Alternating Current (AC)
$\displaystyle \small \bullet$ AC is an electric current which changes its voltage and direction periodically.
$\displaystyle \small \bullet$ AC keeps changing with time and this change is sinusoidal.
$\displaystyle \small \bullet$ Property of AC is that its value increases from zero to peak value and then returns to zero, not only in positive half but also in negative half of the cycle.
$\displaystyle \small \bullet$ The voltage of Ac increases for duration 0 to T/4, and is highest at T/4.
$\displaystyle \small \bullet$ For duration T/4 to T/2, voltage reduces and attains zero level at T/2.
$\displaystyle \small \bullet$ For T/2 to 3T/4, voltage increases again and attains highest level at 3T/4.
$\displaystyle \small \bullet$ For 3T/4 to T, it reduces again and becomes zero at T.
Terms related to AC
Cycle
A complete sinusoidal change of alternating current’s voltage and direction is called a cycle. The upper half of the cycle is called positive and lower half is known as negative cycle.
Frequency
The number of complete cycles per second in AC is called frequency.
It is denoted as f and is inversely proportional to time period (when frequency increases, time period decreases and when frequency decreases time period increases).
Its unit is cycles per second or Hz.
$\displaystyle \small f=\frac{1}{Time\; Period(T)}$
Time period
The taken to complete one cycle in AC is called time period. It is denotes as T and its unit is second.
$\displaystyle \small Time\; Period=\frac{1}{Frequency(f)}$
Root Mean Square Value
It is the DC current when passed through a resistance for a given time produces the same amount of energy as the alternating current does in the same resistance in the same time. To find RMS value, its instant values are captures in a cycle at various instants. These values are squared, added and average is calculated.
$\displaystyle \small I_{rms}=\sqrt{\frac{I_{1}^{2}+I_{2}^{2}+I_{3}^{2}...I_{n}^{2}}{N}}$
$\displaystyle \small I_{rms}=0.707\times I_{max}$
Similarly,
$\displaystyle \small E_{rms}=0.707\times E_{max}$
Peak Value
The highest positive and negative value obtained in sinusoidal wave of AC signal is called peak value.
Peak to Peak Value
The distance between peak value of positive half cycle and peak value of negative half cycle in sinusoidal AC signal is called peak to peak value.
Instantaneous Value
Value observed at a time instant is sinusoidal AC signal is called instantaneous value.
$\displaystyle \small I=I_{0}\sin (\omega t+\phi )$
where,
$\displaystyle \small I_{0}$ = Peak value of current
Average Value
The average of instant current or voltage in half cycle of AC is known as average value of AC voltage or current.
$\displaystyle \small E_{average}=\frac{e_{1}+e_{2}+e_{3}+...+e_{n}}{n}$
$\displaystyle \small E_{average}=0.637E_{max}$
Form factor
The ratio of RMS value of AC and the average value is called form factor. Its denoted by K.
$\displaystyle \small K=\frac{RMS\; Value}{Average\; Value}=1.11$
Peak factor/ Amplitude factor
It is the ratio of highest value and RMs value of AC.
$\displaystyle \small Peak\;Factor=\frac{Highest\; Value}{RMS\; Value}=1.414$
Phase
The relative position of components of two ACs is called phase. Current and voltage are the two components. This relation gives information about which component is greater or lesser.
In-phase
When voltage and current in a phase attain their highest and lowest values in the same direction after beginning from zero at the same time, it is called in phase. Electric angle in alternating quantities should be zero.
Out-of-phase
When voltage and current in alternating quantities do not begin at same time and do not attain their highest and lowest values in the same direction at same time instant, it is called out of phase.
Phase difference
When alternating quantities reach their highest and lowest values from zero at different time intervals, then this difference in time is called phase difference. It is denoted by Ï•.
$\displaystyle \small e=OQ\sin \omega t$
$\displaystyle \small i=OP\sin (\omega t-\phi )$
Wavelength
Straight distance covered by a wave in a cycle is called wavelength. It I denoted by λ. Its unit is meter.
Speed of Wave
The straight distance covered by a wave in one second is called its speed. It is denoted as v and unit is metre/second.
Speed of radio wave, light wave, heat wave = $\displaystyle \small 3\times 10^{8}$ m/sec
Speed of sound wave = 330 m/sec
$\displaystyle \small v=f.\lambda$
where,
v = speed of wave in m/sec
f = frequency in Hz
$\displaystyle \small \lambda$ = wavelength in meter
Pure Resistive Circuit
Resistance is the opposition that a substance offers to the flow of electric current. AC circuit in which the value of reactance is negligible is called pure resistive circuit. Here the value of impedance (Z) is almost equal to resistance.
Power consumption of circuit is given by,
$\displaystyle \small P=VI=\frac{V^{2}}{R}=I^{2}R$
where,
P = power consumption in Watts
V = RMS voltage in Volts
I = RMS current in Ampere
R = resistance in Ohm
If there is negligible reactance in pure resistive circuit, voltage and current are in-phase.
Inductor
When the conductor is curled in the form of a coil and given a shape of an inductor with definite inductance, it is known as choke or inductor.
Inductance
$\displaystyle \small \bullet$ When AC flows through a coil, due to the flow of current a magnetic field of alternating property is generated around it.
$\displaystyle \small \bullet$ Due to this magnetic field, an electromotive force is generated in another coil. This effect is called induction.
$\displaystyle \small \bullet$ The property of AC circuit because of which it opposes the changes in current is called inductance.
$\displaystyle \small \bullet$ It is denoted by ‘L’.
$\displaystyle \small \bullet$ Its unit is Henry (H).
Inductive Reactance
Inductive reactance is the opposition offered by a conductor coil against the flow of AC.
It is denoted by $\displaystyle \small X_{L}$.
Its unit us Ohm (Ω).
$\displaystyle \small X_{L}=2\pi fL=\frac{V}{I}$
where,
$\displaystyle \small X_{L}$ =inductive reactance in ohm
f = frequency in Hz
L=inductance in Henry
I = electric current in Ampere
V = voltage in Volt
Due to the opposing induced electromotive force generated in pure inductive circuit, electric current lags behind voltage by $\displaystyle \small 90^{0}$.
$\displaystyle \small \tan \theta =\frac{X_{L}}{R}$
θ = lagging angle of electric current in degrees.
Susceptance: the reciprocal of inductive reactance in an AC circuit is called susceptance.
$\displaystyle \small B=\frac{1}{X_{L}}$
Coefficient of Coupling
$\displaystyle \small K=\frac{M}{\sqrt{L_{1}L_{2}}}$
where,
K = coefficient of coupling
M = mutual inductance in Henry
$\displaystyle \small L_{1},L_{2}$ = inductance of individual coil in Henry
Time Constant of an Inductor
The time taken by electric current to reach 63.3% of the highest value from zero in an inductor is called its time constant.
$\displaystyle \small T=\frac{L}{R}$
where,
T = time constant in sec
L = inductance in Henry
R = resistance in ohm
Capacitance
Capacitor: when two conductors (plates/sheets) and insulators are given shape of a component to provide capacitance of definitive value, it is called capacitor or condenser.
Its main purpose is to store power in the form of electric charge.
Capacitance: the property of AC circuits which opposes the changes in voltage is called capacitance.
It’s denoted by C.
Its unit is farad(f).
$\displaystyle \small C=\frac{Q}{V}$
where,
Q = electric charge in Coulomb
V = voltage in Volt
C = capacitance in farad
Electrostatic energy stores in capacitor:
$\displaystyle \small E=\frac{1}{2}CV^{2}$
where,
E = stored electrostatic energy in Joule
C = capacitance in farad
V = potential difference or voltage in Volts
Capacitive Reactance
$\displaystyle \small \bullet$ The capacitors opposition to AC flow is called capacitive reactance.
$\displaystyle \small \bullet$ It is denoted by $\displaystyle \small X_{C}$
$\displaystyle \small \bullet$ Its unit is ohm(Ω)
$\displaystyle \small X_{C}=\frac{1}{2\pi fC}$
where,
$\displaystyle \small X_{C}$ = capacitive reactance in ohm
f = frequency in Hertz
C = capacitance in farad
$\displaystyle \small \bullet$ Due to stored electrostatic energy in pure capacitive circuit, voltage lags behind current by $\displaystyle \small 90^{0}$.
$\displaystyle \small \tan \theta =\frac{X_{C}}{R}$
where,
θ = lagging angle of voltage in degrees
Time constant of a Capacitor
The time taken to transmit potential difference to 63.3% of the highest value from zero by a capacitor is called its time constant.
$\displaystyle \small t=CR$
where,
t = time constant in sec
C = capacitance in farad
R = resistance in ohm
Impedance
$\displaystyle \small \bullet$ The resistance offered by an AC circuit to electric current is called impedance.
$\displaystyle \small \bullet$ It is denoted by Z.
$\displaystyle \small \bullet$ Its unit is ohm(Ω).
$\displaystyle \small \bullet$ The reciprocal of impedance in an AC circuit is called admittance. Its unit is mho(℧).
$\displaystyle \small Y=\frac{1}{Z}$
Series R-L circuit
When resistor and inductor are connected in series, then total potential difference of circuit is equal to total amount of resistance potential difference $\displaystyle \small V_{R}$ and inductor potential difference $\displaystyle \small V_{L}$, however equal current flows in both.
Total potential difference of circuit
$\displaystyle \small V=\sqrt{(V_{R})^{2}+(V_{L})^{2}}$
Total impedance of circuit
$\displaystyle \small Z=\sqrt{R^{2}+(X_{L})^{2}}$
Current in circuit
$\displaystyle \small I=\frac{V}{Z}$
Power factor of circuit (lagging angle of current)
$\displaystyle \small \cos \phi =\frac{R}{Z}$
Total power consumption of circuit
$\displaystyle \small P=VI\cos \phi$
Series R-C circuit
When resistor and capacitor are connected in series, then total potential difference of circuit is equal to total amount of resistance potential difference $\displaystyle \small V_{R}$ and capacitor potential difference $\displaystyle \small V_{C}$, however current in both is of equal magnitude.
Total potential difference of circuit
$\displaystyle \small V=\sqrt{(V_{R})^{2}+(V_{C})^{2}}$
Total impedance of circuit
$\displaystyle \small Z=\sqrt{R^{2}+(X_{C})^{2}}$
Current in circuit
$\displaystyle \small I=\frac{V}{Z}$
Power factor of circuit (lagging angle of voltage)
$\displaystyle \small \cos \phi =\frac{R}{Z}$
Total power consumption of circuit
$\displaystyle \small P=VI\cos \phi$
Series L-C-R circuit
When resistor, inductor and capacitor are connected in series, then total potential difference of circuit is equal to sum of resistance potential difference $\displaystyle \small V_{R}$, inductor potential difference $\displaystyle \small V_{L}$ and capacitor potential difference $\displaystyle \small V_{C}$. However equal current flows in them.
Total potential difference of circuit
$\displaystyle \small V=\sqrt{V_{R}^{2}+(V_{C}-V_{L})^{2}}$
Total impedance of circuit
$\displaystyle \small Z=\sqrt{R^{2}+(X_{C}-X_{L})^{2}}$
Current in circuit
$\displaystyle \small I=\frac{V}{Z}$
Power factor of circuit
$\displaystyle \small \cos \phi =\frac{R}{Z}$
Parallel R-L circuit
In parallel R-L circuit, current in both components will flow according to the value of components. However, the potential difference in both will be equal.
Total current in parallel circuit
$\displaystyle \small I_{T}^{2}=I_{R}^{2}+I_{L}^{2}$
Total impedance of circuit
$\displaystyle \small Z=\frac{RX_{L}}{\sqrt{R^{2}+X_{L}^{2}}}$
Power factor of circuit
$\displaystyle \small \cos \phi =\frac{Z}{R}$
Parallel R-C circuit
In parallel R-C circuit, current in both components will flow according to the value of components. However, the potential difference in both will be equal.
Total current in parallel circuit
$\displaystyle \small I_{T}^{2}=I_{R}^{2}+I_{C}^{2}$
Total impedance of circuit
$\displaystyle \small Z=\frac{RX_{C}}{\sqrt{R^{2}+X_{C}^{2}}}$
Power factor of circuit
$\displaystyle \small \cos \phi =\frac{Z}{R}$
Parallel R-L-C circuit
In parallel R-L-C circuit, current flowing in all three components will be of varying magnitude and is equal to difference between total current IR and current in other two. However, the potential difference in all will be equal.
Total current in parallel circuit
$\displaystyle \small I^{2}=I_{R}^{2}+(I_{L}-I_{C})^{2}$
Total impedance of circuit
$\displaystyle \small \frac{1}{Z}=\sqrt{\frac{1}{R^{2}}+\left ( \frac{1}{X_{L}}-\frac{1}{X_{C}} \right )^{2}}$
$\displaystyle \small Z=\frac{L}{CR}$
Power factor
$\displaystyle \small \cos \phi =\frac{Z}{R}$
Power consumption
$\displaystyle \small P=VI\cos \phi$
Resonance Frequency
The frequency at which the value of inductive reactance ($\displaystyle \small X_{L}$) becomes equivalent to capacitive reactance ($\displaystyle \small X_{C}$) is called resonance frequency.
$\displaystyle \small X_{L}=X_{C}$
Resonance frequency, $\displaystyle \small f_{r}=\frac{1}{2\pi \sqrt{LC}}$
If R is greater in parallel resonant circuit,
$\displaystyle \small f_{r}=\frac{1}{2\pi }\sqrt{\frac{1}{LC}-\frac{R^{2}}{L^{2}}}$
Series Resonant circuit: $\displaystyle \small Z=R$, i.e. impedance is minimum and current is maximum
Parallel Resonant circuit: $\displaystyle \small Z=\frac{X_{L}^{2}}{R}$, i.e. impedance is maximum and current is minimum.
Circuit Q Factor
$\displaystyle \small Q=\frac{Reactance}{Resistance}=\frac{X_{L}}{R}$
where,
Q = ratio
$\displaystyle \small X_{L}$ = inductive reactance in Ohm
R = resistance in Ohm
Polyphase
$\displaystyle \small \bullet$ The supply of more than one phase is called polyphaser.
$\displaystyle \small \bullet$ There are many single phase voltage of equal degree, same frequency and at a difference of equal angle from one another (2-phase, 3-phase, 6-phase etc.).
$\displaystyle \small \bullet$ The phase angles depends upon number of phases.
$\displaystyle \small \bullet$ The value of phase angle in 2-phase is $\displaystyle \small 90^{0}$.
$\displaystyle \small \bullet$ The value of phase angle in 3-phase is $\displaystyle \small 120^{0}$.
$\displaystyle \small \bullet$ Circuit connections in 3-phase system
1. Star Connection
It has 3 coils, 6 terminals with phase angle $\displaystyle \small 120^{0}$. Three ends of these are connected at one point called star point or neutral point and supply is obtained from remaining three ends.
$\displaystyle \small E_{L}=\sqrt{3}E_{p}$
$\displaystyle \small I_{L}=I_{p}$
where,
$\displaystyle \small E_{L}$ or $\displaystyle \small V_{L}$ = line voltage
$\displaystyle \small E_{p}$ or $\displaystyle \small V_{p}$ = phase voltage
$\displaystyle \small I_{L}$ = line current
$\displaystyle \small I_{p}$ = phase current
Total power $\displaystyle \small =\sqrt{3}V_{L}I_{L}\cos \phi =3V_{p}I_{p}\cos \phi$
2. Delta connection
The 6 terminals are connected in a way that the trailing end of first coil is connected to initial end of second coil and trailing end of second coil is connected to initial end of first coil. Supply is obtained from each intersection.
$\displaystyle \small E_{L}=E_{p}$
$\displaystyle \small I_{L}=\sqrt{3}I_{p}$
where,
$\displaystyle \small E_{L}$ or $\displaystyle \small V_{L}$ = line voltage
$\displaystyle \small E_{p}$ or $\displaystyle \small V_{p}$ = phase voltage
$\displaystyle \small I_{L}$ = line current
$\displaystyle \small I_{p}$ = phase current
Total power $\displaystyle \small =\sqrt{3}V_{L}I_{L}\cos \phi =3V_{p}I_{p}\cos \phi$
Combination of Resistors
1. Series combination
2. Parallel combination
3. Series-Parallel combination
Series Combination
$\displaystyle \small \bullet$ Equal current flows from each resistor.
$\displaystyle \small \bullet$ Single direction of current flow.
$\displaystyle \small \bullet$ Voltage drop is according to the value of resistance.
$\displaystyle \small \bullet$ Amount of voltage drop in each resistor is equal to the supply voltage given in the circuit.
Total current, $\displaystyle \small I=I_{1}=I_{2}=I_{3}$
Total voltage, $\displaystyle \small V=V_{1}+V_{2}+V_{3}$
Total resistance, $\displaystyle \small R=r_{1}+r_{2}+r_{3}$
Voltage source in series combination
Voltage source, $\displaystyle \small V_{T}=V_{S1}+V_{S2}+V_{S3}$
When polarity of one cell is inverted then voltage decrease in series,
Total voltage, $\displaystyle \small V_{T}=V_{S1}-V_{S2}+V_{S3}$
Parallel Combination
$\displaystyle \small \bullet$ Current is distributed in each branch or resistor.
$\displaystyle \small \bullet$ Many direction of current flow.
$\displaystyle \small \bullet$ Amount of current flowing through all branches is equal to total current.
$\displaystyle \small \bullet$ Voltage is equal at terminals of each resistor and is equal to the supply voltage.
Total current, $\displaystyle \small I=I_{1}+I_{2}+I_{3}$
Total voltage, $\displaystyle \small V=V_{1}=V_{2}=V_{3}$
Total resistance, $\displaystyle \small \frac{1}{R}=\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}$
[Total Conduction, $\displaystyle \small G=\frac{1}{R}$]
Series-Parallel Combination
Current is divided. Voltage drop and current is calculated with the help of Kirchhoff’s law.
Series and Parallel Combination of Capacitor
Capacitance in series:
Let, $\displaystyle \small C_{1}, C_{2}, C_{3}$ are capacitors connected in series and $\displaystyle \small V_{1}, V_{2}, V_{3}$ are their voltages respectively. Then,
$\displaystyle \small V_{1}=\frac{Q}{C_{1}}$, $\displaystyle \small V_{2}=\frac{Q}{C_{2}}$, $\displaystyle \small V_{3}=\frac{Q}{C_{3}}$
$\displaystyle \small V=V_{1}+ V_{2}+ V_{3}$
$\displaystyle \small \frac{1}{C}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}$
By connecting capacitors in series, the value of total capacitance decreases.
Capacitance in Parallel:
Let, $\displaystyle \small C_{1}, C_{2}, C_{3}$ are capacitors connected in parallel, V is total voltage and Q is total charge. Then,
$\displaystyle \small Q_{1}=VC_{1}$, $\displaystyle \small Q_{2}=VC_{2}$, $\displaystyle \small Q_{3}=VC_{3}$
$\displaystyle \small Q=Q_{1}+Q_{2}+Q_{3}$
$\displaystyle \small C=C_{1}+C_{2}+C_{3}$
By connecting capacitors in parallel, the value of total capacitance increases.
Series and Parallel combination of Inductor
Series combination:
(i) When two induction coils are in series combination and flux is in common direction
Let $\displaystyle \small L_{1},L_{2}$ are inductance of first and second coil respectively and M is mutual inductance.
Automatically induced emf in coil A, $\displaystyle \small e_{LA}=-\frac{L_{1}di}{dt}$
Mutually induced emf in coil A, $\displaystyle \small e_{MA}=-\frac{Mdi}{dt}$
Automatically induced emf in coil B, $\displaystyle \small e_{LB}=-\frac{L_{2}di}{dt}$
Mutually induced emf in coil B, $\displaystyle \small e_{MB}=-\frac{Mdi}{dt}$
Resultant induced emf = $\displaystyle \small -(L_{1}+L_{2}+2M)\frac{di}{dt}$ $\displaystyle \small =-\frac{Ldi}{dt}$
[$\displaystyle \small L=L_{1}+L_{2}+2M$]
If coupling is not present at the middle of inductor,
$\displaystyle \small L_{T}=L_{1}+L_{2}$
(ii) When two induction coils are in series combination and flux is in opposite direction
Let $\displaystyle \small L_{1},L_{2}$ are inductance of first and second coil respectively and M is mutual inductance.
Automatically induced emf in coil A, $\displaystyle \small e_{LA}=-\frac{L_{1}di}{dt}$
Mutually induced emf in coil A, $\displaystyle \small e_{MA}=+\frac{Mdi}{dt}$
Automatically induced emf in coil B, $\displaystyle \small e_{LB}=-\frac{L_{2}di}{dt}$
Mutually induced emf in coil B, $\displaystyle \small e_{MB}=+\frac{Mdi}{dt}$
Resultant induced emf = $\displaystyle \small -(L_{1}+L_{2}-2M)\frac{di}{dt}$ $\displaystyle \small =-\frac{Ldi}{dt}$
[$\displaystyle \small L=L_{1}+L_{2}-2M$]
Parallel Combination:
Let $\displaystyle \small L_{1},L_{2}$ are inductance of first and second coil respectively and M is mutual inductance.
Automatically induced emf in coil A, $\displaystyle \small e_{LA}=-\frac{L_{1}di}{dt}$
Mutually induced emf in coil A, $\displaystyle \small e_{MA}=+\frac{Mdi}{dt}$
Resultant emf in coil A = $\displaystyle \small e_{LA}+e_{MA}$ $\displaystyle \small =-\left ( L_{1}\frac{di_{1}}{dt}+M\frac{di_{2}}{dt} \right )$
Similarly, Resultant emf in coil B = $\displaystyle \small e_{LB}+e_{MB}$ $\displaystyle \small =-\left ( L_{2}\frac{di_{1}}{dt}+M\frac{di_{2}}{dt} \right )$
Since coils is in parallel combination, resultant emf of both coils will be same.
$\displaystyle \small L=\frac{L_{1}+L_{2}-2M}{L_{1}-M}$
If coupling is not present in between inductor,
$\displaystyle \small \frac{1}{L_{T}}=\frac{1}{L_{1}}+\frac{1}{L_{2}}$
Power Factor
$\displaystyle \small \bullet$ In AC circuit, the cos of angle between current and voltage is called power factor.
OR
$\displaystyle \small \bullet$ The ratio of real power and apparent power is called power factor.
$\displaystyle \small \bullet$ Normally in AC circuit, there is phase difference (Ï•) in the middle of current and voltage. The degree of cos Ï• in the circuit is power factor.
$\displaystyle \small \bullet$ Consider an inductive circuit wherein the current is behind supply voltage and back angle is Ï•.
$\displaystyle \small \bullet$ Circuit current can be divided into two components which are perpendicular to each other.
(i) Active/Watt full component: $\displaystyle \small I\cos \phi$
Voltage is in the form of vector
(ii) Reactive/Watt less component: $\displaystyle \small I\sin \phi$
Voltage is $\displaystyle \small 90^{0}$ (perpendicular) to vector.
$\displaystyle \small \bullet$ Power factor can never be more than 1.
Consider power triangle
$\displaystyle \small OA=VI\cos \phi$ [Active power in Watts or kiloWatts]
$\displaystyle \small AB=VI\sin \phi$ [Reactive power in VAR or kVAr]
$\displaystyle \small OB=VI$ [Apparent power in VA or kVA]
$\displaystyle \small (kVA)^{2}=(kW)^{2}+(kVAr)^{2}$
Power factor,
$\displaystyle \small \cos \phi =\frac{kW}{kVA}$
$\displaystyle \small \sin \phi =\frac{kVA}{kVAr}$
$\displaystyle \small kVAr=kW\tan \phi$
$\displaystyle \small \bullet$ If the power factor is unity, then the electric current is 100% useful.
$\displaystyle \small \bullet$ If the power factor is 0.5 then only 50% of the total current is useful.
$\displaystyle \small \bullet$ AC is an electric current which changes its voltage and direction periodically.
$\displaystyle \small \bullet$ AC keeps changing with time and this change is sinusoidal.
$\displaystyle \small \bullet$ Property of AC is that its value increases from zero to peak value and then returns to zero, not only in positive half but also in negative half of the cycle.
$\displaystyle \small \bullet$ The voltage of Ac increases for duration 0 to T/4, and is highest at T/4.
$\displaystyle \small \bullet$ For duration T/4 to T/2, voltage reduces and attains zero level at T/2.
$\displaystyle \small \bullet$ For T/2 to 3T/4, voltage increases again and attains highest level at 3T/4.
$\displaystyle \small \bullet$ For 3T/4 to T, it reduces again and becomes zero at T.
Terms related to AC
Cycle
A complete sinusoidal change of alternating current’s voltage and direction is called a cycle. The upper half of the cycle is called positive and lower half is known as negative cycle.
Frequency
The number of complete cycles per second in AC is called frequency.
It is denoted as f and is inversely proportional to time period (when frequency increases, time period decreases and when frequency decreases time period increases).
Its unit is cycles per second or Hz.
$\displaystyle \small f=\frac{1}{Time\; Period(T)}$
Time period
The taken to complete one cycle in AC is called time period. It is denotes as T and its unit is second.
$\displaystyle \small Time\; Period=\frac{1}{Frequency(f)}$
Root Mean Square Value
It is the DC current when passed through a resistance for a given time produces the same amount of energy as the alternating current does in the same resistance in the same time. To find RMS value, its instant values are captures in a cycle at various instants. These values are squared, added and average is calculated.
$\displaystyle \small I_{rms}=\sqrt{\frac{I_{1}^{2}+I_{2}^{2}+I_{3}^{2}...I_{n}^{2}}{N}}$
$\displaystyle \small I_{rms}=0.707\times I_{max}$
Similarly,
$\displaystyle \small E_{rms}=0.707\times E_{max}$
Peak Value
The highest positive and negative value obtained in sinusoidal wave of AC signal is called peak value.
Peak to Peak Value
The distance between peak value of positive half cycle and peak value of negative half cycle in sinusoidal AC signal is called peak to peak value.
Instantaneous Value
Value observed at a time instant is sinusoidal AC signal is called instantaneous value.
$\displaystyle \small I=I_{0}\sin (\omega t+\phi )$
where,
$\displaystyle \small I_{0}$ = Peak value of current
Average Value
The average of instant current or voltage in half cycle of AC is known as average value of AC voltage or current.
$\displaystyle \small E_{average}=\frac{e_{1}+e_{2}+e_{3}+...+e_{n}}{n}$
$\displaystyle \small E_{average}=0.637E_{max}$
Form factor
The ratio of RMS value of AC and the average value is called form factor. Its denoted by K.
$\displaystyle \small K=\frac{RMS\; Value}{Average\; Value}=1.11$
Peak factor/ Amplitude factor
It is the ratio of highest value and RMs value of AC.
$\displaystyle \small Peak\;Factor=\frac{Highest\; Value}{RMS\; Value}=1.414$
Phase
The relative position of components of two ACs is called phase. Current and voltage are the two components. This relation gives information about which component is greater or lesser.
In-phase
When voltage and current in a phase attain their highest and lowest values in the same direction after beginning from zero at the same time, it is called in phase. Electric angle in alternating quantities should be zero.
Out-of-phase
When voltage and current in alternating quantities do not begin at same time and do not attain their highest and lowest values in the same direction at same time instant, it is called out of phase.
Phase difference
When alternating quantities reach their highest and lowest values from zero at different time intervals, then this difference in time is called phase difference. It is denoted by Ï•.
$\displaystyle \small e=OQ\sin \omega t$
$\displaystyle \small i=OP\sin (\omega t-\phi )$
Wavelength
Straight distance covered by a wave in a cycle is called wavelength. It I denoted by λ. Its unit is meter.
Speed of Wave
The straight distance covered by a wave in one second is called its speed. It is denoted as v and unit is metre/second.
Speed of radio wave, light wave, heat wave = $\displaystyle \small 3\times 10^{8}$ m/sec
Speed of sound wave = 330 m/sec
$\displaystyle \small v=f.\lambda$
where,
v = speed of wave in m/sec
f = frequency in Hz
$\displaystyle \small \lambda$ = wavelength in meter
Pure Resistive Circuit
Resistance is the opposition that a substance offers to the flow of electric current. AC circuit in which the value of reactance is negligible is called pure resistive circuit. Here the value of impedance (Z) is almost equal to resistance.
Power consumption of circuit is given by,
$\displaystyle \small P=VI=\frac{V^{2}}{R}=I^{2}R$
where,
P = power consumption in Watts
V = RMS voltage in Volts
I = RMS current in Ampere
R = resistance in Ohm
If there is negligible reactance in pure resistive circuit, voltage and current are in-phase.
Inductor
When the conductor is curled in the form of a coil and given a shape of an inductor with definite inductance, it is known as choke or inductor.
Inductance
$\displaystyle \small \bullet$ When AC flows through a coil, due to the flow of current a magnetic field of alternating property is generated around it.
$\displaystyle \small \bullet$ Due to this magnetic field, an electromotive force is generated in another coil. This effect is called induction.
$\displaystyle \small \bullet$ The property of AC circuit because of which it opposes the changes in current is called inductance.
$\displaystyle \small \bullet$ It is denoted by ‘L’.
$\displaystyle \small \bullet$ Its unit is Henry (H).
Inductive Reactance
Inductive reactance is the opposition offered by a conductor coil against the flow of AC.
It is denoted by $\displaystyle \small X_{L}$.
Its unit us Ohm (Ω).
$\displaystyle \small X_{L}=2\pi fL=\frac{V}{I}$
where,
$\displaystyle \small X_{L}$ =inductive reactance in ohm
f = frequency in Hz
L=inductance in Henry
I = electric current in Ampere
V = voltage in Volt
Due to the opposing induced electromotive force generated in pure inductive circuit, electric current lags behind voltage by $\displaystyle \small 90^{0}$.
$\displaystyle \small \tan \theta =\frac{X_{L}}{R}$
θ = lagging angle of electric current in degrees.
Susceptance: the reciprocal of inductive reactance in an AC circuit is called susceptance.
$\displaystyle \small B=\frac{1}{X_{L}}$
Coefficient of Coupling
$\displaystyle \small K=\frac{M}{\sqrt{L_{1}L_{2}}}$
where,
K = coefficient of coupling
M = mutual inductance in Henry
$\displaystyle \small L_{1},L_{2}$ = inductance of individual coil in Henry
Time Constant of an Inductor
The time taken by electric current to reach 63.3% of the highest value from zero in an inductor is called its time constant.
$\displaystyle \small T=\frac{L}{R}$
where,
T = time constant in sec
L = inductance in Henry
R = resistance in ohm
Capacitance
Capacitor: when two conductors (plates/sheets) and insulators are given shape of a component to provide capacitance of definitive value, it is called capacitor or condenser.
Its main purpose is to store power in the form of electric charge.
Capacitance: the property of AC circuits which opposes the changes in voltage is called capacitance.
It’s denoted by C.
Its unit is farad(f).
$\displaystyle \small C=\frac{Q}{V}$
where,
Q = electric charge in Coulomb
V = voltage in Volt
C = capacitance in farad
Electrostatic energy stores in capacitor:
$\displaystyle \small E=\frac{1}{2}CV^{2}$
where,
E = stored electrostatic energy in Joule
C = capacitance in farad
V = potential difference or voltage in Volts
Capacitive Reactance
$\displaystyle \small \bullet$ The capacitors opposition to AC flow is called capacitive reactance.
$\displaystyle \small \bullet$ It is denoted by $\displaystyle \small X_{C}$
$\displaystyle \small \bullet$ Its unit is ohm(Ω)
$\displaystyle \small X_{C}=\frac{1}{2\pi fC}$
where,
$\displaystyle \small X_{C}$ = capacitive reactance in ohm
f = frequency in Hertz
C = capacitance in farad
$\displaystyle \small \bullet$ Due to stored electrostatic energy in pure capacitive circuit, voltage lags behind current by $\displaystyle \small 90^{0}$.
$\displaystyle \small \tan \theta =\frac{X_{C}}{R}$
where,
θ = lagging angle of voltage in degrees
Time constant of a Capacitor
The time taken to transmit potential difference to 63.3% of the highest value from zero by a capacitor is called its time constant.
$\displaystyle \small t=CR$
where,
t = time constant in sec
C = capacitance in farad
R = resistance in ohm
Impedance
$\displaystyle \small \bullet$ The resistance offered by an AC circuit to electric current is called impedance.
$\displaystyle \small \bullet$ It is denoted by Z.
$\displaystyle \small \bullet$ Its unit is ohm(Ω).
$\displaystyle \small \bullet$ The reciprocal of impedance in an AC circuit is called admittance. Its unit is mho(℧).
$\displaystyle \small Y=\frac{1}{Z}$
Series R-L circuit
When resistor and inductor are connected in series, then total potential difference of circuit is equal to total amount of resistance potential difference $\displaystyle \small V_{R}$ and inductor potential difference $\displaystyle \small V_{L}$, however equal current flows in both.
Total potential difference of circuit
$\displaystyle \small V=\sqrt{(V_{R})^{2}+(V_{L})^{2}}$
Total impedance of circuit
$\displaystyle \small Z=\sqrt{R^{2}+(X_{L})^{2}}$
Current in circuit
$\displaystyle \small I=\frac{V}{Z}$
Power factor of circuit (lagging angle of current)
$\displaystyle \small \cos \phi =\frac{R}{Z}$
Total power consumption of circuit
$\displaystyle \small P=VI\cos \phi$
Series R-C circuit
When resistor and capacitor are connected in series, then total potential difference of circuit is equal to total amount of resistance potential difference $\displaystyle \small V_{R}$ and capacitor potential difference $\displaystyle \small V_{C}$, however current in both is of equal magnitude.
Total potential difference of circuit
$\displaystyle \small V=\sqrt{(V_{R})^{2}+(V_{C})^{2}}$
Total impedance of circuit
$\displaystyle \small Z=\sqrt{R^{2}+(X_{C})^{2}}$
Current in circuit
$\displaystyle \small I=\frac{V}{Z}$
Power factor of circuit (lagging angle of voltage)
$\displaystyle \small \cos \phi =\frac{R}{Z}$
Total power consumption of circuit
$\displaystyle \small P=VI\cos \phi$
Series L-C-R circuit
When resistor, inductor and capacitor are connected in series, then total potential difference of circuit is equal to sum of resistance potential difference $\displaystyle \small V_{R}$, inductor potential difference $\displaystyle \small V_{L}$ and capacitor potential difference $\displaystyle \small V_{C}$. However equal current flows in them.
Total potential difference of circuit
$\displaystyle \small V=\sqrt{V_{R}^{2}+(V_{C}-V_{L})^{2}}$
Total impedance of circuit
$\displaystyle \small Z=\sqrt{R^{2}+(X_{C}-X_{L})^{2}}$
Current in circuit
$\displaystyle \small I=\frac{V}{Z}$
Power factor of circuit
$\displaystyle \small \cos \phi =\frac{R}{Z}$
Parallel R-L circuit
In parallel R-L circuit, current in both components will flow according to the value of components. However, the potential difference in both will be equal.
Total current in parallel circuit
$\displaystyle \small I_{T}^{2}=I_{R}^{2}+I_{L}^{2}$
Total impedance of circuit
$\displaystyle \small Z=\frac{RX_{L}}{\sqrt{R^{2}+X_{L}^{2}}}$
Power factor of circuit
$\displaystyle \small \cos \phi =\frac{Z}{R}$
Parallel R-C circuit
In parallel R-C circuit, current in both components will flow according to the value of components. However, the potential difference in both will be equal.
Total current in parallel circuit
$\displaystyle \small I_{T}^{2}=I_{R}^{2}+I_{C}^{2}$
Total impedance of circuit
$\displaystyle \small Z=\frac{RX_{C}}{\sqrt{R^{2}+X_{C}^{2}}}$
Power factor of circuit
$\displaystyle \small \cos \phi =\frac{Z}{R}$
Parallel R-L-C circuit
In parallel R-L-C circuit, current flowing in all three components will be of varying magnitude and is equal to difference between total current IR and current in other two. However, the potential difference in all will be equal.
Total current in parallel circuit
$\displaystyle \small I^{2}=I_{R}^{2}+(I_{L}-I_{C})^{2}$
Total impedance of circuit
$\displaystyle \small \frac{1}{Z}=\sqrt{\frac{1}{R^{2}}+\left ( \frac{1}{X_{L}}-\frac{1}{X_{C}} \right )^{2}}$
$\displaystyle \small Z=\frac{L}{CR}$
Power factor
$\displaystyle \small \cos \phi =\frac{Z}{R}$
Power consumption
$\displaystyle \small P=VI\cos \phi$
Resonance Frequency
The frequency at which the value of inductive reactance ($\displaystyle \small X_{L}$) becomes equivalent to capacitive reactance ($\displaystyle \small X_{C}$) is called resonance frequency.
$\displaystyle \small X_{L}=X_{C}$
Resonance frequency, $\displaystyle \small f_{r}=\frac{1}{2\pi \sqrt{LC}}$
If R is greater in parallel resonant circuit,
$\displaystyle \small f_{r}=\frac{1}{2\pi }\sqrt{\frac{1}{LC}-\frac{R^{2}}{L^{2}}}$
Series Resonant circuit: $\displaystyle \small Z=R$, i.e. impedance is minimum and current is maximum
Parallel Resonant circuit: $\displaystyle \small Z=\frac{X_{L}^{2}}{R}$, i.e. impedance is maximum and current is minimum.
Circuit Q Factor
$\displaystyle \small Q=\frac{Reactance}{Resistance}=\frac{X_{L}}{R}$
where,
Q = ratio
$\displaystyle \small X_{L}$ = inductive reactance in Ohm
R = resistance in Ohm
Polyphase
$\displaystyle \small \bullet$ The supply of more than one phase is called polyphaser.
$\displaystyle \small \bullet$ There are many single phase voltage of equal degree, same frequency and at a difference of equal angle from one another (2-phase, 3-phase, 6-phase etc.).
$\displaystyle \small \bullet$ The phase angles depends upon number of phases.
$\displaystyle \small \bullet$ The value of phase angle in 2-phase is $\displaystyle \small 90^{0}$.
$\displaystyle \small \bullet$ The value of phase angle in 3-phase is $\displaystyle \small 120^{0}$.
$\displaystyle \small \bullet$ Circuit connections in 3-phase system
1. Star Connection
It has 3 coils, 6 terminals with phase angle $\displaystyle \small 120^{0}$. Three ends of these are connected at one point called star point or neutral point and supply is obtained from remaining three ends.
$\displaystyle \small E_{L}=\sqrt{3}E_{p}$
$\displaystyle \small I_{L}=I_{p}$
where,
$\displaystyle \small E_{L}$ or $\displaystyle \small V_{L}$ = line voltage
$\displaystyle \small E_{p}$ or $\displaystyle \small V_{p}$ = phase voltage
$\displaystyle \small I_{L}$ = line current
$\displaystyle \small I_{p}$ = phase current
Total power $\displaystyle \small =\sqrt{3}V_{L}I_{L}\cos \phi =3V_{p}I_{p}\cos \phi$
2. Delta connection
The 6 terminals are connected in a way that the trailing end of first coil is connected to initial end of second coil and trailing end of second coil is connected to initial end of first coil. Supply is obtained from each intersection.
$\displaystyle \small E_{L}=E_{p}$
$\displaystyle \small I_{L}=\sqrt{3}I_{p}$
where,
$\displaystyle \small E_{L}$ or $\displaystyle \small V_{L}$ = line voltage
$\displaystyle \small E_{p}$ or $\displaystyle \small V_{p}$ = phase voltage
$\displaystyle \small I_{L}$ = line current
$\displaystyle \small I_{p}$ = phase current
Total power $\displaystyle \small =\sqrt{3}V_{L}I_{L}\cos \phi =3V_{p}I_{p}\cos \phi$
Combination of Resistors
1. Series combination
2. Parallel combination
3. Series-Parallel combination
Series Combination
$\displaystyle \small \bullet$ Equal current flows from each resistor.
$\displaystyle \small \bullet$ Single direction of current flow.
$\displaystyle \small \bullet$ Voltage drop is according to the value of resistance.
$\displaystyle \small \bullet$ Amount of voltage drop in each resistor is equal to the supply voltage given in the circuit.
Total current, $\displaystyle \small I=I_{1}=I_{2}=I_{3}$
Total voltage, $\displaystyle \small V=V_{1}+V_{2}+V_{3}$
Total resistance, $\displaystyle \small R=r_{1}+r_{2}+r_{3}$
Voltage source in series combination
Voltage source, $\displaystyle \small V_{T}=V_{S1}+V_{S2}+V_{S3}$
When polarity of one cell is inverted then voltage decrease in series,
Total voltage, $\displaystyle \small V_{T}=V_{S1}-V_{S2}+V_{S3}$
Parallel Combination
$\displaystyle \small \bullet$ Current is distributed in each branch or resistor.
$\displaystyle \small \bullet$ Many direction of current flow.
$\displaystyle \small \bullet$ Amount of current flowing through all branches is equal to total current.
$\displaystyle \small \bullet$ Voltage is equal at terminals of each resistor and is equal to the supply voltage.
Total current, $\displaystyle \small I=I_{1}+I_{2}+I_{3}$
Total voltage, $\displaystyle \small V=V_{1}=V_{2}=V_{3}$
Total resistance, $\displaystyle \small \frac{1}{R}=\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}$
[Total Conduction, $\displaystyle \small G=\frac{1}{R}$]
Series-Parallel Combination
Current is divided. Voltage drop and current is calculated with the help of Kirchhoff’s law.
Series and Parallel Combination of Capacitor
Capacitance in series:
Let, $\displaystyle \small C_{1}, C_{2}, C_{3}$ are capacitors connected in series and $\displaystyle \small V_{1}, V_{2}, V_{3}$ are their voltages respectively. Then,
$\displaystyle \small V_{1}=\frac{Q}{C_{1}}$, $\displaystyle \small V_{2}=\frac{Q}{C_{2}}$, $\displaystyle \small V_{3}=\frac{Q}{C_{3}}$
$\displaystyle \small V=V_{1}+ V_{2}+ V_{3}$
$\displaystyle \small \frac{1}{C}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}$
By connecting capacitors in series, the value of total capacitance decreases.
Capacitance in Parallel:
Let, $\displaystyle \small C_{1}, C_{2}, C_{3}$ are capacitors connected in parallel, V is total voltage and Q is total charge. Then,
$\displaystyle \small Q_{1}=VC_{1}$, $\displaystyle \small Q_{2}=VC_{2}$, $\displaystyle \small Q_{3}=VC_{3}$
$\displaystyle \small Q=Q_{1}+Q_{2}+Q_{3}$
$\displaystyle \small C=C_{1}+C_{2}+C_{3}$
By connecting capacitors in parallel, the value of total capacitance increases.
Series and Parallel combination of Inductor
Series combination:
(i) When two induction coils are in series combination and flux is in common direction
Let $\displaystyle \small L_{1},L_{2}$ are inductance of first and second coil respectively and M is mutual inductance.
Automatically induced emf in coil A, $\displaystyle \small e_{LA}=-\frac{L_{1}di}{dt}$
Mutually induced emf in coil A, $\displaystyle \small e_{MA}=-\frac{Mdi}{dt}$
Automatically induced emf in coil B, $\displaystyle \small e_{LB}=-\frac{L_{2}di}{dt}$
Mutually induced emf in coil B, $\displaystyle \small e_{MB}=-\frac{Mdi}{dt}$
Resultant induced emf = $\displaystyle \small -(L_{1}+L_{2}+2M)\frac{di}{dt}$ $\displaystyle \small =-\frac{Ldi}{dt}$
[$\displaystyle \small L=L_{1}+L_{2}+2M$]
If coupling is not present at the middle of inductor,
$\displaystyle \small L_{T}=L_{1}+L_{2}$
(ii) When two induction coils are in series combination and flux is in opposite direction
Let $\displaystyle \small L_{1},L_{2}$ are inductance of first and second coil respectively and M is mutual inductance.
Automatically induced emf in coil A, $\displaystyle \small e_{LA}=-\frac{L_{1}di}{dt}$
Mutually induced emf in coil A, $\displaystyle \small e_{MA}=+\frac{Mdi}{dt}$
Automatically induced emf in coil B, $\displaystyle \small e_{LB}=-\frac{L_{2}di}{dt}$
Mutually induced emf in coil B, $\displaystyle \small e_{MB}=+\frac{Mdi}{dt}$
Resultant induced emf = $\displaystyle \small -(L_{1}+L_{2}-2M)\frac{di}{dt}$ $\displaystyle \small =-\frac{Ldi}{dt}$
[$\displaystyle \small L=L_{1}+L_{2}-2M$]
Parallel Combination:
Let $\displaystyle \small L_{1},L_{2}$ are inductance of first and second coil respectively and M is mutual inductance.
Automatically induced emf in coil A, $\displaystyle \small e_{LA}=-\frac{L_{1}di}{dt}$
Mutually induced emf in coil A, $\displaystyle \small e_{MA}=+\frac{Mdi}{dt}$
Resultant emf in coil A = $\displaystyle \small e_{LA}+e_{MA}$ $\displaystyle \small =-\left ( L_{1}\frac{di_{1}}{dt}+M\frac{di_{2}}{dt} \right )$
Similarly, Resultant emf in coil B = $\displaystyle \small e_{LB}+e_{MB}$ $\displaystyle \small =-\left ( L_{2}\frac{di_{1}}{dt}+M\frac{di_{2}}{dt} \right )$
Since coils is in parallel combination, resultant emf of both coils will be same.
$\displaystyle \small L=\frac{L_{1}+L_{2}-2M}{L_{1}-M}$
If coupling is not present in between inductor,
$\displaystyle \small \frac{1}{L_{T}}=\frac{1}{L_{1}}+\frac{1}{L_{2}}$
Power Factor
$\displaystyle \small \bullet$ In AC circuit, the cos of angle between current and voltage is called power factor.
OR
$\displaystyle \small \bullet$ The ratio of real power and apparent power is called power factor.
$\displaystyle \small \bullet$ Normally in AC circuit, there is phase difference (Ï•) in the middle of current and voltage. The degree of cos Ï• in the circuit is power factor.
$\displaystyle \small \bullet$ Consider an inductive circuit wherein the current is behind supply voltage and back angle is Ï•.
$\displaystyle \small \bullet$ Circuit current can be divided into two components which are perpendicular to each other.
(i) Active/Watt full component: $\displaystyle \small I\cos \phi$
Voltage is in the form of vector
(ii) Reactive/Watt less component: $\displaystyle \small I\sin \phi$
Voltage is $\displaystyle \small 90^{0}$ (perpendicular) to vector.
$\displaystyle \small \bullet$ Power factor can never be more than 1.
Consider power triangle
$\displaystyle \small OA=VI\cos \phi$ [Active power in Watts or kiloWatts]
$\displaystyle \small AB=VI\sin \phi$ [Reactive power in VAR or kVAr]
$\displaystyle \small OB=VI$ [Apparent power in VA or kVA]
$\displaystyle \small (kVA)^{2}=(kW)^{2}+(kVAr)^{2}$
Power factor,
$\displaystyle \small \cos \phi =\frac{kW}{kVA}$
$\displaystyle \small \sin \phi =\frac{kVA}{kVAr}$
$\displaystyle \small kVAr=kW\tan \phi$
$\displaystyle \small \bullet$ If the power factor is unity, then the electric current is 100% useful.
$\displaystyle \small \bullet$ If the power factor is 0.5 then only 50% of the total current is useful.
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