}{\partial t}}.}
It is from this that he derived the law of squares. While Thomson's description of a transmission line is not exactly incorrect, and it is perfectly adequate for the low frequencies involved in a Victorian telegraph cable, it is not the complete picture. In particular, Thomson did not take into account the inductance (L) of the line, or the leakage conductivity (G) of the insulation material. The full description was given by Oliver Heaviside in what is now known as the telegrapher's equations. The law of squares can be derived from a special case of the telegrapher's equations – that is, with L and G set to zero.
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