How to connect LCR circuit in series combination

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How to connect LCR circuit in series combination

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LCR circuit: Suppose a resistance R. an inductance L and a capacitance C are connected in series to a source of alternating emf clip_image002[5]


clip_image004[5] clip_image006[5]peak value of clip_image008[5]

Where clip_image010[5]emf of alternating current

Let I be the current in the series circuit at any instant. Then voltage will be differ due to R, L and C.

Potential across resistor:                   clip_image012[5]               R will be in same phase with clip_image0149

Potential across inductance coil    clip_image0165             Inductance L is ahead of current  clip_image01410  by  clip_image0217

potential across capacitor                 clip_image0255             Capacitance C lags behind the current  clip_image01411 by  clip_image0218

Where    R    =   Resistance  I = current

                 clip_image018[5]  Inductance reactance.

                 clip_image0275=    Capacitive reactance

As clip_image033[5]and clip_image035[5]are in opposite direction their resultant will be clip_image037[7]

The resultant of clip_image039[5] and clip_image037[8]must be equal to the applied emf clip_image042[5]

So using Pythagorean Theorem, we get


Where clip_image046[5]peak value of current

The effective resistance of series LCR circuit which opposes the flow of current is called Impedance. It is denoted by Z.

Impedance clip_image048[5]


Special cases:

(i) Inductive LCR circuit

When      clip_image052[5]     or       clip_image054[5]

The emf is ahead of current by angle clip_image056[7]

clip_image058[5]        or        clip_image060[5]

The Instantaneous current clip_image062[5]

(ii) Capacitive LCR circuit

When      clip_image064[5]    or     clip_image066[5]

The current is ahead of emf by angle clip_image056[8]


The Instantaneous current   clip_image071[5]

(iii) Resonance condition:     clip_image073[5]   or   clip_image075[5]


Clearly the circuit is purely resistive

The current and voltage are in same phase and the current is maximum. This is Resonance condition