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3. SERIES AC CIRCUIT

3.1 When only Resistance is in an AC circuit

Consider a simple ac circuit consisting of a resistor of resistance R and an ac generator, as shown in the figure. According to Kirchhoff's loop law at any instant, the algebraic sum of the potential difference around a closed loop in a circuit must be zero.

Series AC Circuit | Physics Class 12 - NEET

ε - VR = 0

ε - iRR = 0

ε0 sinωt - iRR = 0

iRSeries AC Circuit | Physics Class 12 - NEET sinωt = i0 sin ωt ..(i)

where i0 is the maximum current. i0Series AC Circuit | Physics Class 12 - NEET

From above equations, we see that the instantaneous voltage drop across the resistor

VR = i0R sinωt ...(ii)

We see in equation (i) & (ii), iR and VR both vary as sin wt and reach their maximum values at the same time as shown in figure (a), they are said to be in phase. A phasor diagram is used to represent phase relationships. The lengths of the arrows correspond to V0 and i0. The projections of the arrows onto the vertical axis give VR and iR. In case of the single-loop resistive circuit, the current and voltage phasors lie along the same line, as shown in figure (b), because iR and VR are in phase.

Series AC Circuit | Physics Class 12 - NEET Series AC Circuit | Physics Class 12 - NEET

3.2 When only Inductor is in An AC circuit

Now consider an ac circuit consisting only of an Inductor of inductance L connected to the terminals of an ac generator, as shown in the figure. The induced emf across the inductor is given by Ldi/dt. On applying Kirchhoff's loop rule to the circuit

ε- VL = 0 ⇒ ε - LSeries AC Circuit | Physics Class 12 - NEET

When we rearrange this equation and substitute

ε = ε0 sin ωt, we get   Series AC Circuit | Physics Class 12 - NEET

Series AC Circuit | Physics Class 12 - NEET = ε0 sin ωt ...(iii)

Integration of this expression gives the current as a function of time

iLSeries AC Circuit | Physics Class 12 - NEET = - Series AC Circuit | Physics Class 12 - NEET

For average value of current over one time period to be zero, C = 0

Therefore, iL = - Series AC Circuit | Physics Class 12 - NEET

When we use the trigonometric identity

coswt = - sin(wt - p/2), we can express equation as

iLSeries AC Circuit | Physics Class 12 - NEET ...(iv)

From equation (iv), we see that the current reaches its maximum values when cos wt = 1.

i0Series AC Circuit | Physics Class 12 - NEET = Series AC Circuit | Physics Class 12 - NEET ...(v)

where the quantity XL, called the inductive reactance, is

XL = ωL

The expression for the rms current is similar to equation (v), with ε0 replaced by εrms.

Inductive reactance, like resistance, has unit of ohm.

Series AC Circuit | Physics Class 12 - NEET

We can think of equation (v) as Ohm's law for an inductive circuit.

On comparing result of equation (iv) with equation (iii), we can see that the current and voltage are out of phase with each other by π/2 rad, or 90º. A plot of voltage and current versus time is given in figure (a). The voltage reaches its maximum value one quarter of an oscillation period before the current reaches its maximum value. The corresponding phasor diagram for this circuit is shown in figure (b). Thus, we see that for a sinusoidal applied voltage, the current in an inductor always lags behind the voltage across the inductor by 90º.

Series AC Circuit | Physics Class 12 - NEET Series AC Circuit | Physics Class 12 - NEET

Ex.4 An inductor of inductance L = 5 H is connected to an

AC source having voltage v = 10 sin (10t + Series AC Circuit | Physics Class 12 - NEET)

Find Series AC Circuit | Physics Class 12 - NEET

(i) Inductive Reactance (xL)

(ii) Peak & Rms voltage (V0 & Vrms)

(iii) Peak & Rms current (I0 & Irms)

(iv) Instantaneous current (I(t))

Sol. (i) xL = ωL = 10 × 5 = 50

(ii) v0 = 10

vrmsSeries AC Circuit | Physics Class 12 - NEET

(iii) Series AC Circuit | Physics Class 12 - NEET

IrmsSeries AC Circuit | Physics Class 12 - NEET

(iv) I(t) = Series AC Circuit | Physics Class 12 - NEET

3.3 When only Capacitor is in An AC circuit

Figure shows an ac circuit consisting of a capacitor of capacitance C connected across the terminals of an ac generator. On applying Kirchhoff's loop rule to this circuit, we get

Series AC Circuit | Physics Class 12 - NEET

ε - VC = 0

VC = ε = ε0 sin ωt ...(vi)

where VC is the instantaneous voltage drop across the capacitor. From the definition of capacitance, VC = Q/C, and this value for VC substituted into equation gives

Q = C ε0 sin ωt

Since i = dQ/dt, on differentiating above equation gives the instantaneous current in the circuit.

Series AC Circuit | Physics Class 12 - NEET

Here again we see that the current is not in phase with the voltage drop across the capacitor, given by equation (vi). Using the trigonometric identity cos ωt = sin(ωt + π/2), we can express this equation in the alternative from

Series AC Circuit | Physics Class 12 - NEET ...(vii)

From equation (vii), we see that the current in the circuit reaches its maximum value when cos ωt= 1.

Series AC Circuit | Physics Class 12 - NEET

Where XC is called the capacitive reactance.

Series AC Circuit | Physics Class 12 - NEET

The SI unit of XC is also ohm. The rms current is given by an expression similar to equation with V0 replaced by Vrms.

Combining equation (vi) & (vii), we can express the instantaneous voltage drop across the capacitor as

VC = V0 sin ωt = i0 XC sin ωt

Comparing the result of equation (v) with equation (vi), we see that the current is π/2 rad = 90º out of phase with the voltage across the capacitor. A plot of current and voltage versus time, shows that the current reaches its maximum value one quarter of a cycle sooner than the voltage reaches its maximum value. The corresponding phasor diagram is shown in the figure (b). Thus we see that for a sinusoidally applied emf, the current always leads the voltage across a capacitor by 90º.

Series AC Circuit | Physics Class 12 - NEET Series AC Circuit | Physics Class 12 - NEET

Brain Teaser

What is the reactance of a capacitor connected to a constant DC source ?


Ex.5 A capacitor of capacitive reactance 5? is connected with A.C. source having voltage V = 3 sin (ωt + p/6). Find rms and Peak voltage rms and peak current and instantaneous current

Series AC Circuit | Physics Class 12 - NEET

Sol. On comparing with

Series AC Circuit | Physics Class 12 - NEET ⇒ v0 = 3

Series AC Circuit | Physics Class 12 - NEET ⇒ Series AC Circuit | Physics Class 12 - NEET

Series AC Circuit | Physics Class 12 - NEET ⇒ I(t) = I0 sin Series AC Circuit | Physics Class 12 - NEET

The document Series AC Circuit | Physics Class 12 - NEET is a part of the NEET Course Physics Class 12.
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FAQs on Series AC Circuit - Physics Class 12 - NEET

1. What is an AC circuit?
Ans. An AC circuit refers to an electrical circuit that operates with an alternating current. In such circuits, the flow of electric charge periodically changes direction, usually in the form of a sine wave. AC circuits are commonly used in homes and industries for powering various electrical devices.
2. How does an AC circuit differ from a DC circuit?
Ans. AC circuits differ from DC circuits in terms of the direction of electric current flow. In an AC circuit, the current periodically changes direction, whereas in a DC circuit, the current flows in one direction only. AC circuits typically use transformers to change voltage levels, while DC circuits use batteries or power supplies.
3. What is the significance of series AC circuits?
Ans. Series AC circuits are important in understanding the behavior of electrical components connected in series with an alternating current source. By analyzing series AC circuits, we can determine various parameters such as impedance, current, voltage drops, and power dissipation. This knowledge is crucial for designing and troubleshooting electronic systems.
4. How do we calculate the impedance in a series AC circuit?
Ans. Impedance in a series AC circuit can be calculated using the formula Z = √(R^2 + (XL - XC)^2), where R is the resistance, XL is the inductive reactance, and XC is the capacitive reactance. The impedance represents the combined effect of resistance, inductance, and capacitance in the circuit and is measured in ohms.
5. What is the power factor in a series AC circuit?
Ans. The power factor in a series AC circuit indicates the efficiency of power transfer from the source to the load. It is the ratio of true power (measured in watts) to apparent power (measured in volt-amperes). A power factor of 1 signifies maximum power transfer, while a power factor less than 1 indicates reactive power consumption. Power factor correction techniques are often employed to improve the efficiency of electrical systems.
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