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Prof. Surajit Dhara SACT, Dept. Of Physics, Narajole Raj College C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits ❖ Sinusoidal Voltage Applied to a Series L-C-R Circuit: Let us consider an alternating emf. 𝐸(𝑑)=𝐸

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C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College

Prof. Surajit Dhara SACT,

Dept. Of Physics, Narajole Raj College

C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits

❖ Sinusoidal Voltage Applied to a Series L-C-R Circuit: Let us consider an alternating emf. 𝐸(𝑑) = 𝐸0π‘’π‘—πœ”π‘‘ is applied in a circuit containing a resistance R, an inductance L and a capacitance C in series.

Let q(t) be the charge through the capacitor at any instant t and instantaneous currnet is i , where 𝑖 =π‘‘π‘ž

𝑑𝑑. So the instantaneous voltage, Across the resistance = 𝑅𝑖

Across the inductance = βˆ’π‘‘π‘–

𝑑𝑑

Across the capacitance = π‘ž

⁄𝑐

Thus KVL equation for the circuit, 𝑅𝑖 +π‘ž

⁄ = 𝐸(𝑑) βˆ’ 𝐿𝑐 𝑑𝑖

𝑑𝑑 β‡’ 𝑅𝑖 + 𝐿𝑑𝑖

𝑑𝑑+1

π‘βˆ« 𝑖𝑑𝑑 = 𝐸0π‘’π‘—πœ”π‘‘ …..(1)

➒ Solution:- Let 𝑖(𝑑) = π΄π‘’π‘—πœ”π‘‘ be the trial solution. From the equation (1)

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C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College

𝑅𝐴𝑒𝑗𝑀𝑑+ πΏπ΄π‘—πœ”π‘’π‘—πœ”π‘‘ + 𝐴

π‘–πœ”π‘π‘’π‘—πœ”π‘‘ = 𝐸0π‘’π‘–πœ”π‘‘

β‡’ 𝐴 (𝑅 + π‘—πœ”πΏ + 1

π‘—πœ”π‘) = 𝐸0

β‡’ 𝐴 = 𝐸0

𝑅+𝑗(πœ”πΏβˆ’1

πœ”π‘) …….(2) Thus the current in the circuit,

𝑖(𝑑) = 𝐸0π‘’π‘—πœ”π‘‘ 𝑅 + 𝑗(πœ”πΏ βˆ’ 1

πœ”π‘)

= π‘…βˆ’π‘—(πœ”πΏβˆ’

1 πœ”π‘) 𝑅2+(πœ”πΏβˆ’1

πœ”π‘)2. 𝐸0π‘’π‘—πœ”π‘‘

= 𝐸0

√(𝑅2 + (πœ”πΏ βˆ’ 1 πœ”π‘)2[

𝑅

√(𝑅2+ (πœ”πΏ βˆ’ 1 πœ”π‘)2

βˆ’ 𝑗 (πœ”πΏ βˆ’ 1 πœ”π‘)

√(𝑅2+ (πœ”πΏ βˆ’ 1 πœ”π‘)2]

π‘’π‘—πœ”π‘‘

Put, 𝑅

√(𝑅2+(πœ”πΏβˆ’1 πœ”π‘)2

= π‘π‘œπ‘ πœƒ

(πœ”πΏ βˆ’ 1 πœ”π‘)

√(𝑅2 + (πœ”πΏ βˆ’ 1 πœ”π‘)2

= π‘ π‘–π‘›πœƒ

.: π‘‘π‘Žπ‘›πœƒ = πœ”πΏβˆ’

1 πœ”π‘

𝑅

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C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College

.: 𝑖 = 𝐸0

√(𝑅2+(πœ”πΏβˆ’1 πœ”π‘)2

π‘’βˆ’π‘—πœƒ. π‘’π‘—πœ”π‘‘

β‡’ π’Š = π’ŠπŸŽπ’†π’‹(πŽπ’•βˆ’πœ½) Where 𝑖0 = 𝐸0

√(𝑅2+(πœ”πΏβˆ’1 πœ”π‘)2

𝑖 = 𝑖0[cos (πœ”π‘‘ βˆ’ πœƒ) + 𝑗𝑠𝑖𝑛(πœ”π‘‘ βˆ’ πœƒ)]

𝑖0cos(πœ”π‘‘ βˆ’ πœƒ) β‡’ π‘…π‘’π‘Žπ‘™ π‘π‘Žπ‘Ÿπ‘‘ π‘œπ‘“ π‘π‘’π‘Ÿπ‘Ÿπ‘’π‘›π‘‘

𝑖0𝑠𝑖𝑛(πœ”π‘‘ βˆ’ πœƒ) β‡’ πΌπ‘šπ‘Žπ‘”π‘–π‘›π‘Žπ‘Ÿπ‘¦ π‘π‘Žπ‘Ÿπ‘‘ π‘œπ‘“ π‘π‘’π‘Ÿπ‘Ÿπ‘’π‘›π‘‘

➒ Peak Value of Current:

𝑖0 =|𝑍|𝐸 = 𝐸0

√(𝑅2+(πœ”πΏβˆ’1 πœ”π‘)2

➒ Impedance: 𝑧 = 𝐸

𝑖 = 𝑅 + 𝑗(πœ”πΏ βˆ’ 1

πœ”π‘) 𝑍⃗ = 𝑅⃗⃗ + 𝑗(πœ”πΏβƒ—βƒ— βˆ’ 1

πœ”π‘) |𝑍| = √(𝑅2+ (πœ”πΏ βˆ’ 1

πœ”π‘)2 Inductive reactance 𝑋𝐿 = πœ”πΏ

Capacitive reactance 𝑋𝑐 = 1

πœ”π‘

➒ Impedance diagram and phasor diagram:- a) When πœ”πΏ > 1

πœ”π‘ i.e. inductive reactance is greater than capacitive reactance. Here

βˆ… is positive; which implies that the current lags behind the emf by an angle βˆ…

, where βˆ… = tanβˆ’1πœ”πΏβˆ’

1 πœ”π‘ 𝑅 .

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C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College

b) When 1

πœ”π‘ > πœ”πΏ i.e. when capacitive reactance is greater than inductive reactance.

Here 𝛼 = βˆ’π‘‰π‘’; which implies that the current leads the applied emf by an angle 𝛼 = tanβˆ’1(

1 πœ”π‘βˆ’πœ”πΏ

𝑅 ) c) When πœ”πΏ = 1

πœ”π‘ i.e when inductive reactance is equal to capacitive reactance.

Here πœƒ = 0 i.e the circuit is purely resistive.

➒ Series Resonant Circuit: The circuit containing inductor(L), capacitor(c) and resistor(R) are connected in series and subjected to an alternating emf. Where

(i) The circuit behaves as purely resistive.

(ii) Impedance is minimum.

(iii) Current is maximum

(iv) The emf and current will be in same phase.

Then the circuit is called series resonant circuit.

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C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College

➒ Sharpness of Resonance of a Series L-C-R Circuit: For a series L-C-R circuit the maximum current 𝑖0 = 𝐸0

√(𝑅2+(πœ”πΏβˆ’1 πœ”π‘)2

; At resonance , πœ”πΏ = 1

πœ”π‘ then 𝑖0 =𝐸0

𝑅

The curves plotting the current verses frequency (𝑓 π‘Žπ‘›π‘‘ πœ”) are known as resonance curves. From the figure we see that when the resistance in the circuit is reduced the resonance curve become sharper. The peak value of i shows that, the circuit responds only to the frequency exactly equal to the natural frequency of the circuit 𝝎𝟎 = 𝟏

βˆšπ‘³π’„ . This resonance is sharp. The sharpness of resonance is a measure of the rate of fall of amplitude from its maximum value at resonance frequency on either side of it.

➒ Quality Factor: Sharpness of resonance curve is determined by quality factor, called

β€œQ” of the circuit. Which is defined as the ratio of the reactance (π‘‹πΏπ‘œπ‘Ÿπ‘‹πΆ) to the impedance at resonant frequency is called Quality factor of the circuit. i.e.

𝑄 = 𝑋𝐿

𝑍|

πœ”0 = πœ”0𝐿

𝑅 Or, 𝑄 = 𝑋𝑐

𝑍|

πœ”0 = 1

πœ”0𝑐𝑅

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C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College

❖ Parallel L-C-R Circuit: Let the combination is connected to alternating emf, 𝐸 = 𝐸0π‘’π‘—πœ”π‘‘. If Z be the total impedance for the combination then,

1

𝑍 = 1

π‘πΏβˆ’π‘… + 1 𝑍𝑐

β‡’ 1

𝑍 = 1

𝑅 + π‘—πœ”πΏ+ π‘—πœ”π‘

β‡’ 𝑧 = 𝑅 + π‘—πœ”πΏ

(1 βˆ’ πœ”2𝐿𝐢) + 𝑗(πœ”π‘π‘…)

= 𝑅 + π‘—πœ”πΏ

(1 βˆ’ πœ”2𝐿𝐢) + 𝑗(πœ”π‘π‘…)Γ—(1 βˆ’ πœ”2𝐿𝐢) βˆ’ 𝑗(πœ”π‘π‘…) (1 βˆ’ πœ”2𝐿𝐢) βˆ’ 𝑗(πœ”π‘π‘…)

. : 𝑧 = 𝑅 + 𝑗[πœ”πΏ βˆ’ πœ”π‘(πœ”2𝐿2+ 𝑅2)]

(1 βˆ’ πœ”2𝐿𝐢)2+ πœ”2𝑐2𝑅2

= 𝑅

(1βˆ’πœ”2𝐿𝐢)2+πœ”2𝑐2𝑅2+ π‘—πœ” πΏβˆ’π‘(πœ”2𝐿2+𝑅2)

(1βˆ’πœ”2𝐿𝐢)2+πœ”2𝑐2𝑅2

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C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College

= 𝑅0+ π‘—πœ”πΏπ‘œ (say)

Where 𝑅0 = 𝑅

(1βˆ’πœ”2𝐿𝐢)2+πœ”2𝑐2𝑅2 ; effective resistance of the coil.

πΏπ‘œ = πΏβˆ’π‘(πœ”2𝐿2+𝑅2)

(1βˆ’πœ”2𝐿𝐢)2+πœ”2𝑐2𝑅2 ; effective inductance of the coil.

➒ Magnitude of Impedance:-

|𝑧| = βˆšπ‘…02+ πœ”2𝐿02

|𝑧| = [ 𝑅2+ πœ”2𝐿2

(1 βˆ’ πœ”2𝐿𝐢)2+ πœ”2𝑐2𝑅2]

1⁄2

➒ Resonant Frequency :- The parallel at which the circuit is purely resistive or current will be in same phase with the emf is known as resonant frequency. At πœ” = πœ”0 , 𝐿0 = 0

β‡’ 𝐿 βˆ’ 𝑐(πœ”02𝐿2+ 𝑅2) = 0 β‡’ πœ”02𝐿2+ 𝑅2 = 0

β‡’ πœ”02𝐿2 =𝐿 𝑐 βˆ’ 𝑅2 β‡’ πœ”02 = 1

πΏπ‘βˆ’π‘…2

𝐿2

β‡’ πœ”0 = √1

πΏπ‘βˆ’π‘…2

𝐿2

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C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College

Parallel resonant frequency, 𝑓0 = 1

2πœ‹βˆš1

πΏπ‘βˆ’π‘…2

𝐿2

➒ Condition of Parallel Resonance: For parallel resonance , πœ”0= real (must) β‡’ √1

πΏπ‘βˆ’π‘…2

𝐿2 > 0 β‡’ 1

πΏπ‘βˆ’π‘…2

𝐿2 > 0

β‡’ 𝑅2 < 𝐿 𝑐 β‡’ 𝑅 < √𝐿

𝑐

Hence for parallel resonance resistance R should be kept as low as possible.

Referensi

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