Answer

Verified

394.2k+ views

**The power factor, $\cos \phi $ determines the average power dissipated in a LCR circuit, An LCR circuit, capacitance, resistance and inductance will affect the power dissipated within the circuit.**

__Hint:____Let’s define the data given in the question:__

**Step by step solution:**It is given that voltage applied to the LCR circuit, $V = {V_0}\sin \omega t$

From this equation, we get the current flowing in the LCR circuit by the formula:

$I = {I_0}\sin (\omega t - \phi )$

Here $\phi $ is the phase lag of the current with respect to the applied voltage $V = {V_0}\sin \omega t$

And ${I_0}$ is given by equation, \[{I_0} = {V_0}Z\phi = {\tan ^{ - 1}}({X_C} - {X_L})R\]

Now we are heading to the power dissipated in the LCR circuit,

The power dissipated in the circuit is given by two equations,

$P = {I^2}R$ And $P = VI$

Here we are taking second equation, that is:

The power dissipated, $P = VI$

Applying the values $V$and $I$ in the above equation, we get,

$P = {V_0}\sin \omega t \times {I_0}\sin (\omega t - \phi )$

Applying trigonometric conversions we get,

$ \Rightarrow P = \dfrac{{{V_0}{I_0}}}{2}\left[ {\sin \omega t - \sin (\omega t - \phi )} \right]$

$ \Rightarrow P = \dfrac{{{V_0}{I_0}}}{2}\left[ {\cos \phi - \cos (2\omega t - \phi )} \right]$

Now we get the power dissipated in the LCR circuit

For the average power dissipated we are taking the average of the above value.

Therefore, average power for one complete cycle= average of $\left\{ {\dfrac{{{V_0}{I_0}}}{2}\left[ {\cos \phi - \cos (2\omega t - \phi )} \right]} \right\}$

In the above expression the average of the second term over a complete cycle will be equal zero.

That is, $\dfrac{{{V_0}{I_0}}}{2}\cos (2\omega t - \phi )$ will be equal to zero

Therefore, the average power dissipated over one complete circle will be equal to

$\dfrac{{{V_0}{I_0}}}{2}\cos \phi $ . (Here $\cos \phi $ is the power factor and is given by$\cos \phi = \dfrac{R}{Z}$.)

Now we are applying the two conditions which are given in the question to the average power dissipated, that is

Conditions:

1. No power dissipated even though the current flows through the circuit:

If the power dissipated is zero, the value of $\cos \phi $ should be equal to zero, that is,

$\phi = \dfrac{\pi }{2}$

This is the condition in which the circuit is purely capacitive or inductive.

2. Maximum power dissipated in the circuit

If the power dissipated is maximum, the value of $\cos \phi $ should be equal to one, that is,

$\phi = 0$

This means the circuit is purely resistive.

__There is always loss of power in LCR circuits depending on the quality of materials and electrical conditions where the circuitry operates. The power dissipated in a series resonant circuit can also be expressed in terms of the rms voltage and current.__

**Note:**Recently Updated Pages

Select the smallest atom A F B Cl C Br D I class 11 chemistry CBSE

Cryolite and fluorspar are mixed with Al2O3 during class 11 chemistry CBSE

The best reagent to convert pent 3 en 2 ol and pent class 11 chemistry CBSE

Reverse process of sublimation is aFusion bCondensation class 11 chemistry CBSE

The best and latest technique for isolation purification class 11 chemistry CBSE

Hydrochloric acid is a Strong acid b Weak acid c Strong class 11 chemistry CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Which are the Top 10 Largest Countries of the World?

The provincial president of the constituent assembly class 11 social science CBSE

Write the 6 fundamental rights of India and explain in detail