## 1. AC Voltage to resistor and circuit

Chapter 7: Alternating Current

AC VOLTAGE APPLIED TO A RESISTOR

​​Alternating Currents are used almost as a standard by electricity distribution companies. In India, 50 Hz Alternating Current is used for domestic and industrial power supply. Many of our devices are in fact nothing but resistances. These resistances cause some voltage drop but since the voltage this time is alternating, these voltage drops are dealt with differently

AC VOLTAGE APPLIED TO AN INDUCTOR

AC voltage is applied to the inductor. In order to find out the equation, we will consider the circuit as shown in the figure below. we have an inductor and an AC voltage V, which is represented by the symbol ~. The voltage produces a potential difference across its terminals that varies using a sinusoidal equation. The difference that is, the AC voltage thus can be given as,

From the equation, we deduce that vis used to signify the amplitude for the oscillating potential to denotes the differences. The angular frequency is given by ω. The current can be calculated by using the Kirchhoff’s loop rule. The equation which forms is as under,

Using the above equation in the given circuit,

The value of current as given by,

Therefore, the integration constant is zero.

Here, the amplitude of the current is given by

The quantity ωL can be said to be equivalent to the resistance of this device and is termed as the inductive resistance. We denote the inductive resistance of the device as XL.

Thus, we can say that the amplitude of current in this circuit is given as

AC Voltage Applied to a Capacitor
An AC voltage source is connected to a capacitor. The expression for the voltage from the voltage source is given by v = vmsin(ωt). A capacitor is an electrical device that stores electrical energy. It is a passive electronic component with two terminals. The effect of the capacitor is known as capacitance. A capacitor when connected to a voltage source draws current from the source so as to charge itself. Once the capacitor is charged, the potential at its plates becomes equal to the potential at the battery. At this point, the current stops flowing into the capacitor. This is called the charging of the capacitor.
At a particular time “t”, denotes the charge on the capacitor by “q”. The instantaneous voltage across the capacitor is given by,

Using the kirchhoff’s rule,

Since the current is continuously changing, to find the current. Derivative of the charge is required,

Differentiating the given equation,

i = vmωC cos(ωt)

Rearranging the above equation,

i = imsin(ωt + π/2)

Here, im = vmωC. It is the amplitude of the oscillating current. It can also be re-written as,

This equation when compared to the ohm’s law gives 1/ωC as resistance. It is called capacitive reactance and it is denoted by XC

Now, the amplitude of the current becomes,

im =

AC VOLTAGE APPLIED TO A SERIES LCR CIRCUIT
we will follow the analytical analysis of the circuit.
Analytical solution
As i

, we can write

Hence, writing the voltage equation in terms of the charge q through the circuit, we can write,

The above equation can be considered analogous to the equation of a forced, damped oscillator. In order to solve the equation, we assume a solution given by,

So,

And

Substituting these values in the voltage equation, we can write,

Here, we have substituted the value of Xc and XL by Xc = 1/ωC and XL = ω L.

As we know,

hence, substituting this value in the above equation, we get,

Now, let

So we can say,

Now, comparing the two sides of the equation, we can write,

And,

Hence, the equation for current in the circuit can be given as,

LC OSCILLATION
The Difference between the Direct and Alternating current is that the direct current (DC), travels only in one direction while the alternating current (AC) is an electric current that alternates direction on occasion and alters its amplitude continuously over time.In LC Oscillations, the charge equation is as follows:
q = qm cos(ωt)
We derive the present equation by differentiating this equation:
i = dq/dt
i = –qm ωsin(ωt)
The formula for calculating the energy stored in a capacitor is:
U= q2/2C
Substituting the equation for a given time interval t;
U= qm2/2C × (ωt)
The formula for calculating the energy stored in an inductor is:
U= 1/2 L i2
Substituting the capacitor’s equation for the same amount of time;
U= 1/2 L qmω(ωt)
Since the angular frequency, ω=1/ √LC
U= qm2/2C × (ωt)
As a result, the LC Oscillations’ total energy will be;
U = U+ UC
U = qm2/2C × (ωt)+qm2/2C × (ωt)
U = qm/ 2C

Transformer
There are usually two coils primary coil and secondary coil on the transformer core. The core laminations are joined in the form of strips. The two coils have high mutual inductance. When an alternating current pass through the primary coil it creates a varying magnetic flux. As per faraday’s law of electromagnetic induction, this change in magnetic flux induces an emf (electromotive force) in the secondary coil which is linked to the core having a primary coil. This is mutual induction.

Overall, a transformer carries the below operations:

1. Transfer of electrical energy from circuit to another
2. Transfer of electrical power through electromagnetic induction
3. Electric power transfer without any change in frequency
4. Two circuits are linked with mutual induction

1. Core

The core acts as a support to the winding in the transformer. It also provides a low reluctance path to the flow of magnetic flux. The winding is wound on the core as shown in the picture. It is made up of a laminated soft iron core in order to reduce the losses in a transformer. The factors such as operating voltage, current, power etc decide core composition. The core diameter is directly proportional to copper losses and inversely proportional to iron losses.

2. Windings

Windings are the set of copper wires wound over the transformer core. Copper wires are used due to:

• The high conductivity of copper minimizes the loss in a transformer because when the conductivity increases, resistance to current flow decreases.
• The high ductility of copper is the property of metals that allows it to be made into very thin wires.

There are mainly two types of windings. Primary windings and secondary windings.

• Primary winding: The set of turns of windings to which supply current is fed.
• Secondary winding: The set of turns of winding from which output is taken.

The primary and secondary windings are insulated from each other using insulation coating agents.

3. Insulation Agents

Insulation is necessary for transformers to separate windings from each other and to avoid short circuit. This facilitates mutual induction. Insulation agents have an influence on the durability and the stability of a transformer.

Following are used as an insulation medium in a transformer:

• Insulating oil
• Insulating tape
• Insulating paper
• Wood-based lamination

Ideal Transformer

The ideal transformer has no losses. There is no magnetic leakage flux, ohmic resistance in its windings and no iron loss in the core.

Vs = (Ns/ Np) Vp

Power at the input end is same as the power at the output end.

Therefore Pintput = Poutput

IpVp =  IsVs

## 1. AC Voltage to resistor and circuit

Introduction:

In the discussion so far we have used direct current (DC), the magnitude and the direction of DC remain the same and do not change with time. In this chapter, we will discuss the Alternating current (AC). In Alternating currents and voltages the magnitude as well as direction change periodically with time.

Most of our electrical appliances today work on AC because electrical energy produced by power companies is transmitted and distributed as Alternating current. This is because electrical energy can be transmitted more economically over long distances in the form of AC. And It can be easily and efficiently converted from one voltage to another by the means of transformers.

Alternating current

An alternating current is that current whose magnitude changes continuously with time and direction reverses periodically.

It is in contrast to a direct current whose magnitude and direction do not change with time.

• Alternating emf is given by E= Eosin(wt)

By ohm’s law  current I= E/R  = Eosin(wt)/ R= (Eo/R)*sin(wt)

Therefore alternating current of resistance is  I= Iosinwt

Where Io=Eo/R   which is called peak current or amplitude of AC.

• Time period of AC is the time taken by the alternating current to complete one cycle of its variation is called time period

T = 2Π/w

• Frequency of AC current is the number of complete cycles per second.

f=w/(2Π)

• The average value of AC over one complete cycle is zero. Therefore we defined the average value of a.c over half a cycle.

Average Value of AC

It is defined as the value of a direct current which sends the same charge in a circuit in the same time as it is sent by the given alternating current in its half-time period. It is denoted by Iav

Iav= 2Io/Π= 0.637 Io

Similarly, average value of Alternating EMF is given by

Eav=(2/Π)*Eo= 0.637 Eo

Root mean square(RMS) value of AC

It is defined as that value of direct current that produces the same heating effect in a given resistor as is produced by the given alternating current when passed for the same time. It is denoted by Irms.

Irms =(Io/2 ) =0.707*Io

Similarly, the root mean square value of Alternating emf is given by

Erms =Eo/2= 0.707 *Io

AC voltage applied to a Resistor

The figure shows a resistor connected to source E of ac voltage. We consider a source that produces sinusoidally varying potential differences across its terminal.

The potential difference is called ac voltage is given by

V=Vm( sinwt) Where Vm is the amplitude of oscillating potential difference and ‘w’ is its angular frequency.

By kirchoff’s voltage rule   Vm sin wt=IR

• So current in the circuit I=(Vm sin wt)/R=Vm/R sin wt=Im sin wt

Where Im= Vm/R  = peak current or current amplitude.

• Ohm’s law works equally well for both ac and dc voltages
• We can conclude that V and I both reach zero, minimum and maximum at the same time. Clearly voltage and current are in phase with each other in a resistor connected with ac source.

Representation of AC current and voltage by rotating vector- Phasors

We have seen that current through a resistor is the phase with the ac voltage. But this is not in the case of the inductor and capacitor or a combination of these circuits.

In order to show the phase relationship between V and I in various ac circuits, we use the representation of the rotating vectors called Phasors.

A phasor is a vector that rotates about the origin with angular speed ‘w’

Both cosine and sine functions can be represented by the phasor diagram

AC voltage applied to Inductor

Consider a circuit having a pure inductor of inductance L  connected with an alternating voltage source.

We have assumed that the inductor has zero resistance.

When an AC source that produces variable current and hence varying magnetic field, is connected with the inductor. Then according to the principle of electromagnetic induction, a back emf is produced in the coil.

When we apply kirchhoff's Voltage law in the circuit we will get

Applied emf  (e)+ back emf (-e’) =0

e+ (-e')=0

From the expression of applied alternating voltage and alternating current, it is clear that current lags behind the voltage by 90 degrees in the purely inductive circuit.

AC circuit having capacitor only

Consider a circuit having a capacitor of capacitance C connected with an alternating voltage source.

The alternating voltage is given by V=Vo sin(wt)

Consider ‘q’ to be the instantaneous charge in the circuit.

The emf across the capacitor will be  V= q/C then according to kirchhoff’s voltage law we have      V-qC= Vo sin(wt)-q/c=0

So we can write the q=CVo sin(wt)

Now we know that current i=dq/dt

Where   io= current amplitude in the capacitive circuit

Xc=(1/wcis the called capacitive reactance

Capacitive reactance is the resistance offered by the capacitor to the flow of alternating current in the circuit

From the expressions of applied alternating voltage and alternating current, we can conclude that current leads the voltage by 90 degrees in the pure capacitive circuit.

Below is the phasor diagram and the wave diagram for a purely capacitive circuit.

Average power for purely capacitive circuit

because  < sin 2wt> =0, an average value of sine over a complete cycle is zero. Hence capacitors do not consume any power from the circuit.

Now before moving to the Series LCR circuit Let’s first recap all we have learned about purely resistive, inductive and capacitive circuits.

Purely resistive circuit

Purely Inductive circuit

Purely capacitive circuit

AC Voltage is applied to a series LCR circuit.

Consider a circuit having a resistor of resistance R, a capacitor of capacitance C and an inductor of inductance L connected in series with an alternating voltage source.

The voltage of the alternating source V=Vo sin(wt)

If ‘q’ is the charge on the capacitor and ‘I’ be the current in the circuit at time ‘t’, then from Kirchoff’s law we have

L di/dt  + IR +  q/C= Vo sin(wt)

When current passes through the series LCR circuit, the voltage of the resistor VR is in phase with the current, while it is out of phase in the case of the inductor and capacitor as shown in the diagram.

• VR is in phase with the current I in the circuit.
• VL leads the current I by 90 degrees.
• VC lags the current I by 90 degrees.

We can represent the voltages across R, C and L and current with the help of phasor diagrams.

The resultant voltage of R, C and L will give the applied Alternating current

when VL > VC

Alternating current in series LCR circuit I=Io sin(wt-ϕ),

Where Io=Vo/Z   is the amplitude of the current in the series LCR circuit.

And ϕ = phase difference between the applied alternating voltage V and alternating current ‘I’ in series LCR circuit and when VL > VC is given by -

In the figure given above Current ‘I’ and alternating applied voltage V along with voltage across R, C and L are shown

When  Vc > VL  then I= Io sin(wt+ϕ) and ϕ=tan^(-1) ((Xc-XL)/R)

Resonance

In series, LCR resonance occurs when the magnitude of inductive reactance and capacitive reactance becomes equal and they are 180 degrees out of phase with each other and hence cancel each other.

Xc = XL   ;    1/wc= wl  ;   w^2=1/(LC)

• At the resonance impedance of the series, LCR is minimum.

Z= R    as  (Xc= XL)

• Current in the circuit is maximum as impedance is minimum

I= E/R

• Series LCR circuits behave like a resistive circuit at resonance as capacitive reactance and inductive reactance cancel each other.
• Current and voltage are in phase with each other in a series LCR circuit at resonance

• Resonant angular frequency is inversely proportional to square root of product of inductance and capacitance.​​​​​​

Wo =  1/LC

• Frequency of oscillation at resonance fr is given by

fr=1/(2ΠLC)

Quality factor: Quality factor is the ratio of the voltage across the capacitor or the inductor at resonance to the applied voltage. It is dimensionless and has no units.

Bandwidth of the

• It signifies the sharpness of the peak at resonance. More the value of Q will be sharper will be the resonance curve.​​​​​​​

• As Quality factor Q is inversely proportional to R, smaller value of R means the greater value of Q and hence the resonance curve will be sharper

• Bandwidth( BW) of series LCR circuits in resonance: The bandwidth of a resonant circuit is defined as the total number of cycles below and above the resonant frequency for which the current is equal to or greater than 70.7% of its resonant value.

• Larger be the Q value of the series LCR circuit, smaller be the bandwidth of the circuit.

Power in AC circuit

Power factor: The power factor is an indicator of efficient utilization of power. In an AC circuit, PF is defined as the ratio of real power flowing to the load to the apparent power in the circuit and is a dimensionless number.

Power in series LCR circuit

In pure inductive circuit and pure capacitive circuit  ϕ=90

So , Pav= Vrms*Irms*cos 90= 0

Therefore current in the inductive and capacitive circuit is called wattless current as there is no power loss across the inductor and capacitor.

LC oscillation

In this discussion we have a capacitor of capacitance C which is fully charged, then we connect an inductor of inductance L in the circuit as shown.

In this circuit, there will be oscillation between charge and current. We can visualize this as an oscillation of electrical energy and magnetic energy in the circuit such that total energy of the circuit remains the same.

• Suppose initially at t=0, the capacitor is fully charged with charge Qand current in the circuit i=0, so all the energy is of electrical nature.   q(t=0)=Q0    i(t=0)=0
• At t>0, the capacitor begins to discharge and current starts rising in the circuit, therefore electrical energy of the capacitor decreases and magnetic energy of the inductor increases by the same amount. Energy is transferred from capacitor to inductor.
• At t= T/4, the capacitor is completely discharged so electrical energy of the capacitor becomes zero and magnetic energy of the inductor becomes maximum and current flows in anticlockwise direction as shown in the figure below.
• When the capacitor is totally discharged and current is now beginning to fall and energy is being transferred from inductor to capacitor and the capacitor is being charged but now in the opposite direction.
• At time t= T/2, the capacitor is now fully charged and current in the circuit is zero.  All of the energy is in electrical form and magnetic energy is zero as the capacitor is fully charged again and current in the circuit is zero.
• After this instant, charge in the capacitor starts to fall and current is now in the clockwise direction as shown. Energy is being transferred from capacitor to inductor. Electrical energy of the capacitor starts to decrease and the magnetic energy of the inductor does increase.
•  At t=3T/4, again the capacitor is now discharged and the current in the circuit is maximum. Electrical energy of the capacitor is zero and the magnetic energy is maximum.​​​​​​​

Transformer: A transformer is a device that changes the voltage of the AC source. A transformer consists of two coils, primary and secondary. Input alternating voltage is given to the primary circuit. The number of turns and the voltage and current in primary and secondary coils is given below. Here we have considered an ideal transformer in which there is no energy loss.

Vp/Vs=Np/Ns=Is/Ip

There are two types of transformers: Step down and step-up transformers.

In a step-down transformer, the number of turns in the primary is more than the number of turns (Np> Ns) in the secondary coil. So

Here  Np > Ns      so , Vs < Vp and Is > Ip.

In the step-up transformer, the number of turns in the secondary is more than the number of turns in the primary.

Here  Ns > Np   so , Vs > Vp and Ip > Is.

Working principle of transformer.

When an alternating voltage is applied to the primary, the resulting current produces an alternating magnetic flux which links the secondary and induces an emf in it. The value of emf depends on the number of turns in the secondary.

In the Ideal transformer we have assumed that there is no energy loss in the process. But In actual transformers, there is small energy loss due to the following reasons.

1. Flux leakage: There is always some flux leakage. Not all the flux due to primary passes through the secondary due to poor design of the core or air gaps between the windings.
2. Resistance of the windings: The wires used for winding also have some resistance and some energy is lost in heating of the wire.
3. Eddy currents:  Alternating magnetic flux induces eddy current in the core and causes heating.
4. Hysteresis: the magnetization of the core is repeatedly reversed by the alternating magnetic field so there is some hysteresis loss also.

A fun thing to try

AC simulation

Click on the link above and you could actually make your own AC circuit using the desired value of R, C and L with any combination. We can see the direction of electron current and conventional current. Also, we can use an ammeter and voltmeter to track the values of the voltages and current. Make as many circuits as you like and learn from them.

Happy learning !!