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Theoretical foundations of electrical engineering

Long line without losses (problems on the line calculation with distributed parameters)

Problem situation
"Calculate the distribution of RMS values of voltage and current in a long line without losses (parameters L0 = 0, 35 μF / m, C0 = 21 pF / m). Harmonic frequency of the transmitted signal is f = 0, 7 GHz. Line operating mode is Rн=2Zв. Instantaneous current is i2(t)=25sin(2πft+40o). Line length is λ=18 cm. Construct diagrams U(y), I(y) and define the coefficient value of the travelling wave". Given
L0=0,35 μF / m;
C0=21 pF / m;
f=0,7 GHz;
Rн=2Zв;
i2(t)=25sin(2πft+40o);
λ=18 cm.

Solution

We find cyclic frequency:


We find the wave resistance of the line, suggesting that the wire resistance is equal to zero, and resistance between wires is infinite (line without losses):


We find the propagation constant:


Then the phase coefficient is equal:


We put the symbol m:


Distribution of the current in the line based on the distance from the end of the line:


Instantaneous values of the voltage:


Distribution of the voltage in the line based on the distance from the end of the line:


Voltage diagram on the line as a function of the distance from the end of the line y:
Long lines (line with distributed parameters) - Voltage diagram on the line

The diagram is constructed further the line beginning (it is indicted by a vertical line) to define minimum and maximum voltage of the combined wave.
The coefficient of the travelling wave is ratio of the minimum voltage of the combined wave to its maximum voltage – define according to the diagram.
Kt.w.=3270/6540=0,5;
Diagram of the current on a line as function of the distance from the end of the line y:
Long lines (line with distributed parameters) - Diagram of the current on the line

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