JEE Main & Advanced Physics Alternating Current / प्रत्यावर्ती धारा Parallel RLC Circuits

\[{{i}_{R}}=\frac{{{V}_{0}}}{R}={{V}_{0}}G\]

\[{{i}_{L}}=\frac{{{V}_{0}}}{{{X}_{L}}}={{V}_{0}}{{S}_{L}}\]  

\[{{i}_{C}}=\frac{{{V}_{0}}}{{{X}_{C}}}={{V}_{0}}{{S}_{C}}\]

(1) Current and phase difference : From phasor diagram current \[i=\sqrt{i_{R}^{2}+{{({{i}_{C}}-{{i}_{L}})}^{2}}}\] and phase difference

\[\varphi ={{\tan }^{-1}}\frac{({{i}_{C}}-{{i}_{L}})}{{{i}_{R}}}={{\tan }^{-1}}\frac{({{S}_{C}}-{{S}_{L}})}{G}\]

(2) Admittance (Y) of the circuit : From equation of current \[\frac{{{V}_{0}}}{Z}=\sqrt{{{\left( \frac{{{V}_{0}}}{R} \right)}^{2}}+{{\left( \frac{{{V}_{0}}}{{{X}_{L}}}-\frac{{{V}_{0}}}{{{X}_{C}}} \right)}^{2}}}\]

\[\Rightarrow \]\[\frac{1}{Z}=Y=\sqrt{{{\left( \frac{1}{R} \right)}^{2}}+{{\left( \frac{1}{{{X}_{L}}}-\frac{1}{{{X}_{C}}} \right)}^{2}}}=\sqrt{{{G}^{2}}+{{({{S}_{L}}-{{S}_{C}})}^{2}}}\]

(3) Resonance : At resonance  (i) \[{{i}_{C}}={{i}_{L}}\]\[\Rightarrow \]\[{{i}_{\min }}={{i}_{R}}\]

(ii) \[\frac{V}{{{X}_{C}}}=\frac{V}{{{X}_{L}}}\]\[\Rightarrow \]\[{{S}_{C}}={{S}_{L}}\,\Rightarrow \,\Sigma \,S=0\]

(iii) \[{{Z}_{\max }}=\frac{V}{{{i}_{R}}}=R\]

(iv) \[\varphi =0\]\[\Rightarrow \]\[\text{p}\text{.f}\text{.}\,\,=\cos \phi =1=\]maximum

(v) Resonant frequency \[\Rightarrow \nu =\frac{1}{2\pi \sqrt{LC}}\]

(4) Parallel LC circuits : If inductor has resistance (R) and it is connected in parallel with capacitor as shown

(i) At resonance

(a) \[{{Z}_{\max }}=\frac{1}{{{Y}_{\min }}}=\frac{L}{CR}\]

(b) Current through the circuit is minimum and \[{{i}_{\min }}=\frac{{{V}_{0}}CR}{L}\]

(c) \[{{S}_{L}}={{S}_{C}}\] \[\Rightarrow \]\[\frac{1}{{{X}_{L}}}=\frac{1}{{{X}_{C}}}\]\[\Rightarrow \]\[X=\infty \]

(d) Resonant frequency \[{{\omega }_{0}}=\sqrt{\frac{1}{LC}-\frac{{{R}^{2}}}{{{L}^{2}}}}\,\frac{rad}{sec}\] or \[{{\nu }_{0}}=\frac{1}{2\pi }\sqrt{\frac{1}{LC}-\frac{{{R}^{2}}}{{{L}^{2}}}}\,Hz\](Condition for parallel resonance is \[R<\sqrt{\frac{L}{C}}\])

(e) Quality factor of the circuit \[=\frac{1}{CR}.\frac{1}{\sqrt{\frac{1}{LC}-\frac{{{R}^{2}}}{{{L}^{2}}}}}\]. In the state of resonance the quality factor of the circuit is equivalent to the current amplification of the circuit.

(ii) If inductance has no resistance : If \[R=0\] then circuit becomes parallel LC circuit as shown

Condition of resonance : \[{{i}_{C}}={{i}_{L}}\]\[\Rightarrow \]\[\frac{V}{{{X}_{C}}}=\frac{V}{{{X}_{L}}}\]

\[\Rightarrow \] \[{{X}_{C}}={{X}_{L}}\].

At resonance current \[i\] in the circuit is zero and impedance is infinite. Resonant frequency : \[{{\nu }_{0}}=\frac{1}{2\pi \sqrt{LC}}Hz\]