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UNIT I - CIRCUIT ANALYSIS TECHNIQUES

PART A

1. Define active and passive element.

The elements which can deliver energy are called active elements. E.g.: Voltage source and current source. The elements which consume energy either by absorbing or storing are called passive elements. E.g. Resistor, Inductor and Capacitor

2. How much energy does a 100W electric bulb consume in two hours?

Power = Energy/Time; Energy = P x t = 100*2*3600 = 720000 = 720 KJ

3. Define series and parallel connection.

If two or more elements are connected such that the current through them are same, then the connection is called series connection. If two or more elements are connected such that the voltage across them is same then the connection is called parallel connection.

4. Define i) charge ii) electric current iii) power iv) network & v) circuit.

i) Charge: Charge is an electrical property of the atomic particles of which matter consists, measured in coulombs(C).

ii) Electric current is the time rate of change of charge, measured in amperes (A). i = dq/dt

A direct current (DC) is a current that remains constant with time.

An alternating current (AC) is a current that varies sinusoidally with time

iii) Power is the time rate of expending or absorbing energy, measured in watts (w).

p = dw/dt ;p- Power in watts (w); w- Energy in joules (J); t - Time in seconds (S)

(or) p = v i ; v - Voltage in volts(V); i - Current in amperes (A)

iv) Network: The inter connection of two or more simple circuit elements forms an electrical network.

v) Circuit: If the network contains at least one closed path, it is an electric circuit.

5. A stove element draws 15 A when connected to a 120V line. How long does it take to consume 30KJ?

Time = Energy , t = w = 30*103 = 16.67 s

Power p 120*15

6. What is the difference between circuit and network?

An electrical circuit is a collection of electrical components which accomplish a specific task such as heating, lighting or running a motor. This collection may or may not form a complete topological loop, depending on whether it is presently connected to power, integrated into a larger device or circuit, or damaged. Sometimes, it is convenient to speak of an electrical circuit as a network, de-emphasizing the return path. Return paths are sometimes omitted from circuit diagrams, making the resulting graphic visually resemble a network topology rather than some sort of loop topology.

7. Draw the characteristics of ideal and practical voltage sources.

[pic]

8. Draw the characteristics of ideal and practical current sources.

9. State Ohm’s law.

Ohm’s law states that, at constant temperature, current passing through the conductor is directly proportional to potential difference between two ends of the conductor.

i.e., i α v , i = V/R, where R is the resistance (Ω).

10. A voltage source of 20 sin π t v is connected across a 5 k Ω resistor .Find the current through the resistor and the power dissipated.

i = v/R= 20 sin πt/5*103 = 4 sin πt ,mA ; P = v i = 80 sin2 πt ,mW

11. State Kirchhoff’s current and voltage laws.

KCL (Kirchoff’s Current Law) states that the algebraic sum of currents entering a node (or a closed boundary) is zero.

KVL (Kirchhoff’s Voltage Law) states that the algebraic sum of all voltages around a closed path (or loop) is zero.

12. Give the voltage - current relations for i) resistance ii) inductance and iii) capacitance.

i) Resistance,R: v=iR

ii) Inductance, L: v = L di/dt

iii) Capacitance, C: v=1/C ∫i dt

13. Find V3 and its polarity if the current in the circuit of Fig.3 is 0.40 A.

[pic]

Assume that V3 has the same polarity as V1

Applying KVL and starting from the lower left corners,V1-5I-V2-20I+V3 = 0;

50-2-10-8+V3 = 0;V3 = -30V;Terminal b is positive with respect to terminal a.

14. Find the voltage between A& B in a voltage divider network shown in Fig.

[pic] Voltage between A & B,VAB= (5+4) *100/ (5+4+1) = 90V

15. Determine the equivalent inductance of the three parallel inductors shown in Fig.

[pic]

16. Convert the voltage sources shown in fig to current source.

[pic]

E = 20v, Rs = 5 Ω; Is = E/R = 20/5 =4 A;

17. What will be the inductance of the coil with 1000turns while carrying a current of 2A and producing a flux of 0.5mwb?

N= 1000, φ = 0.5mwb, I = 2A; Inductance, L = N(/I = 1000 X 0.5 X 10-3/2= 0.25H

18.A steady current of 3A flows through an inductance of 0.2 H. What will be the energy stored in the inductance?

I = 3A, L = 0.2H; Energy stored in the inductance, W = LI2/2 = 0.2 X 32/2 = 0.9 J

19. What will be the length of the copper rod having a cross – section of 1cm2 and a resistance of 1ohm? Take resistivity of copper as 2 x 10-8 ohm –m

[pic] [pic]

20. What is the limitation of super position theorem?

Super position theorem can be applied for finding the current through or voltage across a particular element in a linear circuit containing more than two sources. But this theorem cannot be used for the calculation of the power.

21. A 100microfarad capacitance is charged to a steady voltage of 500V. What is the energy stored in the capacitance?

Energy stored in capacitance, E = CV2/2 = 100 x 10-6x 5002/2 = 12.5 J*

22. A 10A current source has a source resistance of 100ohm. What will be the equivalent voltage source? V =IR = 10 x 100 = 1000 V

[pic]

23. What is the equivalent resistance across A – B in the network shown in fig.

[pic]

24. What will be the equivalent inductance across A – B in the network shown in fig.

25. What will be the equivalent capacitance across A – B in the network given below? [pic]

26. Determine the currents I1 and I2 in the circuit as shown in figure.

[pic]

27. State Substitution theorem.

The substitution theorem states that any impedance branch of a circuit can be substituted by a new branch without disturbing the voltages and current in the entire circuit provided the new branch has same set of terminal voltage and current as that of original circuit.

28. Where and why maximum power transfer theorem is applied.

In a certain applications it is desirable to have a maximum power transfer from source to load. The maximum power transfer to load is possible only if the source and load has matched impedance. E.g.: TV/Radio receiver

29. What are the limitations of Thevenin’s Theorem?

The limitations of Thevenin’s theorem are,

1. Not applicable to the circuits consisting of nonlinear elements.

2. Not applicable to unilateral networks.

3. There should no be magnetic coupling between the load and circuit to be replaced by Thevenin’s theorem.

4. In the load side, there should not be controlled sources, controlled from some other part of the circuit.

30. State Maximum power transfer theorem.

For a given Thevenin’s equivalent circuit, maximum power transfer occurs when RL = RTH, that is, when the load resistance is equal to the thevenin’s resistance.

31. What is the condition to obtain maximum power when an ac source with internal impedance is connected to a load with variable resistance and variable reactance?

Maximum power transferred from source to load, when the impedance is equal to complex conjugate of source impedance.

32. Determine the voltages V1 and V2 in the circuit shown in fig.

[pic]

33. State Norton's theorem.

Norton’s theorem states that any circuit with voltage sources, resistances (impedances) and open output terminals can be replaced by a single current source in parallel with single resistance (impedance), where the value of current source is equal to the current passing through the short circuit output terminals and the value of the resistance (impedance) is equal to the resistance seen into the output terminals.

[pic]

34. Write down the formulae for converting Star to Delta.

[pic];[pic];[pic]

35. Write down the formulae for converting Delta to Star.

[pic] ;[pic] ;[pic]

36. A battery of 120 volts having internal resistance of 2 ohm supplies a load resistor RL through a resistance of 1 ohm resistor in series. Find the value of RL so that the power delivered is maximum.

According to maximum power transformer maximum power will be delivered hen the load resistance equals internal resistance.

Hence RL = internal resistances = 1+2 = 3 Ω.

37. How are the following affected by change of frequency of source voltage?(a)resistance

(b) Inductive reactance?(May 2011)

Resistance won`t change due to change of frequency. Inductive reactance increases or

Decreases by increasing or decreasing the frequency.

38. Draw the thevenin’s equivalent circuit for the given circuit.

[pic][pic]

39. A battery of 120 volts having internal resistance of 2 ohm supplies a load resistor RL through a resistance of 1 ohm resistor in series. Find the value of RL so that the power delivered is maximum.

According to maximum power transformer maximum power will be delivered hen the load resistance equals internal resistance. Hence RL = internal resistances = 1+2 = 3 Ω.

40. Find the Thevenin’s equivalent resistance for the given circuit.

[pic]

41. Write the voltage division rule, for the two different resistors connected in series to a voltage

Source. (May 2011)

A resistive divider is a special case where both impedances, Z1 and Z2, are purely resistive. As in the

general case, R1 and R2 may be any combination of series/parallel resistors.

[pic]

PART – B

1) A 18Ω resistor is connected in parallel with a series combination of resistor of 10 Ω and 26 Ω. If the drop across the 10 ohms is 50V, find the total applied voltage and the total current.

2) Reduce the network of Fig.8 into a single resistance.

3) For the circuit shown in Fig.10, Find a) V1 and V2 b) the Power dissipated in the 3KΩ and 20KΩ resistors and c) the power supplied by the current source.

4) In the circuit shown in Fig.11, determine vx and the power absorbed by the 12Ω resistor.

5) Derive the expression for the energy stored in the capacitor and Inductor.

6) (a) Derive the expression for converting from Star connected resistance to Delta.

(b)If resistance in each branch of delta is 30Ω, 25Ω and 40Ω, find the resistance in each branch of star

7) Determine the power dissipated by 5 ohm resistor in the circuit shown in fig below.

[pic]

8) Using source transformation, find the power delivered by the 50V voltage source in the circuit . [pic]

9) By using source transformation, source combination resistance combination converts the circuit shown in fig. into a single voltage source and single resistance.

[pic]

10) Find the current through the 23Ω resistance using superposition theorem.

[pic]

11(a) Find the current through the 3Ω resistor by super position theorem (8)

[pic]

11(b) Find the value of R if it receive the maximum power and also find its maximum power

[pic]

12) Find the power dissipated in R in the circuit shown using Norton theorem, if the value of R is 12Ω.

13) Find the power dissipated in R in the circuit shown using thevenin’s theorem, if the value of R is 20Ω. [pic]

14(a) Find the equivalent resistance between C and B

[pic]

14(b) Find the equivalent resistance between A and B

[pic]

15 (a) (i). For the circuit of figure below, find voltage drop across X-Y terminals. (May 2011).

(ii) Using Kirchhoff’s laws, calculate the current through 6Ω resistance, for the circuit shown below. (May 2011)

15 (b) (i) Using Thevenin`s theorem find the current passing through 10 Ω resistor in the circuit shown below. (May 2011)

(ii)Obtain the Norton`s equivalent circuit for the network shown below and find the current

through AB( .(May 2011)

UNIT II- TRANSIENT RESONANCE IN RLC CIRCUITS

PART - A

1. Define the term time constant of a circuit (May 2011)

In a circuit in which the current is increasing to a final steady value, the time (T) taken to reach 63.2% of the final value is called the time constant of the circuit.

2. Define time constant of a decaying circuit.

For a decaying circuit, the time constant is defined as the time required reaching 36.8% of the initial value.

3. Define transient state and transient time.

In a network containing energy storage elements, with change in excitation the currents and voltage change from one state to another state. The behaviour of the voltage or current when it is changed from one state to another state is called the transient state. The time taken for the circuit to change from one steady state to another steady state is called the transient time.

4. Write down the voltage equation of a series RLC transient circuit excited by a dc source, E.

Applying KVL to the circuit, the voltage equation becomes,[pic]

5. What is the quality factor of a series RC circuit?

Q = ωnL/R = 1 /R( L/C

6. Define damping ratio. Give the damping ratio of RLC series circuit.

[pic];

For RLC series circuit, [pic]

7. Give the natural frequency [pic]and damped frequency β of a series RLC circuit.

Natural frequency [pic];Damped frequency [pic]=[pic]

8. Write the condition for different cases of damping in a series RLC circuit.

If damping ratio, ( = 1, it corresponds to critical damping;( >1, it corresponds to over damping &( < 1, it corresponds to under damping.

9. A DC voltage is applied to a series RL circuit by closing a switch. The voltage across

L is 100 volts at t=0 and drops to 13.5 volts at t = 0.02 sec. If L = 0.1 H, find the value of R.

eL = E e-Rt/L ;At t = 0, eL = E e-0 = E = 100;At t = 0.02, eL =100 E e-0.02R/0.1 = E = 13.5 ;100e-0.2 R = 13.5

Taking natural logarithm on both sides,ln e-0.2R = ln 0.135; -0.2 R = - 2; R = 10 Ω.

10. Write down the voltage equation of a series RLC circuit excited by a source.

[pic]m sin ωt

11. Define the following terms i) sinusoid ii) Time period iii) frequency iv) phasor

• A sinusoid is a signal that has the form of the sine or cosine functions

• The time taken for any wave to complete one full cycle is called the time period (T).

• The frequency of the wave is defined as the number of cycles that a sine wave completes in one second.

• A phasor is complex number that represents the amplitude and phase of a sinusoid.

12. Find the amplitude, phase, time period and frequency of the sinusoid. v(t)=12cos(50t+15º)

We know,v(t)=Vmcos(ω t+Ф); v(t)= 12cos(50t+15º); ω = 50 rad/sec; Amplitude, Vm=12V ; Phase=15º; Time period=2π/ω = 2π/50 = 0.1257 sec; Frequency = 1/T = 7.958Hz.

13. Define RMS value of a sinusoidal current.

The effective or RMS value of an alternating current is given by the steady current (DC) which, when flowing through a given circuit for a given time, produces the same amount of heat as produced by the alternating current which when flowing through the same circuit for the same time.

14. A 200volt ,50Hz source supplies a series RC circuit R=30ohms and C=79µF.Find a)the impedance, b)the current, c)power factor d)power

a) Z = ((R2+XC2) = ((302+402) = 50 ohms;Xc = 1/2πfC = 40Ω;b) I = E/Z = 200/50 = 4A

c) p.f = cos Ф = R/Z = 30/50 = 0.6;d) Power = EI cos Ф = 200×4×0.6 = 480 watts.

15. Find the reactance of a 0.2H inductor at 50Hz frequency, At what frequency is the reactance 500 0hms.

XL=2πfL=2π×50×0.2=62.8318 0hms.When XL=500Ω, f’=?;500=2π f’L;

f’=500/(2π L)=500/(2π ×0.2)=397.887 Ω

16. Define Lumped circuit.

The circuits in which the elements are separated physically like resistors, capacitors and inductors.

17. Define Resonance.

Resonance is defined as a phenomenon in which applied voltage and resulting current are in-phase. In other words, an AC circuit is said to be in resonance if it exhibits unity power factor condition, that means applied voltage and resulting current are in phase.

18. What is a resonant frequency?

The frequency at which resonance occurs is called resonant frequency i.e. X L=XC.

19. .What is a parallel resonance?

The parallel circuit is said to be in resonance, when the power factor is unity. This is true when the imaginary part of the total admittance is zero.

20. Define Q - factor or Figure of Merit, Q.(May 2011)

The quality factor, Q of a resonant circuit is the ratio of its resonant frequency to its bandwidth.

The Q - factor of a circuit can also be defined as,

Q = [pic]

21. What are the resonant conditions?

i) The total impedance Z is minimum and equal to R.;ii) The circuit will be purely resistive circuit.

iii) Power factor of the circuit is unity;iv) Circuit element, I max = V/R; v) Power at resonance, Pr = I2R.

22. What is the series resonance?

The inductive reactance increases as the frequency increases (XL=ωl) but the capacitive reactance decreases with frequency (XC=1/ωc). Thus inductive and capacitive reactances have opposite properties. So, for any LC combination there must be one frequency at which XL =XC. This case of equal and opposite reactance is called series resonance.

23. Define Bandwidth, selectivity, half power frequencies

The difference between the half power frequencies f1 and f2 at which power is half of its maximum is called bandwidth; B.W= f2-f1

It can be observed that at two frequencies f1 and f2 the power is half of its maximum value. These frequencies are called half power frequencies. Out of the two half power frequencies, the frequency f2 is called upper cut-off frequency while the frequency f1 is called lower cut-off frequency.

The selectivity is defined as the ratio of the resonant frequency to the bandwidth.

Selectivity = fr/B.W

24. Show that in a series RLC circuit, f1f2 = fr2 where fr is the resonant frequency and f1, f2 are the half power frequencies.

[pic];[pic];[pic]Hence,[pic]

25. What are the classifications of tuned circuits and state the applications of it.

1. Single tuned circuits;2.Double tuned circuits

Double tuned circuits are used in radio receivers to produce uniform response to modulated signals over a specified bandwidth; double tuned circuits are very useful in communication system.

Part -B

1. Derive an expression for transient current, voltages and the energy stored in inductor of a RL transient circuit excited by a DC source.

2. Derive an expression for transient current of a RL .Decay transient excited by a DC source.

3. A series RL circuit with R = 100 ohms and L =20H has a DC voltage of 200v applied through a switch at t = 0. Find a) the equation for the current and voltage across the different elements, b) the current at t = 0.5 seconds, c) the current at 1 sec and d) the time at which eR =eL.

4. Derive the expression for transient current, voltages and the energy stored in the capacitor of a series RC circuit, excited by a DC source.

5. Derive the transient current equation of a series RLC transient circuit excited by a DC source.

6. A series RLC circuit with R=300ohms, L=1H and C=100µF has a constant voltage of 50V applied to it at t=0.Find the maximum current value .Assume zero initial conditions.

7. Derive the expression for transient current of the series RL circuit excited by an AC voltage source.

8. Calculate (a) impedance of the entire circuit, (b) the total current, (c) the current in each parallel branch, (d) the total power supplied, (e) the power factor.

9. A non-inductive resistance in series with an ideal condenser is connected to a 125V, 50 Hz supply. The current in the circuit is 2.2A and the power loss in resistor is 96.8W. Calculate the resistance and capacitance and draw a vector diagram for the circuit.coil of resistance 10 ohms and inductance 0.1H is connected in series with a 150 micro farad capacitor across a 200V, 50Hz supply. Calculate (i) the inductive reactance, (ii) the capacitive reactance, (iii) the impedance, (iv) the current, (v) the power factor, (vi) the voltage across the coil and the capacitor respectively.

10. In series R-L-C circuit f = 500Hz, L =10mH, C = 5(F with applied voltage of 200V. Phase difference between current & voltage is 50(. Find R and the voltage across L & C. Also draw the phase diagram.

11. What is time constant? Explain time constant in case of series R-L and series R-C circuit.

12. A RLC series circuit with a resistance of 10Ω impedance of 0.2 H and a capacitance of 40 μF is supplied with a 100V supply at variable frequency. Find the following w.r.t the series resonant circuit:-

i) the frequency at resonance ii) the current iii) power iv) power factor v) voltage across R,L,C at that time vi) quality factor of the circuit vii) half power points viii) phasor diagram.

13. A series RLC circuit consists of a resistance of 1kΩ and an inductance of 100mH in series with capacitance of 10pF. If 100V is applied as input across the combination, Determine,

i) The resonant frequency ii) Maximum current in the circuit.

iii) Q-factor of the circuit iv) The half-power frequencies.

14. Prove that for series resonant circuit, the resonant frequency is the geometric mean of two half power frequencies.

15. Derive the expression of resonant frequency and bandwidth of a series resonant circuit.

16. Derive the expression for transfer function and maximum voltage amplification of a single tuned circuit.

17. Derive the expression for transfer function and maximum voltage amplification of a double tuned circuit.

18(a) A RC series circuit is given an excitation of V cos (ωt+ϕ). Derive the expression for the complete solution of the current response. Briefly explain the significance of phase angle in the solution. (May 2011).

18(b) A single tuned circuit is shown in figure below.

[pic]

The input to primary is 25mV with a frequency of 1.1 MHZ. The secondary is tuned to this frequency. Calculate (1) the value of capacitance C,(2) the input impedance, (3) the secondary current, (4) the voltage across the capacitor.(May 2011)

-----------------------

E

(V)

I (A)

Ideal

Practical

Ideal

V (V)

Is

(A)

Practical

A

B

5 [pic]

20V V

+

-

4 A

5 [pic]

A

B

100 [pic]

A

B

10 A

A

B

100 [pic]

1000 V

+

-

A

B

5 [pic]

6 [pic]

3 [pic]

B

A

C1

C3

C2

A

B

5 [pic]

5 [pic]

10 V

V1

V2

COMPLEX

ACTIVE

NETWORK

A

Rth

Isc

A

B

B

-

+

Vth= 5 V

B

A

Rth = 2.5[pic]

5 [pic]

+

-

E

1 [pic]

1 [pic]

20 [pic]

5 [pic]

¨Í®Í10 Ω

52.273V

Is

30 [pic]

+

-

[pic]

[pic]

[pic]

[pic]

[pic]

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

Leq = L1 + [L2L3 / (L2+L3)]

[pic]

Fig.3

4 [pic]

2 [pic]

4 [pic]

5A

2A

2V

5V

Q

P

-

+

-

+

-

+

47 Ω

4 Ω

20 A

23 Ω

27 Ω

200VΩ

9 Ω

9 Ω

27Ω

27Ω

27Ω

A

B

X

4Ω

2Ω

2Ω

5Ω

10V

5V

2V

Y

20V

10V

2Ω

4Ω

1Ω

6Ω

4Ω

+

-

10V

BA

A

1050Ω

200Ω

2500Ω

500Ω

30A

1Ω

1Ω

2Ω

5Ω

10Ω

5Ω

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