Basic Electrical

ABBREVATIONS

RESISTANCE

OHM’s LAW

V=I*R where V=Voltage

I=Current

R=Resistance

DC Resistance:

R= ρ*l/A Where ρ = Resistivity of teh material

l =Length of the conductor

A = Area of the conductor

ρ= 1 / Conductivity=Resistivity

Temperature dependance of a resistor

R =R₀[α *(T-T0)+1] where R­­₀=Resistance at Temperature T0

α =% change in resistivity / unit temperature

T =Temperature

AC resistance = 1.6*Dc resistance
note : AC resistance would be always greater than DC resistance because of 'SKIN EFFECT'

Resistance in series

When R1,R2, R3 are connected in series…..

Req=R1+R2+R3

Ø Voltages divides & Current remains same in a series circuit.

Resistance in parallel

When R1,R2,R3 are connected in parallel

1/Req=(1/R1+1/R2+1/R3) =

Req=(R1*R2*R3)/(R1+R2+R3)

Ø Current divides & Voltage remains the same in a parallel circuit.

Ø The equivalent resistance of two or more resistance is always less than the least value of the individual resistor.

VOLTAGE DIVIDER

When two resistance are connected as shown

1) The voltage across Resistor R2 is given as

Vout = R2/(R1+R2)*Vin Where Vout = Output voltage

Vin = Input voltage

The power P dissipated by a resistance R carrying a current I with a voltage drop V is:

P= (I^2)*R = VI = (V^2)/R

The energy W consumed over time t due to power P dissipated in a resistance R carrying a current I with a voltage drop V is:

W=(I^2)*R*t

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CAPACITANCE

C=Q/V Where C= Capacitance

Q= Charge

V= Voltage

C=(Ebsilon0)*(Ebsilonr)*A/D Where C=Capacitance

A=Area

D=Distance b/w the plates

The Energy stored in a capacitor

W(stored)= C(V^2)/2 = (Q^2)/2C= QV/2

Current-voltage relationship…

I(t)=C dV/Dt Where (dV/dT)=Change in Voltage

The power P transferred by a capacitance C holding a changing voltage V with charge Q is:
P = VI = CV(dv/dt) = Q(dv/dt) = Q(dq/dt) / C

Capacitance in series….

When capacitances C1,C2,C3 are connected in series Equivalent capacitance is

1/Ceq=(1/C1+1/C2+1/C3)

Ceq= (C1*C2*C3)+(C1+C2+C3)

Capacitance in parallel….

When capacitances C1,C2 ,C3 are connected in parallel , the Equivalent capacitance is

Ceq=C1+C2+C3

Ø Capacitor blocks DC but passes AC

The power P transferred by a capacitance C holding a changing voltage V with charge Q is:
P = VI = CV(dv/dt) = Q(dv/dt) = Q(dq/dt) / C

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INDUCTANCE

Voltage-Current relationship…

V=L dI/dT where (dI/dT)=Change in current

Inductance in series…

When Inductors L1, L2, L3 are connected in series the equivalent inductance

Leq=L1+L2+L3

When two coupled inductances L₁ and L₂ with mutual inductance M are connected in series, the total inductance L_t is:
L_t = L₁ + L₂ ± 2M
The plus or minus sign indicates that the coupling is either additive or subtractive, depending on the connection polarity.

Inductance in parallel...

When inductance L1,L2,L3 are connected in parallel ,their equivalent inductance

1/Leq = (1/L1+1/L2+1/L3)

Leq = (L1*L2*L3)/L1+L2+L3

The power P transferred by an inductance L carrying a changing current I with magnetic linkage Y is:
P = VI = LI(di/dt) = Y(di/dt) = Y(dy/dt) / L

The energy W stored in an inductance L carrying current I with magnetic linkage Y is:
W = LI² / 2 = YI / 2 = Y² / 2L

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Voltmeter Multiplier

The resistance R_S to be connected in series with a voltmeter of full scale voltage V_V and full scale current drain I_V to increase the full scale voltage to V is:
R_S = (V - V_V) / I_V

The power P dissipated by the resistance R_S with voltage drop (V - V_V) carrying current I_V is:
P = (V - V_V)² / R_S = (V - V_V)I_V = I_V²R_S

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Ammeter Shunt

The resistance R_P to be connected in parallel with an ammeter of full scale current I_A and full scale voltage drop V_A to increase the full scale current to I is:
R_P = V_A / (I - I_A)

The power P dissipated by the resistance R_P with voltage drop V_A carrying current (I - I_A) is:
P = V_A² / R_P = V_A(I - I_A) = (I - I_A)²R_P

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Wheatstone Bridge

The Wheatstone Bridge consists of two resistive potential dividers connected to a common voltage source. If one potential divider has resistances R₁ and R₂ in series and the other potential divider has resistances R₃ and R₄ in series, with R₁ and R₃ connected to one side of the voltage source and R₂ and R₄ connected to the other side of the voltage source, then at the balance point where the two resistively divided voltages are equal:
R₁ / R₂ = R₃ / R₄

If the value of resistance R₄ is unknown and the values of resistances R₃, R₂ and R₁ at the balance point are known, then:
R₄ = R₃R₂ / R₁

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Kennelly's Star-Delta Transformation

A star network of three impedances Z_AN, Z_BN and Z_CN connected together at common node N can be transformed into a delta network of three impedances Z_AB, Z_BC and Z_CA by the following equations:
Z_AB = Z_AN + Z_BN + (Z_ANZ_BN / Z_CN) = (Z_ANZ_BN + Z_BNZ_CN + Z_CNZ_AN) / Z_CN
Z_BC = Z_BN + Z_CN + (Z_BNZ_CN / Z_AN) = (Z_ANZ_BN + Z_BNZ_CN + Z_CNZ_AN) / Z_AN
Z_CA = Z_CN + Z_AN + (Z_CNZ_AN / Z_BN) = (Z_ANZ_BN + Z_BNZ_CN + Z_CNZ_AN) / Z_BN

Similarly, using admittances:
Y_AB = Y_ANY_BN / (Y_AN + Y_BN + Y_CN)
Y_BC = Y_BNY_CN / (Y_AN + Y_BN + Y_CN)
Y_CA = Y_CNY_AN / (Y_AN + Y_BN + Y_CN)

In general terms:
Z_delta = (sum of Z_star pair products) / (opposite Z_star)
Y_delta = (adjacent Y_star pair product) / (sum of Y_star)

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Kennelly's Delta-Star Transformation

A delta network of three impedances Z_AB, Z_BC and Z_CA can be transformed into a star network of three impedances Z_AN, Z_BN and Z_CN connected together at common node N by the following equations:
Z_AN = Z_CAZ_AB / (Z_AB + Z_BC + Z_CA)
Z_BN = Z_ABZ_BC / (Z_AB + Z_BC + Z_CA)
Z_CN = Z_BCZ_CA / (Z_AB + Z_BC + Z_CA)

Similarly, using admittances:
Y_AN = Y_CA + Y_AB + (Y_CAY_AB / Y_BC) = (Y_ABY_BC + Y_BCY_CA + Y_CAY_AB) / Y_BC
Y_BN = Y_AB + Y_BC + (Y_ABY_BC / Y_CA) = (Y_ABY_BC + Y_BCY_CA + Y_CAY_AB) / Y_CA
Y_CN = Y_BC + Y_CA + (Y_BCY_CA / Y_AB) = (Y_ABY_BC + Y_BCY_CA + Y_CAY_AB) / Y_AB

In general terms:
Z_star = (adjacent Z_delta pair product) / (sum of Z_delta)
Y_star = (sum of Y_delta pair products) / (opposite Y_delta)

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KIRCHOFF’S LAW

Kirchhoff's Current Law
At any instant the sum of all the currents flowing into any circuit node is equal to the sum of all the currents flowing out of that node:
SI_in = SI_out

Similarly, at any instant the algebraic sum of all the currents at any circuit node is zero:
SI = 0

Kirchhoff's Voltage Law
At any instant the sum of all the voltage sources in any closed circuit is equal to the sum of all the voltage drops in that circuit:
SE = SIZ

Similarly, at any instant the algebraic sum of all the voltages around any closed circuit is zero:
SE - SIZ = 0

AC CIRCUITS

Impedance

The impedance Z of a resistance R in series with a reactance X is:
Z = R + jX

Rectangular and polar forms of impedance Z:
Z = R + jX = (R² + X²)^½Ðtan^-1(X / R) = |Z|Ðf = |Z|cosf + j|Z|sinf

Addition of impedances Z₁ and Z₂:
Z₁ + Z₂ = (R₁ + jX₁) + (R₂ + jX₂) = (R₁ + R₂) + j(X₁ + X₂)

Subtraction of impedances Z₁ and Z₂:
Z₁ - Z₂ = (R₁ + jX₁) - (R₂ + jX₂) = (R₁ - R₂) + j(X₁ - X₂)

Multiplication of impedances Z₁ and Z₂:
Z₁ * Z₂ = |Z₁|Ðf₁ * |Z₂|Ðf₂ = ( |Z₁| * |Z₂| )Ð(f₁ + f₂)

Division of impedances Z₁ and Z₂:
Z₁ / Z₂ = |Z₁|Ðf₁ / |Z₂|Ðf₂ = ( |Z₁| / |Z₂| )Ð(f₁ - f₂)

In summary:
- use the rectangular form for addition and subtraction,
- use the polar form for multiplication and division.

Admittance

An impedance Z comprising a resistance R in series with a reactance X can be converted to an admittance Y comprising a conductance G in parallel with a susceptance B:
Y = Z ^-1 = 1 / (R + jX) = (R - jX) / (R² + X²) = R / (R² + X²) - jX / (R² + X²) = G - jB
G = R / (R² + X²) = R / |Z|²
B = X / (R² + X²) = X / |Z|²
Using the polar form of impedance Z:
Y = 1 / |Z|Ðf = |Z| ^-1Ð-f = |Y|Ð-f = |Y|cosf - j|Y|sinf

Conversely, an admittance Y comprising a conductance G in parallel with a susceptance B can be converted to an impedance Z comprising a resistance R in series with a reactance X:
Z = Y ^-1 = 1 / (G - jB) = (G + jB) / (G² + B²) = G / (G² + B²) + jB / (G² + B²) = R + jX
R = G / (G² + B²) = G / |Y|²
X = B / (G² + B²) = B / |Y|²
Using the polar form of admittance Y:
Z = 1 / |Y|Ð-f = |Y| ^-1Ðf = |Z|Ðf = |Z|cosf + j|Z|sinf

The total impedance Z_S of impedances Z₁, Z₂, Z₃,... connected in series is:
Z_S = Z₁ + Z₁ + Z₁ +...
The total admittance Y_P of admittances Y₁, Y₂, Y₃,... connected in parallel is:
Y_P = Y₁ + Y₁ + Y₁ +...

In summary:
- use impedances when operating on series circuits,
- use admittances when operating on parallel circuits.

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Reactance

Inductive Reactance
The inductive reactance X_L of an inductance L at angular frequency w and frequency f is:
X_L = wL = 2pfL

For a sinusoidal current i of amplitude I and angular frequency w:
i = I sinwt
If sinusoidal current i is passed through an inductance L, the voltage e across the inductance is:
e = L di/dt = wLI coswt = X_LI coswt

The current through an inductance lags the voltage across it by 90°.

Capacitive Reactance
The capacitive reactance X_C of a capacitance C at angular frequency w and frequency f is:
X_C = 1 / wC = 1 / 2pfC

For a sinusoidal voltage v of amplitude V and angular frequency w:
v = V sinwt
If sinusoidal voltage v is applied across a capacitance C, the current i through the capacitance is:
i = C dv/dt = wCV coswt = V coswt / X_C

The current through a capacitance leads the voltage across it by 90°.

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Resonance

Series Resonance
A series circuit comprising an inductance L, a resistance R and a capacitance C has an impedance Z_S of:
Z_S = R + j(X_L - X_C)
where X_L = wL and X_C = 1 / wC

At resonance, the imaginary part of Z_S is zero:
X_C = X_L
Z_Sr = R
w_r = (1 / LC)^½ = 2pf_r
The quality factor at resonance Q_r is:
Q_r = w_rL / R = (L / CR²)^½ = (1 / R )(L / C)^½ = 1 / w_rCR

Parallel resonance
A parallel circuit comprising an inductance L with a series resistance R, connected in parallel with a capacitance C, has an admittance Y_P of:
Y_P = 1 / (R + jX_L) + 1 / (- jX_C) = (R / (R² + X_L²)) - j(X_L / (R² + X_L²) - 1 / X_C)
where X_L = wL and X_C = 1 / wC

At resonance, the imaginary part of Y_P is zero:
X_C = (R² + X_L²) / X_L = X_L + R² / X_L = X_L(1 + R² / X_L²)
Z_Pr = Y_Pr^-1 = (R² + X_L²) / R = X_LX_C / R = L / CR
w_r = (1 / LC - R² / L²)^½ = 2pf_r
The quality factor at resonance Q_r is:
Q_r = w_rL / R = (L / CR² - 1)^½ = (1 / R )(L / C - R²)^½

Note that for the same values of L, R and C, the parallel resonance frequency is lower than the series resonance frequency, but if the ratio R / L is small then the parallel resonance frequency is close to the series resonance frequency.

Three Phase Power

For a balanced star connected load with line voltage V_line and line current I_line:
V_line = Ö3 V_phase
I_phase = I_line
Z_phase = V_phase / I_phase = V_line / Ö3I_line
S_phase = 3V_phaseI_phase = Ö3V_lineI_line = V_line² / Z_phase = 3I_line²Z_phase

For a balanced delta connected load with line voltage V_line and line current I_line:
V_phase = V_line
I_phase = I_line / Ö3
Z_phase = V_phase / I_phase = Ö3V_line / I_line
S_phase = 3V_phaseI_phase = Ö3V_lineI_line = 3V_line² / Z_phase = I_line²Z_phase

Power Triangle

There are Three kind of power i.e

True power is a function of a circuit's dissipative elements, usually resistances (R).

Reactive power is a function of a circuit's reactance (X).

Apparent power is a function of a circuit's total impedance (Z).

The relation between all the three of them are given as

P=True power , Measured in watts (W)

P = (I^2)*R

Q=Reactive power , Measured in Volt-Ampere-Reactive (VAR)

Q=(I^2)*X

S=Apparent power ,Measured in Volt-Amps (VA)

S=(I^2)*Z

TRANSFORMERS

Two types according to construction

1) Core type

2) Shell type

Induced EMF in the primary winding:

E1 = 4.44 * f * N1 * [Bm] * A

= 4.11 * f * N1 * [fi] Where N1=Number of turns in Primary winding

F=Frequency

[fi]=maximum flux in core

A=Area

[fi]=A*[Bm]

Form Factor = R.M.S Value/Average Value = 1.11

Transformation ratio :K=V2/V1

For an ideal two-winding transformer with primary voltage V₁ applied across N₁ primary turns and secondary voltage V₂ appearing across N₂ secondary turns:
V₁ / V₂ = N₁ / N₂
The primary current I₁ and secondary current I₂ are related by:
I₁ / I₂ = N₂ / N₁ = V₂ / V₁

For an ideal step-down auto-transformer with primary voltage V₁ applied across (N₁ + N₂) primary turns and secondary voltage V₂ appearing across N₂ secondary turns:
V₁ / V₂ = (N₁ + N₂) / N₂
The primary (input) current I₁ and secondary (output) current I₂ are related by:
I₁ / I₂ = N₂ / (N₁ + N₂) = V₂ / V₁
Note that the winding current is I₁ through the N₁ section and (I₂ - I₁) through the N₂ section.

For a single-phase transformer with rated primary voltage V₁, rated primary current I₁, rated secondary voltage V₂ and rated secondary current I₂, the volt-ampere rating S is:
S = V₁I₁ = V₂I₂

For a balanced m-phase transformer with rated primary phase voltage V₁, rated primary current I₁, rated secondary phase voltage V₂ and rated secondary current I₂, the voltampere rating S is:
S = mV₁I₁ = mV₂I₂

The primary circuit impedance Z₁ referred to the secondary circuit for an ideal transformer with N₁ primary turns and N₂ secondary turns is:
Z₁₂ = Z₁(N₂ / N₁)²

The secondary circuit impedance Z₂ referred to the primary circuit for an ideal transformer with N₁ primary turns and N₂ secondary turns is:
Z₂₁ = Z₂(N₁ / N₂)²

The voltage regulation DV₂ of a transformer is the rise in secondary voltage which occurs when rated load is disconnected from the secondary with rated voltage applied to the primary. For a transformer with a secondary voltage E₂ unloaded and V₂ at rated load, the per-unit voltage regulation DV_2pu is:
DV_2pu = (E₂ - V₂) / V₂
Note that the per-unit base voltage is usually V₂ and not E₂.

Tests done in a transformer…

1)Open circuit test (OC): Determines the core loss / High voltage left open

2)Closed circuit test (CC):

Losses in a transformer :

1) Core loss (Iron Loss) : Constant loss

1.1) Hysteresis loss: Wh=Bmax^1.6 * f ^ 2

1.2) Eddy Current loss: We=Bamx^2 * f ^ 2

2) Copper loss: I^2 * R loss. It is a variable loss

Voltage regulation