Standing waves appear when a length of line is excited at a frequency for which the electrical line length is a significant part of an electrical wavelength. They result from the constructive and destructive interference of forward and refl ected waves on the line.
The behavior of the line can be determined by solving the applicable differential equations relating the line parameters to the exciting frequency. The solution of the equations for a line with losses is rather complex and adds little to the practical considerations, so the lossless line will be analyzed instead.
In the lossless line, L is the series inductance per unit length, and C is the shunt capacitance. If a differential length, dx , is considered, the inductance for that length is L dx , and the voltage in that length is e _ – L dx ( di/dt ). Since e = ( de/dx ) dx , the equation can be written as dx ( de/dx ) = – L dx ( di/dt ). Similarly, dx ( di/dx ) = – C dx ( de/dt ).
Dividing out the dx terms and substituting partial derivatives, the fundamental forms of transmission line equations result: -∂e/∂x = L ∂i/∂t and -∂i/∂x = C ∂e/∂t
By differentiating with respect to x and then with respect to t , these equations can be solved simultaneously to yield second-order, elliptical, partial differential equations for both e and i individually with respect to t and x . The classical forms then result: LC ∂2i/∂t2 -∂2i/∂x2 = 0 and LC ∂2e/∂t2 -∂2e/∂x2 =0
These equations can now be solved by transforms or classical methods. Explicit solutions can be developed with hyperbolic functions in the complex plane, and these solutions were the only practical means of line analysis until the digital computer was developed.
Fortunately, the computer offers an easier method of analysis by numerical integration, and line losses can be incorporated with relative ease. The difference equations can be solved by simple Euler integration, so the whole process is not nearly as daunting as in earlier years.
These equations allow numerical solutions for the voltages and currents on the line as functions of distance and time. Although it may not be immediately apparent, these difference equations, in the limit, replicate the differential equations.
Before proceeding to typical solutions, several derived parameters should be defined. First, the line has a surge, or characteristic, impedance defined as Z0 = ( L/C ) 1/2 and, second, a velocity of propagation v = 1/( LC ) 1/2 . The characteristic impedance defines the relationship between the line and its attached load, and the velocity of propagation defines the speed of signal transmission along the line and consequently its electrical length.
The electrical length of the line, in terms of wavelengths for any given exciting frequency, is λp / λe = v/c , where λp is the physical line length, λe is the exciting frequency wavelength in free space, v is the velocity of propagation, and c is the speed of light.
The overhead line has a high series inductance and relatively low shunt capacitance that leads to a high surge impedance. It also has a relatively high velocity of propagation because of the low capacitance. In the cable, things are reversed.
Shielded cable has a very high capacitance that makes the surge impedance low, and the velocity of propagation is also low. Note that the physical wavelength of a signal in such shielded cable is less than one-third of the wavelength in free space.
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| Traveling Waves |
In the lossless line, L is the series inductance per unit length, and C is the shunt capacitance. If a differential length, dx , is considered, the inductance for that length is L dx , and the voltage in that length is e _ – L dx ( di/dt ). Since e = ( de/dx ) dx , the equation can be written as dx ( de/dx ) = – L dx ( di/dt ). Similarly, dx ( di/dx ) = – C dx ( de/dt ).
Dividing out the dx terms and substituting partial derivatives, the fundamental forms of transmission line equations result: -∂e/∂x = L ∂i/∂t and -∂i/∂x = C ∂e/∂t
By differentiating with respect to x and then with respect to t , these equations can be solved simultaneously to yield second-order, elliptical, partial differential equations for both e and i individually with respect to t and x . The classical forms then result: LC ∂2i/∂t2 -∂2i/∂x2 = 0 and LC ∂2e/∂t2 -∂2e/∂x2 =0
These equations can now be solved by transforms or classical methods. Explicit solutions can be developed with hyperbolic functions in the complex plane, and these solutions were the only practical means of line analysis until the digital computer was developed.
Fortunately, the computer offers an easier method of analysis by numerical integration, and line losses can be incorporated with relative ease. The difference equations can be solved by simple Euler integration, so the whole process is not nearly as daunting as in earlier years.
These equations allow numerical solutions for the voltages and currents on the line as functions of distance and time. Although it may not be immediately apparent, these difference equations, in the limit, replicate the differential equations.
Before proceeding to typical solutions, several derived parameters should be defined. First, the line has a surge, or characteristic, impedance defined as Z0 = ( L/C ) 1/2 and, second, a velocity of propagation v = 1/( LC ) 1/2 . The characteristic impedance defines the relationship between the line and its attached load, and the velocity of propagation defines the speed of signal transmission along the line and consequently its electrical length.
The electrical length of the line, in terms of wavelengths for any given exciting frequency, is λp / λe = v/c , where λp is the physical line length, λe is the exciting frequency wavelength in free space, v is the velocity of propagation, and c is the speed of light.
The overhead line has a high series inductance and relatively low shunt capacitance that leads to a high surge impedance. It also has a relatively high velocity of propagation because of the low capacitance. In the cable, things are reversed.
Shielded cable has a very high capacitance that makes the surge impedance low, and the velocity of propagation is also low. Note that the physical wavelength of a signal in such shielded cable is less than one-third of the wavelength in free space.
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