State the necessary condition for resonance in a series circuit. Compare series and parallel resonance.

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Q.6 (b) State the necessary condition for resonance in a series circuit. Compare series and parallel resonance.

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Condition for series resonance: in an R-L-C series circuit, resonance occurs when the inductive reactance equals the capacitive reactance, X_L = X_C, so the net reactance is zero, the impedance is purely resistive (Z = R, minimum), current is maximum (V/R) and in phase with the voltage (pf = 1). 2π f_rL = 1/2π f_rC ⇒ f_r = 1/2π√(LC) . ParameterSeries resonance (R-L-C series)Parallel resonance (coil R-L ∥ C) ConditionX_L = X_CReactive (susceptive) component of current = 0, i.e. I_L sin φ_L = I_C Resonant frequency f_r = 1/2π√(LC) f_r = 1/2π√(1/LC-R²/L²) Impedance at resonanceMinimum, Z = RMaximum, Z_D = L/(CR) (dynamic impedance) Current at resonanceMaximum, I = V/RMinimum, I = V/Z_D = VCR/L Power factorUnityUnity MagnificationVoltage magnification: V_L = V_C = QVCurrent magnification: I_C ≈ QI Q factor Q = 1/R√(L/C) Q = ω_rL/R CalledAcceptor circuitRejector circuit Below f_r / above f_rCapacitive / inductiveInductive / capacitive UseTuning (radio receivers), filters that pass a frequencyTank circuits in oscillators, filters that reject a frequency

State the necessary condition for resonance in a series circuit. Compare series and parallel resonance.

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Parameter	Series resonance (R-L-C series)	Parallel resonance (coil R-L ∥ C)
Condition	X_L = X_C	Reactive (susceptive) component of current = 0, i.e. I_L sin φ_L = I_C
Resonant frequency	\( f_r = \frac{1}{2\pi\sqrt{LC}} \)	\( f_r = \frac{1}{2\pi}\sqrt{\frac{1}{LC}-\frac{R^2}{L^2}} \)
Impedance at resonance	Minimum, Z = R	Maximum, Z_D = L/(CR) (dynamic impedance)
Current at resonance	Maximum, I = V/R	Minimum, I = V/Z_D = VCR/L
Power factor	Unity	Unity
Magnification	Voltage magnification: V_L = V_C = QV	Current magnification: I_C ≈ QI
Q factor	\( Q = \frac{1}{R}\sqrt{\frac{L}{C}} \)	\( Q = \frac{\omega_rL}{R} \)
Called	Acceptor circuit	Rejector circuit
Below f_r / above f_r	Capacitive / inductive	Inductive / capacitive
Use	Tuning (radio receivers), filters that pass a frequency	Tank circuits in oscillators, filters that reject a frequency