Resonance in R-L-C circuit
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The document discusses resonance in R-L-C series and parallel circuits. In series circuits, resonance occurs when the inductive reactance (XL) equals the capacitive reactance (XC), resulting in maximum current. The resonant frequency is 1/2π√LC. In parallel circuits, resonance occurs when the current through the inductor equals the current through the capacitor, resulting in minimum current. The resonant frequency is 1/2π√(L/C - R2/L). Key differences between series and parallel resonance are also summarized.
Resonance in R-L-C circuit
- 1. RESONANCE IN R-L-C CIRCUIT 1. SERIES CIRCUIT 2. PARALLEL CIRCUIT
- 2. RESONANCE IN R-L-C SERIES CIRCUIT
- 3. Let us consider as R-L-C series circuit We know that the impedance in R-L-C series circuit is Where XL =2πfL & XC = 1 2πfL Such a circuit shown in figure is connected to an a.c. source of constant supply voltage V but having variable frequency. The frequency can be varied from zero, increasing and approaching infinity. 22 | | L CZ R X X 𝓋
- 4. • Since XL and Xc are functions of frequency, at a particular frequency of the applied voltage, XL and Xc will became equal in magnitude. Since XL = Xc XL - Xc = 0 ∴ 𝑍 = 𝑅2 + 0 = 𝑅 The circuit, when XL = Xc and hence 𝑍 = 𝑅 , is said to be in resonance. In a series circuit current I remains the same throughout we can write,
- 5. 𝐼𝑋 𝐿 = 𝐼𝑋 𝐶 i.e. 𝑉𝐿 = 𝑉𝐶 So, at resonance 𝑉𝐿 𝑎𝑛𝑑 𝑉𝐶 will cancel out each other. ∴ the supply voltage 𝑉 = 𝑉𝐿 2 + (𝑉𝐿 − 𝑉𝐶)2 𝑉 = 𝑉𝑅 2 ∴ 𝑉 = 𝑉𝑅 𝑖. 𝑒. The entire supply voltage will drop across the resistor R
- 6. Resonant frequency • At resonance XL = Xc ∴ 2π𝑓𝑟 𝐿 = 1 2π 𝑓𝑟 𝐶 (𝑓𝑟 is the resonant frequency) ∴ 𝑓𝑟 2 = 1 (2π)2 L𝐶 ∴ 𝑓𝑟 = 1 2π L𝐶 Where L is the inductance in henry, C is the capacitance in farad and 𝑓𝑟 the resonant frequency in Hz
- 7. Under resonance condition the net reactance is zero . Hence the impedance of the circuit. This is the minimum possible value of impedance. Hence, circuit current is maximum for the given value of R and its value is given by The circuit behaves like a pure resistive circuit because net reactance is zero . So, the current is in phase with applied voltage .obviously, the power factor of the circuit is unity under resonance condition. as current is maximum it produces large voltage drop across L and C. 2 2 0 0L CZ R X R X orX X m V V I Z R Z R
- 8. Voltage across the inductance at resonance is given by 2 2 1 2 2 L m L m r m r m m m m V I X I L I f L L L I L I I LCLC LC L I C At resonance, the current the current flowing in the circuit is equal to V R 2L V L L V V R C CR
- 9. Similarly voltage across capacitance at resonance is given by Thus voltage drop across L and C are equal and many times the applied voltage. Hence voltage magnification occurs at the resonance condition.so series resonance condition is often refers to as voltage resonance. 2 1 1 2 1 1 2 2 C m C m m r r m m m m V I X I I C f C I LC LC I I C CC LC L V L I C R C
- 10. Q-FACTOR IN R-L-C SERIES CIRCUIT Q-FACTOR: In case of R-L-C series circuit Q-Factor is defined as the voltage magnification of the circuit at resonance. Current at resonance is given by And voltage across inductance or capacitor is given by = OR Voltage magnification = voltage across L or C /applied voltage OR OR m m V I V I R R m LI X m CI X LV V CV V L CV V m L m I X I R m C m I X I R CL XX OR R R
- 11. Thus Q-factor 2 2 1 2 1 2 2 1 12 2 2 1 1 1 CL r r r r XX OR R R L OR R CR f L OR R f CR L OR LCR CR LC L LC OR R LC R C L R C 21 1 2 r r f LL Q factor R C R f CR
- 12. Effects of series resonance 1. When a series in R-L-C circuit attains resonance 𝑋 𝐿 = 𝑋 𝐶 i.e., the next reactance of the circuit is zero. 2. 𝑍 = 𝑅 𝑖. 𝑒. , the impedance of the circuit is minimum. 3. Since Z is minimum, 𝐼 = 𝑉 𝑍 will be minimum. 4. Since I is maximum, the power dissipated would be maximum 𝑃 = 𝐼2 R . 5. Since 𝑉𝐿 = 𝑉𝐶 , 𝑉 = 𝑉𝑅. 𝑖. 𝑒., the supply voltage is in phase with the supply current
- 13. RESONANCE IN R-L-C PARALLEL CIRCUIT
- 14. In R-L-C series circuit electrical resonance takes place when the voltage across the inductance is equal to the voltage across the capacitance. Alternatively, resonance takes place when the power factor of the circuit becomes unity. this is the basic condition of resonance. It remains the same for parallel circuits also .thus resonance will occur in parallel circuit when the power factor of the entire circuit becomes unity . let us consider R-L-C parallel circuit
- 15. For parallel circuit, the applied voltage is taken as reference phasor. The current drawn 𝑑𝑦 𝑑𝑥 an inductive coil lags the applied voltage by an phase angle 𝜃 . The current drawn by capacitor leads the applied voltage by 90° . Now power factor of the entire circuit is in phase with the applied voltage. This will happen when the current drawn by the capacitive branch, equals to the reactive component of current of inductive branch.
- 16. Hence the resonance takes place the necessary condition is ……………(1) Current in a capacitive branch, and ………...(2) Current in inductive branch, Where = impedance of the inductive branch angle of lag and Now …………….(3) sinC LI I C C V I X L L V I Z LZ 2 2 L LR jX R X 1 2 tan sin sin L L L L L L L L L X R X Z X VXV I Z Z Z
- 17. 2 2 22 2 2 2 2 2 1 1 L C L L L C r r r r r VXV X Z Z X X L R L L C C L L R C L R L C 2 2 2 2 2 2 2 2 2 1 1 1 2 1 1 2 r r r r R LC L R LC L R f LC L R f LC L Now substituting the equations 2 and 3 in equation 1,we get
- 18. Where 𝑓𝑟 is a resonant frequency in Hertz. The expression is different from that of series circuit. However if the resistance (R) of the coil is negligible the expression of resonant frequency reduces to 1 1 1 2 2 rf LC LC
- 19. From the phasor diagram it is clear that, the current in the inductive and capacitive branches may be many times greater then the resultant current under the condition of resonance. As the reactive component is 0 under resonance condition in order to satisfy the condition of unity power factor, the resultant current is minimum under this condition. From the above, it is observed that the current taken from the supply can be greatly magnified by means of a parallel resonant circuit. This current magnification is termed as Q-factor of the circuit. sin tan cos C L L I I Q factor I I
- 20. At resonance the resultant current drawn by parallel circuit is in phase with the applied voltage. resultant current But under the condition of resonance Resultant current 2L rX f L R R 2 cos .L L L L V R VR I I Z Z Z 2 1 L L C r r L Z X X L C C / / VR VR L C L CR
- 21. Thus the impedance offered by a parallel resonant circuit . This impedance is purely resistive and generally known as equivalent or dynamic impedance of the circuit.as the current at resonance is minimum, the dynamic impedance represents the maximum impedance offered by the circuit at resonance . So the parallel circuit is consider as a condition of maximum impedance or minimum admittance. The current at resonance is minimum , hence the circuit is sometimes called as rejecter circuit because it rejects that frequency to which it resonant. The phenomenon of resonance is parallel circuits is of great importance because it forms the basis of tuned circuits in electronics. L CR
- 23. Description Impedance at resonance Current at resonance Resonant frequency When 𝒇 < 𝒇 𝒓 When 𝒇 > 𝒇 𝒓 Power factor at resonance Q- factor It magnifies at resonance Series circuit Minimum given by 𝒁 = 𝑹 Maximum = 𝑽 𝑹 𝒇 𝒓 = 𝟏 𝟐𝝅 𝑳𝑪 Circuit is capacitive (as the net reactance is negative) Circuit is inductive (as the net reactance is positive) Unity 𝑿 𝑳 𝑹 voltage Parallel circuit Maximum given by 𝒁 = 𝑳 𝑪𝑹 Minimum = 𝑽 𝑳 𝑪𝑹 𝒇 𝒓 = 𝟏 𝟐𝝅 𝟏 𝑳𝑪 − 𝑹 𝟐 𝑳 𝟐 Circuit is inductive (as the net susceptance is negative) Circuit is capacitive (as the net susceptance is positive) Unity 𝑿 𝑳 𝑹 current
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