Let's look at the circuit as shown below. We have a resistor and an alternating voltage V, represented by the symbol ~, which produces a sinusoidally varying potential difference between its terminals. Here, the potential difference, or alternating voltage, can be represented as v= vm sin ωt
Here vm is the amplitude of the vibrational potential difference and the angular frequency is given by ω.
An alternating voltage is applied to the resistor
The current through a resistor due to a voltage source can be calculated using Kirchhoff's circuit rule as below
Using this equation, we can write,
Here, R is the resistance of the given resistor,
so we can write, i = imsinωt
From Ohm's law we can write,
Because it works equally well with AC and DC voltage.
We have seen that the voltage across a resistor and the current through it are both sinusoidal quantities and are represented in the figure above.
Both quantities are in phase with each is given as follows:
Here the quantities im and R are constant, so the above equation is calculated as:
Using trigonometry, we have
sin2ωt=1/2(1-cos2ωt) , and
we can write
And because the integration of cos 2ωt over the whole cycle is zero.
So we can write cos2ωt = 0 ,
we can write, sin2ωt=1/2
So we can also write, P = 1/2im2R
It is important to note that alternating current can also be expressed as direct current if we describe the current as rms current or square current, which is obtained as:
Alternating Current supplied to the Inductor:
An inductor is a passive device that stores energy in a magnetic field when an electric current passes through it. An inductor can prevent or block the flow of alternating current through it. The inductor gains or losses charge. The flux through the inductor changes to match the current through it. The voltage across an inductor can be measured as the magnitude of the electromotive force (EMF) generated when the current changes. Example: Suppose an inductor produces an EMF of 1 volt when current passes through the coil. We can say that the inductance of the coil is 1 Henry and it changes at a rate of 1 amp per second.
AC voltage and Inductor
Here we learn more about the operation of an electric circuit when an AC voltage is applied to the coil. To find out the equation, let's look at the circuit as shown in the figure below. According to the figure, we have a coil and an alternating voltage V, indicated by the symbol ~. A voltage creates a potential difference across its terminals that varies according to a sinusoidal equation. The difference, or alternating voltage, can therefore be represented as follows: v=vmsinωt From the equation, we conclude that vm means the amplitude of the vibrational potential, which means differences. The angular frequency is given by ω. The current can be calculated using Kirchhoff's circuit rule. The resulting equation is as follows, ∑V(t)=0
AC voltage applied to the inductor Using the above equation in the given circuit, we can write,
Here the second term is the self-induced Faraday emf of the inductor and L is the term given to the self-inductance of the coil.
The above two equations, viz. the inductor voltage equation and the equation derived from Kirchhoff's law for the given circuit, gives the following equation:
Here from the above equation we can see that the equation of change of current as a function of time is sinusoidal in nature and in phase with the source voltage and its amplitude is given by
We can calculate the current through the inductor by integrating the above quantity over time as follows:
The above equation leads us to the present value which gives
The constant of integration has a measure of current and is independent of time. Since the source has an EMF that oscillates symmetrically around zero, the current it stores also oscillates symmetrically around zero, so there is no constant or time-independent component. Therefore, the integration constant is zero.
Here the current amplitude is obtained
It can be said that a large ωL corresponds to the resistance of this device and is called inductive resistance. We mark the inductive resistance of the device with XL.
XL = ωL
So we can say that the current amplitude in this circuit is given as
AC Voltage Source Applied Across a Capacitor
We know that a capacitor is a passive electronic device with two terminals. It has the capacity to store electrical energy in an electrical field. In a DC circuit, when a capacitor is connected to a voltage source, the current will flow for the short time required to charge the capacitor
AC Voltage Source Applied Across a Capacitor
Let us consider the electric circuit shown below. We have a capacitor and an AC voltage V, represented by the symbol ~, that produces a potential difference across its terminals that varies sinusoidally. Here, the potential difference or the AC voltage can be given as,
Here, vm is the amplitude of the oscillating potential difference and the angular frequency is given by ω. The current through the resistor due to the present voltage source can be calculated using the Kirchhoff’s loop rule, as under,
For the given capacitor we can write,
According to Kirchhoff’s rule, we can write from the above circuit,
The current through the circuit can be calculated using the relation,
[Using the relation, cos ωt=sin(ωt+π/2)]
Here the amplitude of the current can be written as,
or else we can write it as,
Here, we can see that the term 1/ωC can be said to be equivalent to the resistance of this device and is termed as the capacitive reactance. We denote the capacitive reactance of the device as XC.
And thus, we can say that the amplitude of the current in this circuit is given as,
In the above equations, the dimension of the capacitive In the above equations, the dimension of the capacitive reactance can be seen to be the same as that of resistance, and also, the SI unit of capacitive reactance is given as ohm. The capacitive reactance restricts the passage of current in a purely capacitive circuit in the same way as resistance hinders the passage of current in a purely resistive circuit.
Here we say, that the capacitive reactance is inversely proportional to the frequency and the capacitance. We also see from the above equations that the current in a capacitive circuit is π/2 ahead of the voltage across the capacitor.
The instantaneous power supplied to the capacitor can be given in terms of the current passing through the capacitor as
Here, the average power supplied over a complete cycle can be given as
AC Voltage Applied to Series LCR Circuit: Analytical Solution
An LCR circuit is also a resonant or tuned circuit. It consists of an inductor-L, capacitor-C, and resistor-R connected in either series or parallel.
Derivation of AC Voltage Applied to Series LCR Circuit:
Consider the circuit shown above. Here, we have an inductor, a resistor, and a capacitor connected through a series connection across an AC voltage source given by V. Here, the voltage is sinusoidal in nature and is given by the equation,
Here, vm is the amplitude of the voltage, and ω is the frequency.
If q is the charge on the capacitor and i the current at time t, we have, from Kirchhoff’s loop rule:
Here, q is the charge held by the capacitor, I is the current passing through the circuit, R is the resistance of the resistor and C is the capacitance of the capacitor. To determine the instantaneous current or the phase of the relationship, we will follow the analytical analysis of the circuit.
, we can write
Hence, by 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. To solve the equation, we assume a solution given by,
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,
So we can say,
Now, comparing the two sides of the equation, we can write,
Hence, the equation for current in the circuit can be given as,
A.C. Circuits Containing Resistance and Inductance in Series:-
A.C circuit contains only inductance. In practice, inductance is always associated with a certain amount of resistance. Some devices, such as choking coil or choke coil have small resistance and comparatively large inductance. But no coil has inductance without resistance.
Resistance and inductance do not exist as two separate items in the circuit. They remain associated together. But for the sake of study, they are shown separately. Since the same current flows through resistance and inductance at every instant and since the sum of voltage drops in resistance and in inductance gives the total applied voltage across the coil, these two items are considered to be connected in series. Thus, an inductive coil may be considered to be consisting of two a.c. circuits connected in series, one containing resistance only and the other inductance only.
An a.c. circuit containing a resistance of R ohm in series with an inductance of L henry is shown in fig below
I = r.m.s. value of current flowing through the circuit,
VR = r.m.s. voltage dropped across the resistance
= IR volt in phase with I,
VL = r.m.s. voltage dropped across the inductance
= IXL volt leading I by 90°, and
V = r.m.s. voltage applied across the whole circuit and is the vector sum of VR and VL.
The vector diagram of the circuit is shown in fig. (c). from this diagram we find that the current lags behind the applied voltage by an angle θ (or the applied voltage leads the circuit current by an angle θ) such that tan θ = XL/R. This is also shown in the wave diagram in fig. (b).
In the vector diagram VR, VL and V have formed a right-angled triangle in which VR represents the adjacent side, VL represents the opposite side or the perpendicular and V represents the hypotenuse.
√R2 + XL2 is the opposition to the flow of current offered jointly by the resistance and reactance of the circuit. This opposition is known as Impedance of the circuit. It is usually denoted by Z and is expressed in ohm.
Power absorbed by the circuit:
Since inductance does not absorb any power, in the circuit only resistance absorbs power. Again, power absorbed by resistance is equal to the product of voltage across resistance and the current flowing through it. Therefore the power absorbed by the whole circuit is given by-
P = VRI watt.
Also VR = IR volt,
P = IR x I = I2R watt.
From the vector diagram,
cosθ = VR/V
or, VR = Vcos θ volt.
... P = Vcos θ x I = VI cos θ watt.
cos θ is the power factor of the circuit. In this circuit since the current lags behind the applied voltage, the power factor of the circuit is a lagging power factor, and the phase angle θ is called a lagging angle.
Power factor cos θ = VR/V = IR/IZ = R/Z lagging.
A. C. Circuits Containing Resistance and Capacitance:
Let an a.c. circuit contain a resistance of R ohm in series with a capacitor of capacitance C farad. This circuit is shown in fig.
A. C. Circuits Containing Resistance and Capacitance in Series
Let, I = r.m.s. value of current flowing though the circuit,
VR = r.m.s. voltage across the resistance
= IR is phase with I,
Vc = r.m.s. voltage across the capacitor
= IXC lagging I by 90°,
And v = r.m.s. voltage applied across the whole circuit and is the vector sum of VR and VC.
The vector diagram of the circuit is shown in fig. (c). In this diagram, the current leads the applied voltage (or the voltage lags behind the current) by an angle θ, such that,
tan θ = XC/R
This is also shown in the wave diagram in fig. (b).
In the vector diagram VR, VC and V have formed a right-angled triangle in which VR represents the adjacent side or the base, VC represents the opposite side or the perpendicular and V represents the hypotenuse.
is the opposition to the flow of current offered jointly by the resistance and the capacitive reactance of the circuit and is called the Impedance of the circuit. It is denoted by Z and is expressed in ohm. Thus,
Impedance of the Circuit,
Current flowing through the circuit,
I = V/Z ampere, and
Voltage applied across the circuit,
V = IZ volt.
Power absorbed by the Circuit:
Since the average power absorbed by a capacitor is nil, power absorbed by the whole circuit is equal to power absorbed by the resistance alone. Again, voltage across the resistance is VR volt and the current flowing through it is I ampere. Therefore,
Power absorbed by the circuit = power absorbed by the resistance,
or P = VRI watt.
But VR = IR volt.
.. P = IR x I = I2R watt.
From the vector diagram,
cos θ = VR/V
or, VR = V cos θ volt.
Hence, P = Vcos θ x I= VI cos θ watt.
cos θ is the power factor of the circuit. Since the current leads the applied voltage, the phase angle θ is a leading angle and the power factor cos θ is a leading power factor.
From the vector diagram,
cos θ = VR/V = IR/IZ = R/Z leading.
Mean Value of Alternating Current:
Mean value of an alternating current is the total charge flown for one complete cycle divided by the time taken to complete the cycle i.e. time period T. The value of an alternating current changes with time. Also, its direction changes after every half cycle. Hence, for a full cycle, the average value is zero. We find the average value for current.
When we have a DC circuit, the current of the circuit is constant. Hence, there is no problem in specifying the magnitude of the current.
In an AC circuit, the current in the circuit is an alternating current. Therefore, its value continuously changes with respect to time. Therefore, we have the difficulty of specifying the magnitude of the current as it is not a constant value. Hence, we calculate the average value of the alternating current.
Mean value of an alternating current is the total charge flown for one complete cycle divided by the time taken to complete the cycle i.e. time period T.
The magnitude of an alternating current changes from time to time and its direction also reverses after every half cycle. This means that the current is positive in one-half cycle and it is negative in the other half cycle. Because of this, the net charge flown is zero.
Therefore, the mean value of an alternating current for a complete cycle is zero. But, of zero value as no meaning and not useful for calculations. As a result, we measure the mean value of an alternating current only for the positive half cycle.
Let derive an expression for the average value of sinusoidal alternating current over a positive half cycle. A sinusoidal alternating current is given as
im is the amplitude of the sine wave (maximum current), ωω is the angular frequency of the function and t is time.
Average current for half cycle is given as
, where T is the time period
Therefore, the mean value of the alternating current is
Other than the mean value, we use another value of the alternating current called root mean square value.
The root mean square value is the root of the mean of the square of the alternating current over one full cycle.
The root mean square (rms) value of a sinusoidal alternating current is equal to
Mean Value of Alternating Emf:
Peak and R.M.S Value of an Alternating Current / Voltage:
What Is an Alternating Current?
An alternating current is defined as a current whose magnitude changes with time and also reverses its direction periodically.
The general equation is given by,
Where, Io is termed as peak value of an alternating current.
Considering the above equation the current changes at any instantaneous time, if the current is passed through the electric circuit and it can be assumed to remain constant for any small time dt. As current is passed for a short time a small amount of charge is flown through the circuit in time dt and it is represented as:
If the current I is indicated as a sine function then,
At half the period of an alternating current, the amount of charge passed through the circuit at time T/2 is given by:
The mean value of the alternating current is given by,
On equating equation (1) and equation (2), we get:
RMS Value of Alternating Current
RMS stands for Root-Mean-Square of instantaneous current values. The RMS value of alternating current is given by direct current which flows through a resistance. The RMS value of AC is greater than the average value. The RMS value of the sine current wave can be determined by the area covered in a half-cycle. This is applicable to all the waves which include sinusoidal, non-sinusoidal, symmetrical, and asymmetrical. It is denoted by Irms or Iv.
RMS Value of AC Formula
Following is the formula of RMS value of AC:
RMS Value of AC Derivation:
The heat produced in a time T/2 is given by:
The rms value of AC is represented as:
By equating equation (3) and equation (4), we get:
These values are measured by an ammeter and voltmeter that are used in the circuit.
Generally, many circuits in electrical and electronics are made up of a combination of resistors, inductors and capacitors. If a circuit is made up of resistors and capacitors, then it is known as an RC circuit. Similarly, when the circuit is containing capacitors and inductors, then it is known as an LC circuit. Therefore, an LR circuit is a circuit which is made up of pure resistors and pure inductors. That means that the circuit is both resistive and inductor and is operated by a voltage source in series or by a current source in parallel. An LR Circuit is also known as an LR network or LR filter. When an inductor is connected in series with the resistor, there will be some changes happening in the circuit due to the presence of inductance. The inductor which is present in the circuit opposes the change in magnetic flux, thereby opposing the change in current flowing in the circuit. As we know, according to Faraday's law of magnetic induction, when the current is present in the circuit, there will be a formation of the magnetic field which causes the magnetic flux through the circuit.When there is any change in the flow of the current, the magnetic flux also changes. Now, an EMF is induced by the variation of the magnetic field around the inductor. Therefore, this EMF is induced by the variation of the own magnetic field of the inductor, so it is known as self-induced EMF. Let us learn how the current in the RL circuit flows and have a look at the LR circuit derivation in detail.
Growth of Current
An LR Circuit is analysed in three ways. The first one is the initial state, which is present at the instant of closing the switch or opening the switch in the circuit. The second one is the transient state, which appears at any instant after closing or opening the switch. The third one is steady-state, which appears after a long time after closing and opening the switch. Let’s start with the initial and steady states of an LR circuit.
Let us assume a circuit of EMF E has the inductance ’L’ and the resistance ‘R’, as shown in the figure.
The voltage drop across the inductor is VL and the voltage drop across the resistor is VR.
Initial Stage of an LR Circuit at t=0 .At t = 0, the inductor offers an infinite opposition to the current flow and hence there is no current flow in the circuit at the time of closing the switch. Due to high opposition to the current flow, the voltage is dropped entirely at the inductor and there is no voltage drop across the resistor. I.e., at t = 0, VR= 0 and VL = E.
At a certain point of time, say t = ∞∞, the current in the inductor does not vary with time after closing or opening the switch for a long period. We can see that the current has reached its maximum value and therefore the inductor does not offer any position to the current flow. So, the voltage drop across the inductor becomes zero and the entire voltage drops across a resistor. At t=∞, VR = E and VL = 0.
In this state, the voltage is dropped both across the resistor and the inductor. At any instant t = 0 and t = ∞ is taken for this state. We know that the voltage drop across the inductor is equal to the inductance multiplied by the rate of change in current across the inductor. I.e.,
And the voltage across the resistor is given by VR = IR. In the transient state, when the switch is closed gradually, the current starts increasing across the inductor. Due to the increase in the current, there will be a self-induced EMF in the inductor which opposes the change of the current in the circuit. Let us take an instant at t = t, the current flowing in the circuit is I as shown in the figure.
Transient State of LR Circuit at Time t=t
Let us apply Kirchhoff's voltage law in this circuit.
Taking the integration from t = 0 to t = t and I = 0 to I = I:
Here, E/R becomes the maximum current when there is no inductor opposing the current flow. And L/R is called the time constant of the LR circuit represented by τ\tau. It is the time for the current to reach 63% of the final current flowing in the circuit. Therefore, putting these values in the equation, we get the final current equation for the growth of the current in the circuit.
This is the instantaneous current at time t = t flowing through the circuit. Let us draw the graph between current and time and see how the current is increasing with time in the growth of the current state.
Graph between Current and Time in the Growth Stage:
Let us similarly derive the current equation in the decaying state of the current. Decay of Current .In the Decay of current, the source EMF is removed from the circuit. The current flowing in the circuit will be maximum at the time of this connection of the source. Let maximum current be I0 flowing through the circuit.
At t=0, the current flowing in the circuit is I0 and at t=t current flowing is I. By applying Kirchhoff's voltage law and using integration, we obtain:
Therefore, this will be the equation of current decay in the LR circuit. We derived the LR circuit formula for growth and decay of current in the RL circuit. This LR circuit derivation is very similar in both the cases of growth and decay of current in the LR circuit. Also here, L/R is known as the time constant and it is denoted by τ\tau. Let us solve a problem with this concept.
When the impedances or admittances of circuit elements cancel each other such that the current becomes maximum, electrical resonance is said to occur. It occurs in an electric circuit at a specific resonant frequency. The best example of electrical resonance is turning on a radio. When we turn the knob on the radio to tune in a station; we are changing the natural frequency of the receiver's electrical circuit to match the transmission frequency of the radio station.
Impedance is defined as the combination of resistance and reactance. It is basically resistance to the flow of electrons in an electrical circuit. Z is the symbol used to describe impedance. The unit of resistance is the ohm. Another way to define impedance with respect to a pointer is that it is the ratio of the cell voltage to the pointer current through the cell.
An LCR circuit is a circuit consisting of a resistor R, an inductor L, and a capacitor C. These three electrical components are connected either in series or in parallel. An LCR circuit is also known as an RLC circuit and this circuit takes its name from the symbols of the main electronic components used. Other names for an LCR circuit are resonant circuit and tuned circuit.
Derivation Of The Expression For The Impedance Of A Series LCR Circuit
In the below circuit diagram, let R, L and C be the resistance, inductance, and capacitance that is connected in series with an alternating current source.
Derivation of impedance of a series LCR circuit
The voltage applied across the LCR series circuit is given as:
v = vo sinωt
v is the instantaneous value
vo is the peak value
ω = 2πf
f is the frequency of alternating current
Let i be the instantaneous current at the time t such that the instantaneous voltage across R, L, and C are iR, iXL, and iXC, respectively. The vector sum of the voltage amplitudes across R, L, C and the amplitude vo of the applied voltage are equal.
The voltage amplitudes across R, L, and C are given as vR, vL, and vC, respectively. The current amplitude is given as Io.
Then, vR = IoR is in the phase with io.
The above equation leads io by 90:
The above equation lags behind io by 90
We can notice that the current in a pure resistor is in phase with the voltage while the current in a pure inductor and the voltage lag by 90 degrees. The current in a pure capacitor and the voltage are led by 90 degrees.
When vL> vC, the phase angle (Φ) between the voltage and the current is positive.
OAP is the right-angled triangle, from which we get,
Here, Z is the impedance of the circuit
The phase angle between v and i is given as:
This gives us:
Quality factor of resonance is a dimensionless parameter that describes how underdamped an oscillator or resonator is, and characterizes a resonator bandwidth relative to its center frequency.
X Quality factor in series LCR circuit
As quality factor is dimensionless, thus it has no SI unit.
Power factor of RLC Circuit
An alternating current circuits' power factor is the ratio of true power dissipation to apparent power dissipation.
The power factor of an RLC circuit indicates how close the circuit is to expending all of its power.
Power factor is also the resistance to impedance ratio of an LCR circuit.
The power factor of an LCR circuit is the resistance to the total impedance ratio of the circuit.
The magnitude of the phasor sum of the resistance, capacitive reactance, and inductive reactance is the total impedance.
Energy stored in the inductor:
An inductor stores energy in magnetic field form when an electric current is flowing through it.
The magnetic field starts to weaken and release energy as the current is gradually reduced, turning the inductor into a current generator.
The equation of magnetic energy, , stored in the inductor can be written as:
where is the current running through the wire, presuming we have an electrical circuit comprising a power source and a solenoid of inductance .
LC oscillations- The electric current and the charge on the capacitor in the circuit undergo electrical LC oscillations when a charged capacitor is connected to an inductor. The electrical energy stored in the capacitor is its initial charge which is named as qm.
It is represented by,
The inductor contains zero energy. Charge varies sinusoidally with respect to time.
When the switch is turned on, the current in the circuit starts to increase, and the charge on the capacitor keeps decreasing. The current induced in the circuit produces a magnetic field in the inductor. When the current increase to its maximum level Im, the magnetic energy in the circuit is represented as:
The magnetic field starts decreasing with time as there is no further change in current through the inductor. The current is induced in the circuit due to a decreasing magnetic field. With reverse polarization, the current starts charging the capacitor. This process is repeated again once the capacitor in the circuit is fully charged with regard to its previous states. Due to this reason, energy in the whole system oscillates between the capacitor and the inductor.
On applying Kirchhoff’s law to the circuit, we conclude that the charge oscillates with a natural frequency.
However, practically this whole process is not possible and is never achieved. The reason for the discrepancy is some loss of energy due to resistance in the circuit and also due to radiations in the form of electromagnetic waves.