A telegraph line of length 100km has a capacity of $0.01 μ F / k m$ and it carries an alternating current at 0.5 kilo cycle per second. If minimum impedance is required, then the value of the inductance that need to be introduced in series is _________mH. (if π = $\sqrt[]{10}$ )

3 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

Payal Gupta | Contributor-Level 10

2 months ago

C = 10^-6 f

f = 0.5 × 10³ = 500

$z = \sqrt[]{x_{R}^{2} + {(x_{L} - x_{C})}^{2}}$

for minimum impedance :

x_L = x_C

$\Rightarrow ω^{2} = \frac{1}{L C}$

$\Rightarrow L = \frac{1}{π^{2}} = \frac{1}{10} \times 1000 = 100 m H$

C = 10-6 ff = 0.5 × 103 = 500 <math> <mrow> <mi>z</mi> <mo>=</mo> <mroot> <mrow> <msubsup> <mrow> <mi>x</mi> </mrow> <mrow> <mi>R</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>L</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>C</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow></mrow> </mroot> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1813757069"> </o:OLEObject></xml><! [endif]-> for minimum impedance :xL = xC <math> <mrow> <mo>⇒</mo> <msup> <mrow> <mi>ω</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>L</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </math> <math> <mrow> <mo>⇒</mo> <mi>L</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <msup> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mn>0</mn> </mrow> </mfrac> <mo>×</mo> <mn>1</mn> <mn>0</mn> <mn>0</mn> <mn>0</mn> <mo>=</mo> <mn>1</mn> <mn>0</mn> <mn>0</mn> <mi>m</mi> <mi>H</mi> </mrow> </math>                  <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1813757070"> </o:OLEObject></xml><! [endif]->              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1813757071"> </o:OLEObject></xml><! [endif]->