The frequency of the output signal becomes ________ times by doubling the value of the capacitance in the LC oscillator circuit.

In the L-C circuits

For the first condition

${ f }_{ 1 }=\dfrac { 1 }{ 2\pi } \sqrt { \dfrac { 1 }{ LC } }$ .....(i)

For the second condition, when $C=$ doubling

${ f }_{ 2 }=\dfrac { 1 }{ 2\pi } \sqrt { \dfrac { 1 }{ 2LC } }$ ....(ii)

From the equation (i), we get

${ f }_{ 2 }=\dfrac { 1 }{ 2\pi } \sqrt { \dfrac { 1 }{ LC } } \times \dfrac { 1 }{ \sqrt { 2 } } \Rightarrow { f }_{ 2 }={ f }_{ 1 }\times \dfrac { 1 }{ \sqrt { 2 } }$

So, the frequency of the output signal becomes ${ 1 }/{ \sqrt { 2 } }$ times by doubling the value of the capacitance in the L-C oscillator circuit.