Alternating Current Revision Notes

A quick revision of all the important concepts

ALTERNATING CURRENT

Alternating Current: An alternating current (a.c.) is a current which continuously, changes in magnitude and periodically reverses in direction.

i = I_{o} s i n ω t = I_{o} s i n (2 π / T) t

Here

I_{o}

is the peak value of a.c.
a.Current,

I = I_{o} s i n ω t

b. Angular frequency,

ω = 2 π n

(

n

is the frequency of a.c.)
c.

I = I_{o} s i n 2 π n t

Mean value of A.C or D.C. value of A.C.: Mean value of a.c. is that value of steady current which sends the same amount of charge, through a circuit, in same time as is done by a.c. in one half-cycle.

(I_{a v}) h a l f c y c l e = (\frac{2}{π}) I_{o}

Thus, mean value of alternating current is

2 / π

times (0.637 times) its peak value.

(V_{a v}) h a l f c y c l e = (\frac{2}{π}) V_{o}

Average value of A.C. over a complete cycle:

I_{a v} = 0

The average value of a.c. taken over the complete cycle of a.c.is zero.

Root mean square value of a.c. or virtual value of a.c
:R.m.s value of alternating current is defined as that value of steady current which produces same heating effect, in a resistance, in a certain time as is produced by the alternating current in same resistance in same time. The r.m.s value of a.c.is also called its virtual value.

I_{r m s} = I_{0} / 2

I_{r m s} = \frac{\int _{t_{1}}^{t_{2}} I ^{2} d t}{t _{2} - t _{1}}

Root mean square value of alternating current is

1 / 2

times (or 0.707 times) the peak value of current.
Similarly,

V_{r m s} = \frac{\int _{t_{1}}^{t_{2}} V ^{2} d t}{t _{2} - t _{1}} = V_{0} / 2

Here

V_{0}

is the peak value of e.m.f.

Form Factor: Ratio of rms value to average value.
i.e.Form Factor =

\frac{V _{r m s}}{V _{a v g}} = \frac{V _{o} / 2}{2 V _{o} / π} = \frac{π}{2 2}

Power consumed and supplied in an A.C circuit
Average power consumed in a cycle =

\frac{\int _{0}^{2 π / ω} P d t}{2 π ω} = \frac{V _{m} I _{m} c o s ϕ}{2} = \frac{V _{m}}{2} \times \frac{I _{m}}{2} c o s ϕ = V_{r m s} I_{r m s} c o s ϕ

Here

c o s ϕ

is the power factor.

SOME DEFINITIONS:
In AC circuits, the power factor is the ratio of the real power that is used to do work and the apparent power that is supplied to the circuit. The factor

c o s ϕ

is called Power factor.

Alternating current through pure L or pure C, which consumes no power for its maintenance in the circuit is called Idle current or wattless current.

I_{m} s i n ϕ

is called wattless current.

Impedance Z is defined as Z =

\frac{V _{m}}{I _{m}} = \frac{V _{r m s}}{I _{r m s}}

ω L

is called inductive reactance and is denoted by

X_{L}

\frac{1}{ω C}

is called capacitive reactance and is denoted by

X_{C}

.

PURELY RESISTIVE CIRCUIT:

I = \frac{V _{s}}{R} = \frac{V _{m} s i n ω t}{R} = I_{m} s i n ω t

I_{m} = \frac{V _{m}}{R}

I_{r m s} = \frac{V _{r m s}}{R}

V_{r m s} . I_{r m s} c o s ϕ = \frac{V _{r m s} ^{2}}{R}

PURELY CAPACITIVE CIRCUIT:

I = \frac{V _{m}}{1 / ω C} c o s ω t = \frac{V _{m}}{X _{C}} c o s ω t = I_{m} c o s ω t

X_{C} = \frac{1}{ω C}

and is called capacitive reactance.

PURELY INDUCTIVE CIRCUIT:

V_{L} = L (d I / d t) = L I_{0} ω c o s ω t

I = (V_{0} / L) ω s i n ω t

Here,

I_{0} = V_{0} / ω L

X_{L} = ω L

And the maximum current,

I_{0} = V_{0} / X_{L}

R-L circuit: RL Circuits are those circuits which are purely the combination of inductors and resistors. Current =

I = ϵ / R [1 - e^{- R t / L}]

Voltage =

V = ϵ e^{- R t / L}

L-C Circuit: An LC circuit is defined as an electrical circuit consisting of the passive circuit elements an inductor (L) and a capacitor (C) connected together. It is also called a resonant circuit, tank circuit, or tuned circuit.
Frequency =

f = \frac{1}{2 π L C}

Charge Variation =

q = q_{0} s i n (ω t + ψ)

Current =

I = q_{0} ω s i n (ω t + ψ)

Resonant angular frequency =

ω = 1 / L C

R-C Circuit: A resistor-capacitor circuit (RC circuit), is an electric circuit composed of resistors and capacitors. Charging of an RC circuit

R \frac{d Q}{d t} = \frac{ϵ C - q}{C}

where

ϵ

is the emf of the cell.
Discharging RC circuit

\frac{d Q}{d t} = - \int \frac{d t}{C R}

The relationship between a capacitors voltage and current define its capacitance and its power. We have

Q = C V .

Then

\frac{d Q}{d t} = C \frac{d V}{d t} \Rightarrow i_{c} = C \frac{d V _{c}}{d t}

\frac{1}{2} C V^{2} + \frac{1}{2} L i^{2} = c o n s t a n t = \frac{1}{2} C V_{0}^{2} = \frac{1}{2} L i_{0}^{2}

Series in L-C-R Circuit:
An LCR circuit, also known as a resonant circuit, tuned circuit, or an RLC circuit, is an electrical circuit consisting of an inductor (L), capacitor (C) and resistor (R) connected in series or parallel. The LCR circuit analysis can be understood better in terms of phasors. A phasor is a rotating quantity.

V = I Z

The modulus of impedance,

∣ Z ∣ = [R^{2} + (ω L - 1 / ω C)^{2}]

The potential difference lags the current by an angle,

ψ = t a n^{- 1} [ω L - 1 / ω C) / R]

Circuit elements with A.C:

Circuit elements	Amplitude relation	Circuit quantity	Phase of V
Resistor	$V_{o} = i_{o} R$	$R$	In phase with i
Capacitor	$V_{o} = i_{o} X_{C}$	$C$	Lags i by $9 0^{\circ}$
Inductor	$V_{o} = i_{o} X_{L}$	$X_{L} = w_{L}$	Leads i by $9 0^{\circ}$

Resonance:
A circuit in which inductance L, capacitance C and resistance R are connected in series, and the circuit admits maximum current corresponding to a given frequency of a.c., is called series resonance circuit.

Z = [R^{2} + (ω L - 1 / ω C)^{2}]

At very low frequencies, inductive reactance

X_{L} = ω L

is negligible, but capacitive reactance

(X_{C} = 1 / ω C)

is very high.
As frequency of alternating e.m.f. applied to the circuit is increased,

X_{L}

goes on increasing and

X_{C}

goes on decreasing. For a particular value of

ω

(

= ω_{r}

, say)

X_{L} = X_{C}

Power Factor of an A.C. Circuit: It is the ratio of true power upon apparent power i.e

Power factor = \frac{True power}{Apparent power} = P_{a v} / E_{r m s} I_{r m s} = c o s ϕ

Transformer: A transformer which increases the a.c. voltage is called a step-up transformer, A transformer which decreases the a.c. voltages is called a step-down transformer.
A transformer consists of a rectangular soft iron core made of laminated sheets, well insulated from one another, figure. Two coils

P_{1}

P_{2}

(the primary coil) and

S_{1}

S_{2}

(the secondary coil) are wound on the same core, but are well insulated from each other. Note that both the coils are also insulated from the core. The source of alternating e.m.f. (to be transformed) is connected to the primary coil

P_{1}

P_{2}

and a load resistance R is connected to the secondary coil

S_{1}

S_{2}

through an open switch

S

. Thus, there can be no current through the secondary coil so long as the switch is open.

\frac{E _{p}}{E _{s}} = \frac{N _{p}}{N _{s}} = \frac{I _{s}}{I _{p}}

Here,

E

= Emf ,

N

= No. of coils and

I

= electric current

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