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- 1. Welcome to Classes BYJU’S Alternating Current What you already know What you will learn S9: LC Oscillations 1 . Pure AC circuits 2 . RC and L R AC circuits 3 . Impedance 4. Powe r in AC circuits 5 . Re sonance 1 . Radio tuning 2 . Bandwidth 3 . Sharpne ss 4. Quality factor 5 . Eve rything ab out L C oscillations
- 2. CASE 3 If 𝑋𝐶 = 𝑋𝐿 , 𝜙 = 0 𝒕𝒂𝒏 𝝓 = 𝑿𝑳 − 𝑿𝑪 𝑹 CASE 1 If 𝑋𝐶 > 𝑋𝐿 , 𝜙 ≠ 0 Circuit is predominantly capacitive Current leads the source voltage CASE 2 If 𝑋𝐶 < 𝑋𝐿 , 𝜙 ≠ 0 Circuit is predominantly inductive Current lags the source voltage
- 3. Resonance If 𝜔 is varied, then at a particular frequency (𝜔0), 𝑋𝐶 = 𝑋𝐿 Impedance is minimum (𝑍 = 𝑅2 + 02 = 𝑅) and purely resistive circuit Current is maximum (𝑖𝑚 = 𝑉𝑚 /𝑅) For resonance condition, 𝑋𝐶 = 𝑋𝐿 1 𝜔0𝐶 = 𝜔0𝐿 ⇒ 𝜔0 = 1 𝐿𝐶 ⇒ 𝑓0 = 1 2𝜋 𝐿𝐶 Resonant frequency
- 6. | BIG FM (92.7) RADIO MIRCHI RADIO MANGO To hear one particular radio station, radio tuning is required.
- 7. 𝑅 𝑣 = 𝑣𝑚 sin(𝜔𝑡) 𝐶 𝐿 | To hear a particular radio station, tuning of radio is necessary i.e., vary the capacitance of a capacitor in tuning circuit of the radio such that the resonant frequency of the circuit becomes nearly equal to the frequency to that of radio signal received. At this condition, the amplitude of the current with the frequency of the signal of the particular radio station in the circuit is maximum.
- 8. | 𝑖 𝑚 𝜔(𝑟𝑎𝑑/𝑠) 𝜔0 2∆𝜔 𝑖𝑚 𝑚𝑎𝑥 𝑖𝑚 𝑚𝑎𝑥 2 𝜔1 𝜔2 Bandwidth (2∆𝜔)
- 9. | 𝑖 𝑚 𝜔(𝑟𝑎𝑑/𝑠) 𝜔0 2∆𝜔 𝑖𝑚 𝑚𝑎𝑥 𝑖𝑚 𝑚𝑎𝑥 2 𝜔1 𝜔2 𝑃𝑚𝑎𝑥 = (𝑖𝑚𝑎𝑥 )2 𝑅/2 At resonance, Power will be maximum. 𝑃 = 𝑖𝑚𝑎𝑥 2 2 𝑅 2 = 𝑃𝑚𝑎𝑥 2 𝜔1 and 𝜔2 are called as half-power points. • The current will be 1/ 2 times the maximum value of current at 𝜔1 and 𝜔2.
- 10. | 𝑖 𝑚 𝜔(𝑟𝑎𝑑/𝑠) 𝜔0 2∆𝜔 The quantity (𝜔0/2∆𝜔) is regarded as a measure of sharpness of resonance. GOOD TUNNING HIGH SHARPNESS LOW BANDWIDTH SMALL ∆𝜔 ∆𝜔 = 𝑅 2𝐿
- 11. | ⇒ 𝜔0 2∆𝜔 = 𝜔0𝐿 𝑅 Sharpness of resonance Quality Factor, 𝑄 ∆𝜔 = 𝑅 2𝐿 𝑖 𝑚 𝜔 (𝑟𝑎𝑑 /𝑠) 𝜔0 2∆𝜔 The quantity (𝜔0 /2∆𝜔) is regarded as a measure of sharpness of resonance. At resonance, 𝑄 can also be defined as, 𝑄 = 𝑉𝐿 𝑉𝑅 = 𝑉𝐶 𝑉𝑅 𝑄 = 𝑖0𝜔0𝐿 𝑖0𝑅 = 𝑖0 𝜔0𝐶𝑖0𝑅 𝑄 = 𝜔0𝐿 𝑅 = 1 𝜔0𝑅𝐶 = 1 𝑅 𝐿 𝐶
- 12. | 𝑖 𝑚 𝜔(𝑟𝑎𝑑/𝑠) 𝜔0 2∆𝜔 𝑄 = 𝜔0 2∆𝜔 𝑄 = 𝜔0 𝐿 𝑅 𝑄 = 1 𝑅 𝐿 𝐶 𝑄 = 1 𝜔0 𝑅𝐶
- 13. Which of the following combination should be selected for better tuning of an L -C-R circuit used for communication? a b c 𝑅 = 25 Ω, 𝐿 = 1.5 𝐻, 𝐶 = 45 𝜇𝐹 𝑅 = 20 Ω, 𝐿 = 1.5 𝐻, 𝐶 = 35 𝜇𝐹 𝑅 = 25 Ω, 𝐿 = 2.5 𝐻, 𝐶 = 45 𝜇𝐹 𝑅 = 15 Ω, 𝐿 = 3.5 𝐻, 𝐶 = 30 𝜇𝐹 d
- 14. 𝐿 should be high. For better tuning, the quality factor of the circuit should be high. 𝑄 = 1 𝑅 𝐿 𝐶 Value of R and C should be minimum 𝑄 = 1 𝑅 𝐿 𝐶 a b c d 𝑅 = 25 Ω, 𝐿 = 1.5 𝐻, 𝐶 = 45 𝜇𝐹 𝑅 = 25 Ω, 𝐿 = 1.5 𝐻, 𝐶 = 35 𝜇𝐹 𝑅 = 25 Ω, 𝐿 = 2.5 𝐻, 𝐶 = 45 𝜇𝐹 𝑅 = 15 Ω, 𝐿 = 3.5 𝐻, 𝐶 = 30 𝜇𝐹
- 16. 𝐿 𝐶 −𝑞𝑂 +𝑞𝑂 ++ + 𝑖 𝐵=0 Step 1 𝐿 𝐶 𝐵 = 𝐵𝑂 𝑖 = 𝑖𝑂 Step 2 𝐿 𝐶 𝐵 = 𝐵𝑂 Step 3 𝑖 = 𝑖𝑂 Now the same oscillation and steps will occur.
- 17. 𝐿 𝐶 −𝑞 +𝑞 At 𝑡 = 0, Current, 𝑖 = 0 Charge on capacitor, 𝑞 = 𝑞𝑚 Induced emf across the inductor, 𝐿 𝑑𝑖 𝑑𝑡 = 0 ( )) At time, 𝑡 Current = 𝑖 Charge on capacitor = 𝑞 Potential difference = 𝑞 𝑐 Induced emf across the inductor, 𝐿 𝑑𝑖 𝑑𝑡 = 0 𝐿 𝐶 −𝑞 +𝑞
- 18. Applying Kirchhoff's Law 𝑞 𝐶 − 𝐿 𝑑𝑖 𝑑𝑡 = 0 … (1) 𝑖 = − 𝑑𝑞 𝑑𝑡 𝑞 𝐶 − 𝐿 𝑑 − 𝑑𝑞 𝑑𝑡 𝑑𝑡 = 0 ⇒ 𝑞 𝐿𝐶 + 𝑑2𝑞 𝑑𝑡2 = 0 𝑑2𝑞 𝑑𝑡2 + 𝑞 𝐿𝐶 = 0 𝐿 𝐶 −𝑞 +𝑞 + − ( ))
- 19. For a particle in SHM 𝑑2 𝑥 𝑑𝑡2 + 𝜔𝑜 2𝑥 = 0 … (3) 𝜔 = 𝑘 𝑚 𝜔2 = 1 𝐿𝐶 ⇒ 𝜔 = 1 𝐿𝐶 𝑑2 𝑞 𝑑𝑡2 + 𝑞 𝐿𝐶 = 0 … (2) 𝐿 𝐶 −𝑞 +𝑞 ( ))
- 20. 𝑑2 𝑥 𝑑𝑡2 + 𝜔𝑜 2𝑥 = 0 … (3) 𝜔 = 1 𝐿𝐶 𝑥 = 𝐴 cos(𝜔𝑜𝑡 + 𝜙) 𝑞 = 𝑞𝑚 cos 𝜔𝑜𝑡 + 𝜙 … (4) At 𝑡 = 0, 𝑞 = 𝑞𝑚 ⇒ 𝑞𝑚= 𝑞𝑚 cos 𝜙 ⇒ cos 𝜙 = 1 ⇒ 𝜙 = 0 𝑞 = 𝑞𝑚 cos 𝜔𝑜 𝑡 𝐿 𝐶 −𝑞 +𝑞 ( ))
- 21. 𝜔 = 1 𝐿𝐶 𝑞 = 𝑞𝑚 cos 𝜔𝑜𝑡 … (5) 𝑖 = − 𝑑𝑞 𝑑𝑡 ⇒ 𝑖 = − 𝑑 𝑑𝑡 [𝑞𝑚 cos 𝜔𝑜𝑡 ] ⇒ 𝑖 = 𝜔𝑜𝑞𝑚 sin 𝜔𝑜𝑡 𝑖𝑚 = 𝜔𝑜𝑞𝑚 𝑖 = 𝑖𝑚 sin 𝜔𝑜 𝑡 𝐿 𝐶 −𝑞 +𝑞 ( ))
- 22. 𝜔 = 1 𝐿𝐶 𝑞 = 𝑞𝑚 cos 𝜔𝑜𝑡 … (5) 𝑖𝑚 = 𝜔𝑜𝑞𝑚 𝑖 = 𝑖𝑚 sin 𝜔𝑜𝑡 … (6) 𝑈 = 𝑞2 2𝐶 , 𝑈′ = 1 2 𝐿𝑖2 𝐿 = 1 𝜔𝑜 2 𝐶 𝑈𝑇𝑜𝑡𝑎𝑙 = 𝑈 + 𝑈′ 𝑈𝑇𝑜𝑡𝑎𝑙 = 𝑞2 2𝐶 + 1 2 𝐿𝑖2 𝐿 𝐶 −𝑞 +𝑞 ( ))
- 23. 𝜔 = 1 𝐿𝐶 𝑞 = 𝑞𝑚 cos 𝜔𝑜𝑡 … (5) 𝑖𝑚 = 𝜔𝑜𝑞𝑚 𝑖 = 𝑖𝑚 sin 𝜔𝑜𝑡 … (6) 𝑈𝑇𝑜𝑡𝑎𝑙 = 𝑞2 2𝐶 + 1 2 𝐿𝑖2 𝑈𝑇𝑜𝑡𝑎𝑙 = [𝑞𝑚 cos 𝜔𝑜𝑡 ]2 2𝐶 + 1 2 1 𝜔𝑜 2 𝐶 [𝑖𝑚 sin 𝜔𝑜𝑡 ]2 𝐿 = 1 𝜔𝑜 2 𝐶 𝑈𝑇𝑜𝑡𝑎𝑙 = 𝑞𝑚 2 2𝐶 [ cos2 𝜔𝑜𝑡) + (sin2 𝜔𝑜𝑡) 𝑈𝑇𝑜𝑡𝑎𝑙 = 𝑞𝑚 2 2𝐶 𝐿 𝐶 −𝑞 +𝑞 ( ))