The Transmission Line Wave Equation
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1 1//9 The Transmission Line Wave Equation.doc 1/8 The Transmission Line Wave Equation Let s assume that v ( t, ) and (, ) i t each have the timeharmonic form: j t { V e ω j t = } and i (, t) = Re { I( ) e ω } v ( t, ) Re ( ) The time-derivative of these functions are: vt (, ) t i(, t) t e = Re V( ) = Re j V( ) e t { ω } e = Re I ( ) = Re j Ie ( ) t { ω } Inserting these results into the telegrapher s equations, we find: V ( ) Re e j t Re { ( R j L ) I ( ) e ω = ω } I ( ) Re e = Re ( G j C ) V ( ) e { ω } Simplifying, we have the complex form of telegrapher s equations:
2 1//9 The Transmission Line Wave Equation.doc /8 V( ) = ( R jω L) I( ) I ( ) = ( G jω C ) V( ) Note that these complex differential equations are not a function of time t! * The functions I( ) and V ( ) are complex, where the magnitude and phase of the complex functions describe the j t magnitude and phase of the sinusoidal time function e ω. * Thus, I( ) and V ( ) describe the current and voltage along the transmission line, as a function as position. * Remember, not just any function I( ) and V ( ) can exist on a transmission line, but rather only those functions that satisfy the telegraphers equations. Our task, therefore, is to solve the telegrapher equations and V! find all solutions I( ) and ( )
3 1//9 The Transmission Line Wave Equation.doc 3/8 Q: So, what functions I( ) and V ( ) do satisfy both telegrapher s equations?? A: To make this easier, we will combine the telegrapher V and equations to form one differential equation for ( ) another for I( ). First, take the derivative with respect to of the first telegrapher equation: V( ) = ( R jω L) I( ) V( ) I( ) = = ( R jω L) Note that the second telegrapher equation expresses the V : derivative of I( ) in terms of ( ) I ( ) = ( G jωc ) V( ) Combining these two equations, we get an equation involving V only: ( ) V( ) = ( R jωl)( G jωc ) V( ) We can simplify this equation by defining the complex value : = ( R jωl)( G jωc )
4 1//9 The Transmission Line Wave Equation.doc 4/8 So that: V( ) = V( ) In a similar manner (i.e., begin by taking the derivative of the second telegrapher equation), we can derive the differential equation: I ( ) = I ( ) We have decoupled the telegrapher s equations, such that we now have two equations involving one function only: V( ) = V( ) I ( ) = I ( ) These are known as the transmission line wave equations. Note that value is complex, and is determined by taking the square-root of a complex value. Likewise, is a complex value. Do you know how to square a complex number? Can you determine the square root of a complex number?
5 1//9 The Transmission Line Wave Equation.doc 5/8 Note only special functions satisfy these wave equations; if we take the double derivative of the function, the result is the original function (to within a constant )! Q: Yeah right! Every function that I know is changed after a double differentiation. What kind of magical function could possibly satisfy this differential equation? A: Such functions do exist! For example, the functions V ( ) = e and V ( ) e = each satisfy this transmission line wave equation (insert these into the differential equation and see for yourself!). Likewise, since the transmission line wave equation is a linear differential equation, a weighted superposition of the two solutions is also a solution (again, insert this solution to and see for yourself!): V = V e V e ( ) In fact, it turns out that any and all possible solutions to the differential equations can be expressed in this simple form! Therefore, the general solution to these complex wave equations (and thus the telegrapher equations) are:
6 1//9 The Transmission Line Wave Equation.doc 6/8 V ( ) = V e V e I( ) = I e I e where V,V,I, and I are complex constants. It is unfathomably important that you understand what this result means! It means that the functions V ( ) and I( ), describing the current and voltage at all points along a transmission line, can always be completely specified with just four complex constants ( V,V,I,I )!! We can alternatively write these solutions as: V ( ) = V ( ) V ( ) where: I( ) = I ( ) I ( ) V ( ) V e V ( ) V e I ( ) I e I ( ) I e
7 1//9 The Transmission Line Wave Equation.doc 7/8 The two terms in each solution describe two waves propagating V I ) propagating in the transmission line, one wave ( ( ) or ( ) in one direction ( ) and the other wave ( V ( ) or I ( ) propagating in the opposite direction ( ). ) V ( ) V e V ( ) V e Q: So just what are the complex values V, V, I, I? A: Consider the wave solutions at one specific point on the transmission line the point =. For example, we find that: ( ) V = = V e ( = ) ( ) = V e = V ( 1) = V In other words, V is simply the complex value of the wave V at the point = on the transmission line! function ( ) Likewise, we find:
8 1//9 The Transmission Line Wave Equation.doc 8/8 ( ) V = V = ( ) I = I = ( ) I = I = Again, the four complex values V, I, V, I are all that is needed to determine the voltage and current at any and all points on the transmission line. More specifically, each of these four complex constants completely specifies one of the four transmission line wave V I V I. functions ( ), ( ), ( ), ( ) Q: But what determines these wave functions? How do we find the values of constants V, I, V, I? A: As you might expect, the voltage and current on a transmission line is determined by the devices attached to it on either end (e.g., active sources and/or passive loads)! The precise values of V, I, V, I are therefore determined by satisfying the boundary conditions applied at each end of the transmission line much more on this later!
The Transmission Line Wave Equation
1//5 Th Transmission Lin Wav Equation.doc 1/6 Th Transmission Lin Wav Equation Q: So, what functions I (z) and V (z) do satisfy both tlgraphr s quations?? A: To mak this asir, w will combin th tlgraphr
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