**RESONANCE**

__DEFINITION:__ “A condition exists when an oscillating circuit responds with maximum amplitude of an external source of angular frequency

**‘ω’**called resonance.”**RESONATING CIRCUITS:**

__DEFINITION:__ “The circuits that contain both inductors and capacitors along with a resistor so arranged that the circuit is capable of resonance are called resonating circuits.”

__SERIES RESONATING CIRCUIT:__ Consider a resister, inductor and capacitor which are connected in series with A.C. source whose frequency can be varied. When current flows through inductor, the inductive reactance becomes:

**X**

_{L}= ωLand when current flows through capacitor, the capacitive reactance becomes:

**X**

_{C}= 1/ (ωC)**At High Frequency:**

Inductive Reactance**X**is greater than capacitive reactance_{L}**X**due to which circuit behaves as R-L Series Circuit in which voltage V leads to the current I by 90° or π/2 (As,_{C}**f ∝ X**and_{L}**f ∝ 1/X**. So, when_{C}**f**increases**X**)._{L}> X_{C}

**At Low Frequency:**

Capacitive Reactance**X**is greater than inductive reactance_{C}**X**due to which circuit behaves as R-C Series Circuit in which voltage V lags to the current I by 90° or π/2 (As,_{L}**f ∝ X**and_{L}**f ∝ 1/X**. So, when f decreases_{C}**X**)._{L}< X_{C}

**At Intermediate Frequency**

In between these frequencies, there will be a certain value of frequency at which**X**becomes equal to_{L}**X**and this frequency is called resonance Frequency._{C}

__Characteristics:__- At resonance,
**X**, so they are equal and opposite to each other and cancel each other’s effect and circuit behaves as resistive circuit._{L}= X_{C}

- By changing frequency if
**X**_{L}= X_{C,}This circuit becomes the resonant circuit.

**1/(ω**

_{r}c)=ω_{r}L**ω**

_{r}^{2}=1/LC**ω**

_{r}=1/√(LC)**2πf**

_{r}=1/√(LC)**f**

_{r}=1/√(2π(LC))‘

**f**’ is known as ‘resonant frequency’._{r}- Impedance of circuit becomes minimum and is equal to R (resistive circuit).

- Power loss at resonance is maximum.

**P=IV cosθ**

**θ=0**

Hence, power factor cos0° =1

- The variation of current with frequency is giver by the graph given below:

There are 3 cases :

**V**(At high frequency)_{C}> V_{L}**V**(At resonance frequency)_{C}= V_{L}**V**(At low frequency)_{C}< V_{L}

__At Vc>VL:__

**V=[V**

_{R}

^{2}**+ (V**

_{C}- V_{L})

^{2}**]^1/2**

(At resonance frequency,

**V**)_{C}= V_{L}**V=[(V**

_{R})

^{2 }**+ 0]^1/2**

**V= V**

_{R}

__ii. At Vc<VL:__(At resonance frequency,

**V**)_{C}= V_{L}**V=[(V**

_{R})

^{2 }**+ 0]^1/2**

**V= V**

_{R}Therefore, at resonance, the voltage drop across inductance

**V**_{L }and the voltage drop across capacitance**V**may be much larger than the source voltage_{C}**V**._{R}**Entry # 11**

**By Rabia Khalid**

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