### Solved problems on Thevenin’s Theorem

HomeBasic Electrical Engineering

1) Find current flowing through 1 $\Omega$ resistor.  Calculate $V_{th}$

$130-0.2I-0.2I-110 = 0$

$-0.4I = -20$

$I = 50 A$

$-V_{th} + 110 +0.2I = 0$

$V_{th} = 120 V$ $Calculate R_{th}$

$0.2 || 0.2 = 0.1 \Omega$

0.1 + 0.2 = 0.3 $\Omega$ $R_{th} = 0.3 \Omega$ Thevenin’s equivalent circuit.

$I_{1\Omega} = \dfrac{V_{th}}{R_{th} + R_L} = \dfrac{120}{0.3+1} = 92.3076 A$

### Solved Problems on Superposition Theorem

HomeBasic Electrical Engineering

1) Find Current flowing through $1 \Omega$ resistor by using superposition theorem.  Consider 130 V Voltage source only.

$-0.2I_1-0.2(I_1-I_2)+130 = 0$

$-0.4I_1 +0.2I_2 = -130 \ldots\ldots (1)$

$-I_1-0.2(I_2-I_1)-0.2I_2=0$

$0.2I_1-1.4I_2=0 \ldots\ldots (2)$

$I_1 = 350 A$

$I_2 = 50 A$

$I’_{1\Omega} = 50 A (\downarrow)$ Consider 110 V only.

$-0.2I_1 -0.2(I_1-I_2)-110 = 0$

$-0.4I_1+0.2I_2 = 110 \ldots\ldots(3)$

$-I_2+110-0.2(I_2-I_1)-0.2I_2=0$

$0.2I_1 – 1.4I_2 = -110 \ldots\ldots (4)$

$I_1 = -253.8467 A$

$I_2 = 42.3076 A$

$I’’_{1\Omega} = 42.3076 A (\downarrow)$

$\therefore I_{1\Omega} = I’_{1\Omega} + I’’_{1\Omega} = 50 + 42.3076 = 92.3076 A (\downarrow)$

2) Find current flowing through 5 $\Omega$.  $2-0.1I_1-10I_1-5(I_1-I_2)=0$

$15.1I_1 + 5I_2=-2 \ldots\ldots (1)$

$-20I_1-0.2I_2-5(I_2-I_1)=0$

$5I_1-25.2I_2=0 \ldots\ldots (2)$

$I_1 = 0.1417 A$

$I_2 = 0.02812 A$

$I’_{5\Omega} = I_1-I_2 = 0.1158 A (\downarrow)$ Consider 4 V only.

$-0.1I_1-10I_1-5(I_1-I_2) = 0$

$-15.1I_1+5I_2 = 0 \ldots\ldots (3)$

$20I_2-0.2I_2-4-5(I_2-I_1)=0$

$5I_1-25.2I_2 = 4 \ldots\ldots (4)$

$I_1 = -0.05685 A$
$I_2 = -0.1698 A$

$I’’_{5 \Omega} = I_2-I_1 = 0.1130 A (\downarrow)$

$I_{5 \Omega} = I’_{5\Omega} + I’’_{5\Omega} = 0.2288 A (\downarrow)$

3) Determine current through 1 $\Omega$ resistor.  Consider a 10 V voltage source only.

$-2(I_1-I_2)-2I_1 = 0$

$-4I_1+2I_2=0 \ldots\ldots (1)$

$-10 -2(I_2-I_1)-I_2=0$

$2I_1-3I_2=10 \ldots\ldots (2)$

$I_1 = -2.5 A$

$I_2 =-5 A$

$I’_{1\Omega} = -5 A (\rightarrow)$ Consider a 6 V voltage source only.

$-2I_1-2(I_1-I_2)-6 = 0$

$-4I_1+2I_2=6 \ldots\ldots (3)$

$-I_2+6-2(I_2-I_1)=0$

$2I_1 – 3I_2 =-6 \ldots\ldots (4)$

$I_1 = -0.75 A$

$I_2 =1.5 A$

$I’’_{1\Omega} = 1.5 A (\rightarrow)$ Consider 8 V only.

$8-2I_1-2(I_1-I_2)=0$

$-4I_1+2I_2=-8 \ldots\ldots (5)$

$-I_2-2(I_2-I_1)=0$

$2I_1-3I_2=0 \ldots\ldots (6)$

$I_1 = 3 A$

$I_2 = 2 A$

$I’”_{1\Omega} = 2 A (\rightarrow)$

$Total \; \; current \; \; I_{1 \Omega}=I’_{1 \Omega}+I’’_{1 \Omega}+I’’’_{1 \Omega}$

$I_{1 \Omega}= 2+1.5+(-5) = -1.5 A (\rightarrow)$

$I_{1 \Omega} = 1.5 A (\leftarrow)$

4) Find current through 20 $\Omega$ resistor.  For Mesh 1:

$10-10I_1+10I_2=0$

$-10I_1+10I_2=-10 \ldots \ldots(1)$

For Mesh 2:

$-10I_2+10I_1-I_2-20I_2=0$

$10I_1-31I_2=0 \ldots\ldots(2)$

$I_1 = 1.4761 A$

$I_2 = 0.4761 A$

$I’_{20\Omega} = 0.4761 A (\downarrow)$ Consider 8 V voltage source only.

$-I-8-20I=0$

$-21I = 8$

$I’’_{20\Omega} = -0.3809 A (\downarrow)$ Consider a 12 V voltage source only.

$-12-8I_1+8I_2=0$

$-8I_1+8I_2=12 \ldots\ldots(3)$

$-I_2-8I_2+8I_1-20I_2=0$

$8I_1-29I_2=0 \ldots\ldots(4)$

$I_1=-2.0714 A$

$I_2=-0.5714 A$

$I’’’_{20\Omega} = -0.5714 (\downarrow)$

$I_{20\Omega} = I’_{20\Omega}+I’’_{20\Omega}+I’’’_{20\Omega}$

$I_{20\Omega} = 0.4761+(-0.3809)+(-0.5714)$

$I_{20\Omega} = -0.4762 A (\downarrow)$

$I_{20\Omega} = 0.4762 A (\uparrow)$

### Single Phase Transformer Solved Problems

HomeBasic Electrical Engineering

1)A single phase 90 kVA, 3.2 kV/220 V, 50Hz transformer has 89$\%$ efficiency at unity power factor both at full load and half load. Determine the efficiency at 70$\%$ of full load and 0.8 power factor.

$\eta_{100\%}=\eta_{50\%}$

$\dfrac{VIcos\phi}{VIcos\phi+P_i+P_{cu}}=\dfrac{xVIcos\phi}{xVIcos\phi+P_i+x^2P_{cu}}$

$\dfrac{VIcos\phi}{VIcos\phi+P_i+P_{cu}}=\dfrac{0.5VIcos\phi}{0.5VIcos\phi+P_i+0.5^2P_{cu}}$

$P_i+(0.5)^2P_{cu}=0.5P_i+0.5P_{cu}$

$P_i-0.5P_i=0.5P_{cu}-(0.5)^2P_{cu}$

$P_{cu}=2P_i$

Efficiency at $100\%$

$\eta_{100\%}= \dfrac{VIcos\phi}{VIcos\phi+P_i+P_{cu}}$

$\eta_{100\%}= \dfrac{VIcos\phi}{VIcos\phi+3P_i}$

$\eta_{100\%}= \dfrac{90\times 10^3}{90\times 10^3+3P_i}$

$P_i = 3707.8651 W$

$P_{cu} =2 P_i = 7415.7302 W$

Now efficiency at $70\%$ of full load

$\eta_{70\%} = \dfrac{xVIcos\phi}{xVIcos\phi+P_i+x^2P_{cu}}$

$\eta_{70\%} = \dfrac{0.7\times 90\times 10^3 \times 0.8}{0.7\times 90\times 10^3 \times 0.8+3707.8651+0.7^2 \times 7415.7302}$

$\eta_{70\%}$= 87.28$\%$

### solved problems on electrostatics

HomeBasic Electrical Engineering

## SOLVED PROBLEMS ON CAPACITOR

1) The plate area of parallel plate capacitor is 0.01 $m^2$. The distance between the plate is 2.5 cm. The insulating medium is air. Find capacitance, if the space between the plates is filled with an insulating material of relative permittivity 5?

$C = \dfrac{\epsilon_0 \epsilon_r A }{d}$

$C = \dfrac{8.854 \times 10^{-12} \times 1 \times 0.01}{2.5 \times 10^{-1}}$

C = $3.5416 \times 10^{-12}$ F

for $\epsilon_r = 5$
$C = \dfrac{\epsilon_0 \epsilon_r A }{d}$

$C = \dfrac{8.854 \times 10^{-12} \times 5 \times 0.01}{2.5 \times 10^{-1}}$

$C = 1.7708 \times 10^{-11}$

2) A parallel plate capacitor has a plate area of 500 $cm^2$, a plate separation of 0.15 mm and air as a dielectric. Find the capacitance. Also find the voltage between the plates the electric flux density and electric field strength, if the capacitor has a charge of 2 $\mu$ C.

$C = \dfrac{\epsilon_0 \epsilon_r A }{d}$

$C = \dfrac{8.854 \times 10^{-12} \times 1 \times 500 \times 10^{-4}}{0.15 \times 10^{-3}}$

$C = 2.9513 \times 10^{-9} F$

$Q = CV$

$V = \dfrac{Q}{C}$

$V = \dfrac{2 \times 10^{-6}}{2.9513 \times 10^{-9} }$

$V = 677.6674$ V

$D = \dfrac{2 \times 10^{-6}}{500 \times 10^{-4}}$

$D = 4 \times 10^{-5} \; C/m^2$

$E = \dfrac{V}{d}$

$E = \dfrac{677.6674}{0.15 \times 10^{-3}}$

$E = 4517782.667 V/m$

3) Two parallel metal discs, each 100 mm in diameter, are spaced 1 mm apart, the dielectric being air. Calculate the electric field intensity and capacitance.

$A = \dfrac{\pi d^2}{4}$

A = 0.007854 $m^2$

$E = \dfrac{V}{d} = \dfrac{10^3}{10^{-3}}$

$E = 10^6$

$C = \dfrac{\epsilon_0 \epsilon_r A }{d}$

$C = \dfrac{8.854 \times 10^{-12} \times 1 \times 0.007854}{10^{-3}}$

$C = 6.9539 \times 10^{-11} F$

4) The plate area of a parallel plate capacitor is 0.01 $m^2$. The distance between the plates is 2.5 cm. The insulating medium is air. Find its capacitance. What would be its capacitance, if the space between the plates is filled with an insulating material of relative permittivity 5?

$C = \dfrac{\epsilon_0 \epsilon_r A }{d}$

$C = \dfrac{8.854 \times 10^{-12} \times 1 \times 0.01}{2.5 \times 10^{-2}}$

$C = 3.5416 \times 10^{-12} F$

For $\epsilon_r = 5$

$C = \dfrac{\epsilon_0 \epsilon_r A }{d}$

$C = \dfrac{8.854 \times 10^{-12} \times 5 \times 0.01}{2.5 \times 10^{-1}}$

$C = 1.7708 \times 10^{-11} F$

5) A 5 $\mu$F capacitor is charged to a potential difference of 110 V and then connected in parallel with an uncharged 3 $\mu$F capacitor. Calculate the potential difference across the parallel capacitor.

$Q_1 = C_1 \times V_1 = 5 \times 10^{-6} \times 110 = 5.5 \times 10^{-4} C$

Both the capacitors are connected in parallel. Therefore voltage across both the capacitor will remain same.

$V_1 = V_2$

$\dfrac{q_1}{C_1} = \dfrac{q_2}{C_2}$

Now capacitor $C_2$ is uncharged, therefore the charge will flow from $C_1$ to $C_2$ and this flow of charge will stop when the potential difference across $C_1$ and $C_2$ will become equal i.e. $V_1 = V_2$. Capacitor $C_1$ will loose charge say $q_2$ and capacitor $C_2$ will gain that charge $q_2$. Hence new charge on $C_1$ will become $q_1$.

$\therefore$ Charge on $C_1$ will be $q_1 = Q_1 – q_2$

$\dfrac{Q_1 – q_2}{C_1} = \dfrac{q_2}{C_2}$

$\dfrac{(5.5 \times 10^{-4}) – q_2}{5 \times 10^-6} = \dfrac{q_2}{3 \times 10^-6}$

$q_2 = 2.06525 \times 10^{-4}$

$V_1 = V_2 = \dfrac{q_2}{C_2} = 68.75 V$

Method 2

$Q_1 = C_1 \times V_1 = 5.5 \times 10^{-4} C$

When the capacitors are connected in parallel, the total capacitance is 8 $\mu F$ and the charge of 0.01 C is divided between the two capacitor.

$Q = CV$

$V = \dfrac{Q}{C}$

$V = \dfrac{5.5 \times 10 ^{-4}}{8 \times 10^{-6}}$

$V = 68.75 V$

6) A 2 $\mu F$ capacitor is charged to 100 V and 3 $\mu F$ capacitor to 200 V.  After disconnecting the supply capacitor terminal of similar polarity are connected together calculate the potential difference between the terminal.

Initial charge on each capacitor.

$Q_{1i} = C_1V_1 = 2 \times 10^{-6} \times 100 = 2 \times 10^{-4} C$

$Q_{2i} = C_2V_2 = 3 \times 10^{-6} \times 100 = 6 \times 10^{-4} C$

Now, $V_1 = V_2$

$\dfrac{Q_{1i} + Q}{C_1} = \dfrac{Q_{2i} – Q}{C_2}$

$\dfrac{2 \times 10^{-4} + Q}{ 2 \times 10^{-6}} = \dfrac{6 \times 10^{-4} – Q}{ 3 \times 10^{-6}}$

$Q = 1.2 \times 10^{-4} C$

$V_1 = V_2 = \dfrac{Q_{1i} + Q}{C_1} = \dfrac{( 2 \times 10^{-4} )+(1.2 \times 10^{-4})}{ 2 \times 10^{-6} } = 160 \;V$

Note:  In the above numerical $Q_{1i} < Q_{2i}$, therefore after connecting similar terminals of the capacitors capacitor $C_2$will lose some charge and $C_1$ will gain some charge

## CHARGING AND DISCHARGING OF CAPACITOR

1) A capacitor having capacitance of 4 $\mu F$ is connected in series with a resistance of 1 M$\Omega$ across 200 V DC supply. Find – 1) time constant 2) The initial charging current 3) The time taken by capacitor to raise up to 160 V.

$\tau = RC = 1 \times 10^{6} \times 4 \times 10^{6} = 4 sec$

$I_0 = \dfrac{V}{R} = \dfrac{200}{10^6} = 200 \times 10^{-6} A$

$V_C = V (1 – e^{\frac{-t}{\tau}})$

$160 = 200(1-e^{\frac{-t}{4}})$

$t = 6.4377 sec$

### KVL problems with solutions / Solved problems on Kirchhoff’s Voltage Law (KVL) / Mesh Analysis

Get KVL problems with solutions. Get solved examples on Kirchhoff’s Voltage law. This problem solving technique is known as Mesh Analysis.

## Magnetic circuit with an air gap examples

1) An iron ring has its mean length of flux path as 60 cm and its cross-sectional area as 15 cm². If relative permeability is 500. Find the current required to be passed through a coil of 300 turns wound uniformly around it to produce a flux density of 1.2 T. What would be the flux density with the same current if the iron ring is replaced by an air core.

Given: l = 60 cm; a = 15 cm²; $\mu_r = 500$ ; N = 300; B=1.2 T.

$MMF(F) = \dfrac{Bl}{\mu_0 \mu_r}$

$F = \dfrac{1.2 \times 60 \times 10^{-2}}{4\pi \times 10^{-7}\times 500}$

F = 1145.9155 AT

If iron ring is replaced by air core.

$MMF(F) = \dfrac{B_{air}l}{\mu_0 \mu_r}$

$B_{air} = 2.4 \times 10^{-3} T$

2) A coil of 600 turns and resistance of 10 $\Omega$ is wound uniformly over a steel ring of mean circumference of 30 cm and cross sectional area of 9 $cm^2$. It is connected to a supply of 20 V if the relative permeability of the ring is 1600. Find 1) Reluctance 2) Magnetic field intensity 3) MMF 4) Flux.

i) Reluctance:

$S = \dfrac{l}{\mu_0 \mu_r a}$

$S = \dfrac{30 \times 10^{-2}}{4 \times 10^{-7} \times 1600 \times 9 \times 10^{4}}$

$S = 165786.3991 \; AT/Wb$

ii) Magnetic field intensity:

$I = \dfrac{V}{R}$

$I = \dfrac{20}{10}$

$I = 2 \; A$

$H = \dfrac{NI}{l}$

$H = \dfrac{600 \times 2}{30 \times 10^{-2}}$

$H = 4000 AT/m$

iii) MMF = NI = 600 $\times$ 2 = 1200 AT.

iv) Flux:

$\phi = \dfrac{600 \times 2}{165786.3991}$

$\phi = 7.2382 \times 10^{-3} \; Wb$

3) A ring shaped core is made up of two parts of the same material. Part one is a magnetic path of length 25 cm and with cross sectional area $4 cm^2$, whereas part two is of length 10 cm and cross section area of 6 $cm^2$. The flux density in part two is 1.5 T. If the current through the coil, wound over core is 0.5 A. Calculate the number of turns of coil. Assume $\mu_r$for material is 1000.

Given: $l_1$ = 25 cm; $l_2$ = 10 cm;$a_1$= 4 $cm^2$; $a_2$= 6 $cm^2$; B = 1.5 T; $\mu_r$= 1000

$S = S_1 + S_2$

$S = \dfrac{l_1}{\mu_0 \mu_r A_1} + \dfrac{l_2}{\mu_0 \mu_r A_2}$

$S = \dfrac{1}{4\pi \times 10^{-7} \times{10^3}} \left[ \dfrac{0.25}{4\times 10^{-4}} + \dfrac{0.1}{6\times 10^{-4}} \right]$

S = 629988.3164 AT/Wb.

$B_2 = \dfrac{\phi}{A_2}$

$\phi = 1.5 \times 6 \times 10^{-4} = 9 \times 10^{-4} Wb.$

$F = \phi \times S$

$NI = \phi \times S$

$N = \dfrac{S \phi}{I} = \dfrac{629988.3164 \times 9 \times 10^{-4}}{0.5}$

N = 1133.98 = 1134 T.

4) An iron ring of mean circumference of 150 cm and cross sectional area 12 $cm^2$ is wound with 600 turns of coil. The coil produces flux of 1.25 mWb while carrying a current of 2A. Find the relative permeability of iron.

Given: l = 150 cm; a = 12 $cm^2$; N = 600 turns; $\phi$ = 1.25 mWb; I = 2A.

$F = \phi S$

$S = \dfrac{NI}{\phi}$

$S = \dfrac {600 \times 2}{1.25 \times 10^{-3}}$

S = 960000 Wb/AT

$S = \dfrac{l}{\mu_0 \mu_r a}$

$\mu_r = \dfrac {150 \times 10^{-2}} {4 \times \pi \times 10^{-7} \times 12 \times 10^{-4} \times 960000}$

$\mu_r = 1036.1649$

5. An iron ring with a mean diameter of 25cm and relative permeability of 1000 is uniformly wound with 500 turns. Find the current required to produce a flux density of 1 T in the ring. If an air gap of 1 mm is cut in the ring Calculate the new value of current to maintain the same flux density in the ring.

Given:- diameter(d)=25cm; $\mu_r$=1000; N=500; B=1 T; $l_g$=1mm

$l=\pi \times d$

$=\pi \times 25 \times 10^{-2}$

=0.7853 m

i)$F=\dfrac{Bl}{\mu_0 \mu_r}$

$NI= \dfrac{Bl}{\mu_0 \mu_r}$

$I =\dfrac{1 \times 0.7853}{4 \times \pi \times 10^{-7} \times 1000 \times 500}$

I= 1.2493 A

ii) with an air gap

$F_t =\dfrac{Bl}{\mu_0 \mu_r}$ + $\dfrac{B l_g}{\mu_0 \mu_r}$

$F_t$ = $\dfrac {1 \times 0.7853}{4 \times \pi \times 10^{-7} \times 1000}$ + $\dfrac{1 \times 1 \times 10^{-3}}{4 \times \pi \times 10^{-7} \times 1}$

$F_t$ = 1420.6965 AT

$F_t = NI$

$1420.6965=500 \times I$

I= 2.8413A

## Self Inductance and Mutual Inductance problems

1) Two cols A and B have self-inductance of 10 $\mu$H and 40$\mu$H respectively. A current of 2A in coil A produces a flux linkage of 5$\mu$Wb- turns in coil B. Calculate i) Mutual inductance between the coils. ii) Coefficient of coupling. iii) Average EMF induced in coil B if the current 1A in coil A is reversed at uniform rate in 0.1 seconds.

i) $M = \dfrac{N_2 \phi_2}{I_1}$
$M = \dfrac{ 5 \times 10^{-6}}{2}$=0.0000025 H

ii) $K = \dfrac{M}{\sqrt{L_1 L_2}}$ = 0.125

iii) $e_2 = -M \dfrac{dI_1}{dt}$
$e_2 = -0.0000025 \times \dfrac {-1-1}{0.1}$
$e_2 = 50 \times 10^{-6}$ V

2. A coil of 500 turns is uniformly wound on an iron ring of mean circumference 25 cm having an area of cross-section is 15 $cm^2$. When the coil carries a current of 1 A, produces a flux density of 0.8 T. Calculate i) Magnetizing force (H) ii) Flux iii) Inductance iv) Relative permeability of iron.

Given:- N=500 Turns; l=25 cm; a=15$cm^2$; I= 1 A; B=0.8 T

i) $H= \dfrac{NI}{l}=\dfrac{500 \times 1}{25 \times 10^{-2}}$
H= 2000 AT/m

ii) $\phi = B \times a$
$= 0.8 \times 15 \times 10^{-4}$
= 0.0012 Wb.

iii) $L = \dfrac {N \phi}{I}$

$= \dfrac {500 \times 0.0012}{1}$
$= 0.6 H$

iv) $\mu_r = \dfrac{B}{\mu_0 H}$

$=\dfrac {0.8}{4 \times \pi \times 10^{-7} \times 2000}$
= 318.3098

3. An iron ring of mean diameter 20 cm has a square area of a cross-section of 2 cm $\times$ 2cm & is uniformly wound with 600 turns. The relative permeability of iron is 1000. Calculate i) Self-inductance of coil ii) If the permeability of iron is doubled, find the new value of inductance.

Given:- diameter (d) = 20 cm; N=600 Turns; $\mu_r$=1000; a=2 $\times$ 2 cm= 4 $cm^2$

$l= \pi \times d=0.6283 m$

i) $L=\dfrac{\mu_0 \mu_r a N^2}{l}$

$L = \dfrac{4 \times \pi \times 10^{-7} \times 1000 \times 4 \times 10^{-4} \times 600^2}{0.6283}$

L = 0.288 H

ii) $L \propto \mu_r$

$\therefore \; \dfrac{L_{new}}{L}=\dfrac {\mu_{r new}}{\mu_r}$

$L_{new}= L \times \dfrac{\mu_{r new}}{\mu_r}$

$=0.288 \times \dfrac{2000}{1000}$

=0.576 H

4. Two coils A and B have self-inductance of 120 $\mu H$ and 300 $\mu H$ respectively. A current of 2 A in coil A, produces flux linkage of 200 $\mu$ Wb-turns in coil B. Calculate i) Mutual inductance ii) Coefficient of coupling K, and iii) Average EMF induced in coil B, when the current in coil A is switched off in 0.05 sec.

Given:- $L_A = 120 \mu H$; $L_B = 300 \mu H$; $I_A = 2A$; $N_B \phi_B = 200 \mu Wb$;

i) $M=\dfrac{N_B\phi_B}{I_A}$
$M=\dfrac{200 \times 10^{-6}}{2}$
$M=100\times 10^{-6} H$

ii) $K=\dfrac{M}{\sqrt{L_A L_B}}$

$K= \dfrac{100 \times 10^{-6}}{\sqrt{120 \times 10^{-6}\times 300 \times 10^{-6}}}$

K=0.5270

iii) $e_B = M \dfrac{dI_A}{dt}$

$e_B = 100 \times 10^{-6} \times \dfrac{2}{0.05}$

$e_B= 4 \times 10^{-3}$ V