Notes-Part-1-Class-12-Physics-Chapter-13-AC Circuits-MSBSHSE

Topics to be Learn : Part-1 Introduction AC generator Average and RMS values Phasors Different types of AC circuits Topics to be Learn : Part-2 Power in AC circuit LC oscillations Electric resonance Sharpness of resonance : Q factor Choke coil

Introduction :

There are two types of supplies of electricity:

• DC, the direct current which has fixed polarity of voltage (the positive and negative ends of the power supply are fixed).
• AC, the alternating current for which the polarity of the voltage keeps changing periodically.
• The electricity is typically supplied as AC due to its low cost and ease of transportation.
• Appliances like TVs, computers, transistor radios, and others that we use at home or in the workplace. Utilize a rectifier or similar device to convert AC to DC.
• Some of the devices, use in homes and offices, such as the television, computer, transistor, and radio, convert AC to DC using a device called a rectifier.
• Some household appliances run directly on AC, including fans, refrigerators, air conditioners, induction heaters, coil heaters, etc. These devices almost all contain parts like an inductor and a capacitor.

AC Generator:

Alternating emf  : An alternating emf that varies sinusoidally with time is given by e = e₀ sin ωt, where e₀ is the maximum value of the emf, called the peak value, and ω is the angular frequency of the emf.

ω = 2πf = 2π/T

where f is the frequency of the emf, expressed in Hz, and T is the periodic time of the emf, expressed in second.

From the graph it can be seen that the direction of the emf is reversed after every half revolution of the coil. This type of emf is called the alternating emf and the corresponding current is called alternating current.

Average and RMS values:

Alternating voltages and current go through all values between zero and the peak value in one cycle. Peak value (i0) : Peak value of an alternating current (or emf) is the maximum value of the current (or emf) in either direction.

Remember : AC shock is attractive, while DC shock is repulsive, so 220V AC is more dangerous than 220V DC. Also 220 V AC has a peak value E0 = \(±\sqrt{2}e_{rms}\)  = ± 1.414 × 220 V = ± 311 V but DC has fixed value of 220 V only.

 Phasors:

Phasor :

A phasor is a rotating vector that represents a quantity varying sinusoidally with time.

Different Types of AC Circuits:

(a) AC voltage applied to a resistor:

Below Fig. shows an alternating emf e = e₀ sin ωt applied to a resistor of resistance R.

e₀ is the peak value and co is the angular frequency of the emf. The instantaneous current through the resistor is i = i₀ sin ωt, where i₀ is the peak value of the current.

Here, i and e are always in phase.

For ωt = 0, sin ωt = 0, e=0, i=0;

for ωt = π/2, sin ωt = 1, e = e₀, I = i₀;

for ωt = π, sin ωt =0, e = 0, I = 0;

for ωt =3π/2, sin ωt = —1, e = — e₀, i= — i₀;

for ωt =2π, sin ωt = 0, e = 0, i = 0.

Below Figure shows variation of e and i with ωt.

Phasor diagram:

In the AC circuit containing R only, current and voltage are in the same phase, hence both phasors for i and for e are in the same direction making an angle ωt with OX. Their projections on vertical axis give their instantaneous values.

The phase angle between alternating current and alternating voltage through R is zero as shown in below Fig.

(b) AC voltage applied to an Inductor:

In an AC circuit containing a pure inductor, the voltage is ahead of current by π/2 rad in phase :

Variation of the emf and current with ωt :

Phasor diagram:

The phasor representing peak emf e₀ makes an angle ωt in anticlockwise direction from horizontal axis. As current lags behind the voltage by π/2, so the phasor representing i₀ is turned clockwise with the direction of e₀ as shown in Fig.

In this case, e = e₀ sin ωt and i = \(i_0sin(ωt-\frac{π}{2})\)  where i₀ = \(\frac{e_0}{ωL}\) and L is the inductance of the inductor.

Behaviour of pure inductor :

Inductive reactance = 2πfL.

• If the frequency (f) of the applied emf is very high, the inductive reactance (for reasonable value of inductance L) will be very high. Hence, the current through the inductor will be very low (for reasonable value of peak emf). Hence, it will practically block AC.
• For very low frequency (f), 2πfL is low and hence the inductor will behave as a good conductor.

(c) AC voltage applied to a capacitor:

Let us consider a capacitor with capacitance C connected to an AC source with an emf having instantaneous value e = e₀ sin ωt. This is shown in Fig.

The plates of the capacitor get charged due to the applied voltage. As the alternating voltage is reversed in each half cycle, the capacitor is alternately charged and discharged. If q is the charge on the capacitor, the corresponding potential difference across the plates of the capacitor is V = q/C, ∴ q = CV.

q and V are functions of time, with V = e = e₀ sin ωt. The instantaneous current in the circuit is

i = dq/dt = \(\frac{d}{dt}(CV)\) = \(C\frac{dV}{dt}\)  = \(C\frac{d}{dt}(e_0\,sinωt)\) = ωCe₀ cos ωt

∴ i = \(\frac{e_0}{(1/ωC)}sin(ωt+\frac{π}{2})\) = \(i_0\,sin(ωt+\frac{π}{2})\)

Where i₀ = \(\frac{e_0}{(1/ωC)}\) is the peak value of the current,

From Eq. e = e₀ sin ωt and Eq. \(i=i_0\,sin(ωt+\frac{π}{2})\) we find that in an AC circuit containing a capacitor only, the alternating current i leads the alternating emf e by phase angle of π/2 radian as shown in above graph.

Phasor diagram:

The phasor representing peak emf (e₀) makes an angle ωt in anticlockwise direction with respect to horizontal axis. As current leads the voltage by 90°, the phasor representing i₀ current is turned 90° anticlockwise with respect to the phasor representing emf e₀. The projections of these phasors on the vertical axis gives instantaneous values of e and i.

Behaviour of pure (an ideal) capacitor :

• If the frequency (f) of the applied emf is very low, the capacitive reactance (for reasonable value of capacitance C) will be very high and hence the current through the circuit will be very low (for reasonable value of peak emf).
• For very high frequency, f, Xc is very small. Hence, for very high frequency AC supply, a capacitor behaves like a pure conductor.

(d) AC circuit containing resistance inductance and capacitance in series (LCR circuit):

Now let us consider the total opposition offered by a resistor, pure inductor and capacitor connected in series with the alternating source of emf as shown in Fig.

Let a pure resistor R, a pure inductance L and an ideal capacitor of capacitance C be connected in series to a source of alternative emf. As R, L and C are in series, the current at any instant through the three elements has the same amplitude and phase. Let it be represented by i = i₀ sin ωt.

The voltage across the resistor, e_R = Ri, is in phase with the current. The voltage across the inductor, e_L = X_Li, leads the current by π/2 rad and that across the capacitor, e_c = X_Ci, lags behind the current by π/2 rad. This is shown in the phasor Diagram (Fig.).

From this figure, e₀² = e_R² + (e_L − e_C)²

= R²i₀² + (X_Li₀ − X_Ci₀)² = i₀² [R² + (X_L − X_C)²]

∴ e₀ = \(i_0\sqrt{R^2+(X_L-X_C)^2}\) = i₀Z, Where

Z = \(\frac{e_0}{i_0}\) = \(\sqrt{R^2+(X_L-X_C)^2}\)  is the effective resistance of the circuit. It is called the impedance.

Term impedance :

In an AC circuit containing resistance and inductance and/or capacitance, the effective resistance offered by the circuit to the flow of current is called impedance. It is denoted by Z.

For an LCR series circuit,

Z = \(\sqrt{R^2+(ωL-\frac{1}{ωC})^2}\)

ω = 2πf is the angular frequency and f is the frequency of AC.

Here, in the absence of a capacitor,

Z = \(\sqrt{R^2+ω^2l^2}\), and in the absence of an inductor,

Z = \(\sqrt{R^2+\frac{1}{ω^2C^2}}\)

Q. An alternating emf is applied to an LR circuit. Assuming the expression for the current, obtain the expressions for the applied emf and the effective resistance of the circuit. Assume the inductor and resistor to be ideal. Draw the phasor diagram showing the emf and current.