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Telegrapher's Equations

Module by: Bill Wilson. E-mail the author

Summary: This module introduces and derives the telegrapher's equations, which describe how electrical signals behave as they move along transmission lines.

Let's look at just one little section of the line, and define some voltages and currents Figure 1.

Figure 1
Applying Kirchoff's Laws
Applying Kirchoff's Laws (6_07.png)
For the section of line Δx Δ x long, the voltage at its input is just Vxt V x t and the voltage at the output is Vx+Δxt V x Δ x t . Likewise, we have a current Ixt I x t entering the section, and another current Ix+Δxt I x Δ x t leaving the section of line. Note that both the voltage and the current are functions of time as well as position.

The voltage drop across the inductor is just:

V L =LΔxIxt t V L L Δ x t I x t
(1)
Likewise, the current flowing down through the capacitor is
I C =CΔxVx+Δxt t I C C Δ x t V x Δ x t
(2)
Now we do a KVL around the outside of the section of line and we get
Vxt V L Vx+Δxt=0 V x t V L V x Δ x t 0
(3)
Substituting Equation 1 for V L V L and taking it over to the RHS we have
VxtVx+Δxt=LΔxIxt t V x t V x Δ x t L Δ x t I x t
(4)
Let's multiply by -1, and bring the Δx Δ x over to the left hand side.
Vx+ΔxtVxtΔx=(LIxt t ) V x Δ x t V x t Δ x L t I x t
(5)
We take the limit as Δx0 Δ x 0 and the LHS becomes a derivative:
Vxt x =(LIxt t ) x V x t L t I x t
(6)
Now we can do a KCL at the node where the inductor and capacitor come together.
IxtCΔxVx+Δxt t Ix+Δxt=0 I x t C Δ x t V x Δ x t I x Δ x t 0
(7)
And upon rearrangement:
Ix+ΔxtIxtΔx=(CVx+Δxt t ) I x Δ x t I x t Δ x C t V x Δ x t
(8)
Now when we let Δx0 Δ x 0 , the left hand side again becomes a derivative, and on the right hand side, Vx+ΔxtVxt V x Δ x t V x t , so we have:
Ixt x =(CVxt t ) x I x t C t V x t
(9)
Equation 6 and Equation 9 are so important we will write them out again together:
Vxt x =(LIxt t ) x V x t L t I x t
(10)
Ixt x =(CVxt t ) x I x t C t V x t
(11)
These are called the telegrapher's equations and they are all we really need to derive how electrical signals behave as they move along on transmission lines. Note what they say. The first one says that at some point xx along the line, the incremental voltage drop that we experience as we move down the line is just the distributed inductance LL times the time derivative of the current flowing in the line at that point. The second equation simply tells us that the loss of current as we go down the line is proportional to the distributed capacitance CC times the time rate of change of the voltage on the line. As you should be easily aware, what we have here are a pair of coupled linear differential equations in time and position for Vxt V x t and Ixt I x t

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