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Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

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Series L-R Circuit

  • Now consider an ac circuit consisting of a resistor of resistance R and an inductor of inductance L in series with an ac source generator.
  • Suppose in phasor diagram, current is taken along positive x-direction. The VR is also along positive x-direction and VL along positive y-direction as we know that potential difference across a resistance in ac is in phase with current and it leads in phase by 90º with current across the inductor, and as we know VR = i0R & V0 = i0XL
    i = i0 sin wt
    Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET
    VR(t) = i0 Rsin ωt
    VL(t) = i0 XL sin (ωt + p/2)
    hence we can write
    V(t) = i0R sin ωt + i0 XL sin (ωt + p/2)
    V0 = i0Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET
    where Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET is known as impedence (z) of the circuit.
    now we can write
    Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET
    where tan β= Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET
    hence β = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Example 1. When 100 volt dc is applied across a coil, a current of 1 amp flows through it; when 100 V ac of 50 Hz is applied to the same coil, only 0.5 amp flows. Calculate the resistance of inductance of the coil.

Sol. In case of a coil, i.e., L - R circuit.

i = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET with Z = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

So when dc is applied, ω = 0, so z = R

and hence i = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET i.e., R = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = 200 ?

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEETi.e., ω2L2 = Z2-R2

i.e., (2πfL)2 = 2002 - 1002 = 3 × 104 (as ω = 2πf)

So, L = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = 0.55 H


Example 2. A 12 ohm resistance and an inductance of 0.05/p henry with negligible resistance are connected in series. Across the end of this circuit is connected a 130 volt alternating voltage of frequency 50 cycles/second. Calculate the alternating current in the circuit and potential difference across the resistance and that across the inductance.

Sol. The impedance of the circuit is given by

Z = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

= Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = 13 ohm

Current in the circuit i = E/Z = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = 10 amp

Potential difference across resistance

VR = iR = 10 × 12 = 120 volt

Inductive reactance of coil XL = ωL = 2πfL

Therefore, XL = 2π × 50 × Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = 5 ohm

Potential difference across inductance

VL = i × XL = 10 × 5 = 50 volt

Series C-R Circuit

Now consider an ac circuit consisting of a resistor of resistance R and an capacitor of capacitance C in series with an ac source generator.

Suppose in phasor diagram current is taken along positive x-direction. Then VR is also along positive x-direction but VC is along negative y-direction as potential difference across a capacitor in ac lags in phase by 90º with the current in the circuit. So we can write,

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

VR = I0 R sin ωt Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Potential difference across capacitor

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Potential at any instant t

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

V(t) = V0 sin (ωt + b)

tan α = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Example 3. An A.C. source of angular frequency w is fed across a resistor R and a capacitor C in series. The current registered is i. If now the frequency of the source is changed to ω/3 (but maintaining the same voltage), the current in the circuit is found to be halved. Calculate the ratio of reactance to resistance at the original frequency.

Sol. At angular frequency w, the current in R-C circuit is given by

irms = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET ...(i)

When frequency is changed to ω/3, the current is halved. Thus

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET ...(ii)

From equations (i) and (ii), we have

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Solving this equation, we get Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Hence, the ratio of reactance to resistance is Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Example 4. A 50 W, 100 V lamp is to be connected to an ac mains of 200 V, 50 Hz. What capacitance is essential to be put in series with the lamp?

Sol. As resistance of the lamp R = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = 200 ? and the maximum current i = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET ; so when the lamp is put in series with a capacitance and run at 200 V ac, from V =iZ we have,

Z = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = 400?

Now as in case of C-R circuit, Z = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET,

i.e., R2 + Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = 160000

or, Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = 16 × 104 - (200)2 = 12 × 104

So, Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET × 102

or C = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

i.e., C = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET = 9.2 μF

L.C. Circuit

As shown in figure a capacitor and inductance are connected in series method and alternating voltage is applied across the circuit.

Let Xc is capacitance reactance,

XL is Inductance reactance,

i = i0 sin wt current flowing through the circuit

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

VC(t) = i0 XC sin (ωt - π/2)

VL(t) = i0 XL sin (ωt + π/2)

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

= i0 XC sin wt cos π/2 - i0 XC cos wt sin π/2 + i0 XL sin wt cos π/2 + i0 XL cos wt sin π/2

= i0 cos w t(XL - XC)

V(t) = V0sin (wt + p/2)

V0 = i0Z Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Z = (X- XC)

cos Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET= 0

VCO = i0 XC ; Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Series L-C-R Circuit

Now consider an ac circuit consisting of a resistor of resistance R, a capacitor of capacitance C and an inductor of inductance L are in series with an ac source generator.

Suppose in a phasor diagram current is taken along positive x-direction. Then VR is along positive x-direction, VL along positive y-direction and VC along negative y-direction, as potential difference across an inductor leads the current by 90° in phase while that across a capacitor, lags by 90°.

V = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

L - R - C circuit

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Impedance phasor of above circuit

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

 

& Impedance triangle

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

here B is phase angle By triangle tan β= Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Power factor cosSeries L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET= Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Let I be the current in the series circuit of any instant then

(1) Voltage V(t) = V0 sin (ωt + β) = i0 z sin (ωt + β)

here v0 = i0z & vrms = irmsz

(2) Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET here voltage VL across the inductance is ahead of current I in phase by π/2 rad

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

(3) VC(t) = VO C sin (ωt - π/2)

here voltage VC across the capacitance lags behind the current I in phase by π/2 rad

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

(4) VR(t) = i0 R sin ωt

here voltage VR across the resistor R has same phase as I

VO R = IR

Special Case :

(1) When XL > XC or VL > VC then emf is ahead of current by phase β which is given by

tan β = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET or cos f = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

The series LCR circuit is said to be inductive

(2) When XL < XC or VL < VC then current is ahead of emf by phase angle β which is given by

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET or Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

The series LCR circuit is said to be capacitive

(3) When XL = XC or VL = VC, b = 0, the emf and current will be in the same phase. The series LCR circuit is said to be purely resistive. It may also be noted that

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET or Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET or IRms = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Susceptance : The reciprocal of the reactane of an a.c. circuit is called its susceptance.

Admittance : The reciprocal of the impedance of an a.c. circuit is called its admittance.

Ex.10 Figure shows a series LCR circuit connected to a variable voltage source V = 10 sin (ωt + p/4) ;

xL = 10 ?, XC = 6 ?, R = 3 ?

Calculate Z, i0, irms, vrms, VL O, VC O, VR O, b,

VL Rms, VC Rms, VRms, i(t), VL(t), Vc(t), and VR(t)

 

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEETXL > XC

Sol. V = 10 sin (wt + p/4) so V0 = 10 volt

Vrms = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Therefore, Z = Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET; Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET; Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET; Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

i(t) = 20 sin (ωt + π/4 - 53°)

VL(t) = 20 sin (ωt + π/4 - 53° + π/2)

= 20 sin (ωt + Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET - 53° - Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET) = 12 sin (ωt - Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET - 53°)

Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET

Ex.11 A resistor of resistance R, an inductor of inductance L and a capacitor of capacitance C all are connected in series with an a.c. supply. The resistance of R is 16 ohm and for a given frequency the inductive reactance of L is 24 ohm and capacitive reactance of C is 12 ohm. If the current in the circuit is 5 amp., find

(a) the potential difference across R, L and C (b) the impedance of the circuit

(c) the voltage of a.c. supply (d) phase angle

Sol. (a) Potential difference across resistance

VR = iR = 5 × 16 = 80 volt

Potential difference across inductance

VL = i × (wL) = 5 × 24 = 120 volt

The document Series L-R, C-R & C-R Circuit - Notes | Study Physics Class 12 - NEET is a part of the NEET Course Physics Class 12.
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