Wiley acing the gate electrical engineering part 2

SECTION VII: ELECTRICAL AND ELECTRONIC MEASUREMENTS MARKS DISTRIBUTION FOR GATE QUESTIONS Number of questions Marks Marks Total number of questions 2009 2010 2011 2012 2013 2014 2015 TOPIC DISTRIBUTION FOR GATE QUESTIONS Chapter Theory.indd 613 Year Concepts 2015 Measurement of basic electrical quantities, Electronic measuring instrument 2014 Measuring instruments 2013 PMMC, Moving iron, AC bridges 2012 PMMC, Bridges, Multiplier 2011 Wattmeter, AC bridge, Error analysis 2010 Wattmeter, Ammeter, AC bridge 2009 Dynamometer, Oscilloscope, Wattmeter 3/23/2016 1:21:19 PM Chapter Theory.indd 614 3/23/2016 1:21:19 PM CHAPTER ELECTRICAL AND ELECTRONIC MEASUREMENTS Measurement techniques have played a significant role from the starting from fair exchange of goods in early civilizations to regulation of trade in industrialised societies Better measurement and instrumentation techniques evolved as production of goods became industrialised and advent of computers saw their enormous application to measurement, process control and monitoring In this chapter, we will discuss the instruments used commonly for electrical and electronic measurements and about their error analysis These include instruments for measurement of voltage, current and power; instrument transformers; oscilloscopes and transducers 7.1  CLASSIFICATION OF MEASURING INSTRUMENTS Instruments can be classified based on their mode of operation, manner of energy conversion, measuring techniques and kind of output signal The main instrument types are discussed as follows Chapter Theory.indd 615   1.  Primary (or absolute) and secondary type: Instruments that measure the absolute physical quantity directly in terms of the constants of the instrument and the deflection are called primary or absolute instruments (e.g., tangent galvanometer) If the actual value of the quantity being measured is proportional to some other absolute value of the quantity, the instrument is called secondary The instrument is pre-calibrated using the absolute instrument (e.g., voltmeter, ECG recorder, etc.)   2.  Active and passive type: Instruments that can be directly used for the quantity being measured are known as the active-type If the quantity being measured simply modulates the magnitude of some external power source the instrument is known as passive type   3.  Deflection and null type: In a deflection-type instrument, the physical effect generated by the quantity being measured, produces an equivalent opposing effect in some other part of the instrument, which in turn causes deflection (or mechanical 3/23/2016 1:21:19 PM 616     Chapter 7:  ELECTRICAL AND ELECTRONIC MEASUREMENTS  displacement) which is a measure of the quantity In the null-type instrument the physical effect generated by the physical quantity under measurement is nullified by either a manual or automatic balancing device The equivalent null causing effect is the measure of the quantity   4.  Analog and digital type: In analog instrument, the physical quantities under measurement show continuous (step-less) variation with time In digital instruments, the physical quantities are discrete and vary in steps with time Based on the mode of operation, the secondary instruments are further classified into three types:   1.  Indicating type: In this category the measuring instrument indicates the quantity being measured through a pointer or some type of indicator Majority of the measuring instruments fall under this category, for example, voltmeter, ammeter, etc   2.  Integrating type: In this category of measuring instruments, the measurement is done with the help of integrating device or arrangement over a period of time For example, in the case of energy meter the rotation of disc over a period of time gives the reading of the energy consumed   3.  Recording type: In this category, the measuring instrument is used to record certain quantities to be used for analysis For example, plot chart recorder, ECG, EEG, etc where W is the control weight put at a distance l from the axis of rotation (spindle) of the moving system and kgr is the gravity constant Note: In instruments with spring control graduated scales are used and cramped scales are used for gravity control instruments   3.  Damping torque: In an instrument, the combined effect of deflecting and controlling torques on the movement of the pointer on the scale, causes it to oscillate when indicating the final reading These oscillations are prevented by using a damping mechanism, either by generating air or fluid friction or by action of eddy currents This torque is proportional to the angular velocity of the moving system and hence operated only when the system is in motion Tdamp = kdamp dq dt The effect of damping on deflection (q ) is depicted in Fig (7.1), where graphs I, II and III represent under-damped, critically damped and over-damped instruments, respectively I q II Deflection III 7.1.1  Indicating Type Instruments The different types of torques that function in any indicating measuring instruments are:   1.  Deflecting torque: This torque, also called operating torque, is developed by the magnetic, electrostatic, chemical or thermal effects produced by the quantity to be measured Td ∝ Operating quanity   2.  Controlling torque: The instruments are so designed that the controlling torque acts on its moving part It can (i) control/stop the movement of the pointer beyond the desired reading and (ii) bring the pointer back to its zero position, when the operating quantity is removed The controlling torque (Tc) is usually obtained either by a spring control Tc ∝ q (deflection) ⇒ Tc = kcq where kc is the spring constant or by gravity ­control mechanism Tc ∝ Wl sin q ⇒ Tc = kgr Wl sin q Chapter Theory.indd 616 Time Figure 7.1 |   Effect of damping on deflection 7.2  TYPES OF INDICATING INSTRUMENTS The basic components of all indicating instruments include:   1.  Support for moving system: This can be achieved either by pivoting or suspension   2.  Permanent magnets: These should have constant strength over a period of time   3.  Pointers and scales: The pointers should be light in weight with low constant of inertia A strip of mirror is mounted on the scale beneath the pointer and reading is taken after removing the parallax error between the pointer and its mirror image   4.  Cases: These are the outer covering of the instrument and should be made-up of non-magnetic material 3/23/2016 1:21:25 PM 7.2  TYPES OF INDICATING INSTRUMENTS     617 Depending on the mode of operation of the permanent magnet, the indicating type instruments can be classified into the following types:   1.  Moving coil-type instruments: This is further categorised into: (i) Permanent magnet moving coil: This can be used for direct current and voltage measurements (ii) Dynamometer type: This can be used either directly or through alternating current and voltage measurements   2.  Moving iron-type instruments: This can be used for AC/DC current and voltage measurement These instruments are discussed in detail in the following sections proportional to the measured quantity, that is, voltage or current This electromagnetic torque is counter balanced by the mechanical torque of control springs (bronze hair springs) attached to the movable coil The coil is wound on an aluminium former which moves in the magnetic field of the permanent magnet to provide eddy current damping When the torques are balanced, the pointer attached to the moving coil will stop and its angular deflection will represent the amount of electrical current to be measured against a fixed reference or scale The light weight pointer is carried by the spindle and it moves over this graduated scale The scales of the PMMC instruments are usually linearly spaced as the deflecting torque and hence the pointer deflections are directly proportional to the current passing through the coil The scale and pointer on the pivot are depicted in the top view of PMMC in Fig 7.3 7.2.1  Permanent Magnet Moving Coil Instruments Figure 7.2 shows the construction of a permanent magnet moving coil instrument The instrument has a moving coil of fine wire (circular or rectangular), with N-turns suspended in the uniform, horizontal and radial magnetic field of a permanent magnet in the shape of a horse-shoe It is free to turn about its vertical axis The coil with is placed around an iron core, which is spherical if the coil is circular, and is cylindrical if the coil is rectangular Since the coil is moving and the magnet is permanent, the instrument is called permanent magnet moving coil or a PMMC instrument Scale Pivot Pointer Balance weight Control spring Figure 7.3 |   Top view of PMMC instrument Scale The electromagnetic torque is equal to the multiplication of force with distance to the point of suspension The total deflection torque is given by Control spring Pointer S Permanent magnet Td = NBIlb = NBIA This torque will cause the coil to rotate until an equilibrium position is reached at an angle q with its original orientation At this position, N Rotating coil of N turns Electromagnetic torque = Control spring torque Stationary iron core Td = Tc Figure 7.2 |   Permanent magnet moving coil instrument When the current is passed through the coil it produces another magnetic field and the interaction of this field with the magnetic field of the permanent magnet produces an electromagnetic torque The amount of force experienced by the coil is proportional to the current passing through the coil which again becomes Chapter Theory.indd 617 NBIA = kcq q= NBIA ⇒ q = KI kc   where K = NBA    kc  The angular deflection is thus linearly proportional to the current I 3/23/2016 1:21:32 PM 618     Chapter 7:  ELECTRICAL AND ELECTRONIC MEASUREMENTS  The advantages of PMMC instruments are listed as follows:         1.  Uniform scale 2.  Accurate and reliable 3.  High sensitivity 4.  Free from hysteresis error and not affected by external (stray) magnetic fields   5.  Simple and effective damping mechanism   6.  Low power consumption   7.  Extension into multirange instruments possible 7.2.2  Moving Iron Type Instruments The moving iron (MI) type instruments can measure AC signals at frequencies up to 125 Hz, in addition to DC signals In these instruments, the signal (current) to be measured is allowed to flow through a stationary coil, which produces a magnetic field proportional to the quantity to be measured The moving iron piece (made of soft iron) is fixed with the moving system (a spindle and pointer), gets attracted/repelled proportionately and gives reading on the calibrated scale These instruments are accordingly classified as attraction or repulsion type Figure 7.4(a) shows the schematic of attraction type moving iron type instrument and that for the repulsion type is shown in Fig 7.4(b) Cramped scale and move towards the coil The spindle is rigidly connected to the pointer, controlling weight, moving iron and to the piston The repulsion-type MI instrument consists of two cylindrical soft iron vanes mounted within a fixed current carrying coil One iron vane is kept fixed to the coil frame and other, attached to the pointer shaft, is free to rotate Two irons lie in the magnetic field produced by the coil and current in the coil makes both vanes to become magnetised with the same polarity The repulsion between the similarly magnetised vanes produces a proportional deflection of the pointer In MI type instruments, rotation is opposed by a hairspring that produces the restoring torque The damping is achieved by air or fluid friction damping For an excitation current I carried by the stationary coil, the torque produced that causes the iron disc to move inside the coil is given by Td = I 2dM 2dq where M is the mutual inductance between the coil and the iron disk and q is the angular deflection of the vane Rotation is opposed by a spring connected with the vane and the pointer which produces a backwards torque given by, Ts = kq At equilibrium, the deflecting and the controlling torques are equal, that is, Td = Ts and therefore the deflecting angle q is given by Fixed coil Pointer q= I dM 2k dq Spring Moving iron disc (a) Coil Scale Pointer Spring Moving iron piece (vane) Thus the instrument has a square-law response where the deflection is proportional to the square of the signal being measured Thus, the output reading is a root-mean-squared (rms) quantity As the deflecting torque is proportional to the square of the coil current, these type of instruments possess scales that are non-linear and cramped in the lower range of calibration The advantages of MI-type of instruments are as follows:   1.  Suitable for AC and DC circuits   2.  Simple construction and low cost   3.  Measures voltage in the range of 0−30 V but a series resistance can be inserted in the circuit to measure higher voltages   4.  Accuracy is high   5.  Frictional error is less as torque/weight ratio is high (b) Figure 7.4 |   Moving-iron meter: (a) Attraction type and (b) repulsion type The attraction-type moving-iron meter, the moving iron disc (vane) placed near the coil is free to get attracted Chapter Theory.indd 618 7.2.3  Electrodynamic Type Meters Electrodynamic type meters, also known as dynamometer type meters, can measure both DC as well as AC signals up to a frequency of kHz The schematic diagram 3/23/2016 1:21:38 PM 7.3  BRIDGES AND POTENTIOMETERS     619 for dynamometer type instrument is shown in Fig 7.5 The instrument has a moving circular coil which is placed in the magnetic field produced by two circular stationary coils which are wound separately and connected in series the moving-iron type instrument The limitation of low permissible frequency can be overcome, to an extent, by rectifying the voltage signal and then applying it to a moving-coil meter, as shown in Fig 7.6 Bridge rectifier Scale Moving-coil meter Pointer Moving coil Fixed coils Figure 7.5 |   Schematic representation of electrodynamic meter The deflection torque in this type of wattmeter is produced by the interaction of two magnetic fluxes One of the fluxes is produced by a fixed coil, called current coil, which carries a current proportional to the load current The other flux is created by the moving coil, called the voltage or potential coil, which carries a current proportional to the load voltage The deflecting torque is dependent upon the mutual inductance between the two coils and can be given by dM Td = I1I2 dq where I1 and I2 are the currents flowing in the fixed and moving coils, M is the mutual inductance and q represents the angular displacement between the coils The torque is thus proportional to square of the current If the measured current is alternating, the meter is unable to follow the alternating torque values and instead displays the mean value of square of the current The squared or rms value of the measured current (or any other quantity) can be obtained by suitable modification of the scale Figure 7.6 |   Measurement of high-frequency voltage signals The circuit with bridged rectifier extends the upper limit of measurable frequency to 20 kHz but makes the measurement more sensitive to change in temperature of the environment and resulting non-linear behavioursignificantly impacts measurement accuracy for voltages An alternative method to overcome the low frequency limit is provided by the thermocouple meter 7.3  BRIDGES AND POTENTIOMETERS The AC bridge networks are used for the measurement of inductance and capacitance in the circuits These are modified form of Wheatstone bridge; consisting similarly of four arms, an excitation source and balance detector DC bridges along with potentiometers are used for the measurement of resistance This is depicted in the flow chart shown in Fig 7.7 Bridges The advantages of electrodynamic type meters are as follows:   1.  More accurate than moving-coil and moving-iron instruments but expensive   2.  Voltage, current and power can all be measured by suitable connections of fixed and moving coils   3.  Used to measure voltages in the range of 0−30 V but can be modified by placing a series resistance to measure higher voltages 7.2.4  Measurement of High-Frequency Signals In the instruments discussed in the sections above, the maximum frequency limit is of the order of kHz for the dynamometer type meters and only 100 Hz in the case of Chapter Theory.indd 619 AC Bridges Inductance measurement Maxwell bridge Hay’s bridge Owen’s bridge Anderson’s bridge Capacitance measurement De Sauty’s bridge Schering bridge Wein’s bridge DC Bridges (Resistance measurement) Wheatstone bridge Kelvin double bridge Figure 7.7 |   Bridges for measurement of inductance, capacitance and resistance 3/23/2016 1:21:41 PM 620     Chapter 7:  ELECTRICAL AND ELECTRONIC MEASUREMENTS  The general form of an AC bridge under balance condition is shown in Fig 7.8, where all four arms are considered as impedance (frequency dependent components) If Z1, Z2, Z3 and Z4 are the impedance of AC bridge arms, then at balance point: I1 IC I1R1 e I4wL1 EBA = EBC I1Z1 = I2Z2 I1 = V V and I2 = Z2 + Z4 Z1 + Z3 I4R4 The complex and polar form of equations are: e3 = I3R3 I3 (b) Z1Z4 = Z2Z3  Z1Z4 (∠q1 + ∠q ) = Z2Z3 (∠q2 + ∠q3 )  Figure 7.9 |   Maxwell’s bridge (a) Circuit diagram (b) Phasor diagram (7.1) (7.2) The construction of Maxwell’s bridge shows: B Z1 V A Z2 C D Z3 Z4   1.  One arm consisting of a capacitor C1 in parallel with a resistor R2 Both these variables have adjustable values   2.  The opposite arm consisting of an inductor L1 in series with a resistor R4 Both these variables are unknown values and need to be measured   3.  The other two arms consist of resistors R1 and R3., for which the values are known D Figure 7.8 |   AC bridge under balance condition 7.3.1  Measurement of Inductance Commonly used bridges for measurement of inductance are Maxwell’s bridge, Maxwell—Wien bridge and Hay’s bridge; other’s include Owen’s bridge and Anderson bridge 7.3.1.1 Maxwell’s Inductance-Capacitance Bridge The Maxwell bridge is used to measure unknown inductance in terms of calibrated resistance and capacitance The circuit arrangement for Maxwell’s bridge is shown in Fig 7.9(a) and the corresponding phasor diagram is shown in Fig 7.9(b) The Maxwell bridge measures inductance after adjustment of C1 and R2 such that the current through the bridge between points A and B becomes zero, which occurs when the voltages at points A and B are equal This is known as balancing of circuit When the Maxwell bridge is balanced, the impedances can be written as R Z1 = 3 R1 Z2 (7.3) where Z1 is the impedance of resistor R2 in parallel with capacitor C2, and Z2 is the impedance of inductor L1 in series with resistor R4 Thus, from Eq (7.3), the relation can be mathematically represented as: Z1 = R2 + and Z2 = R4 + jwL1  jwC1 (7.4) Here w = 2pf C1 Thus, when the bridge is balanced, R3 R2 + / ( jwC1 ) = A B L1 R1 R4 (a) Chapter Theory.indd 620 R1 R3 R4 + jwL1 or R1R3 = R2 + / ( jwC1 ) R4 + jwL1  When the bridge is balanced, the negative and positive reactive components cancel out, so 3/23/2016 1:21:53 PM 7.3  BRIDGES AND POTENTIOMETERS     621 R1R3 R2 R1R3 = R2R4 ⇒ R4 = (7.5) From the value of R4 determined by Eq (7.5), L1 can be determined using Eq (7.4) When the Hay’s bridge is balanced, the impedances can be written as R Z1 = 3 (7.6) R2 Z4 where, Z4 is the impedance of the arm containing C4 and R4 while Z1 is the impedance of the arm containing L1 and R1 7.3.1.2 Hay’s Bridge A Hay’s bridge is another AC bridge circuit, which is a modification of Maxwell’s bridge Figure 7.10 shows the circuit diagram of the Hay’s Bridge It can be used for measuring an unknown inductance by balancing its four arms, one of which contains the unknown inductance One of the arms of a Hay’s Bridge has a capacitor of known value, which is the principal component that is used to determine the unknown inductance value Thus, from Eq (7.6), Z4 = R4 + Mathematically, when the bridge is balanced, R3 R1 + jwL1 = R2 R4 +1 / ( jwC4 ) B e1 and Z1 = R1 + jwL1 jwC4 e3 I1 L1 or, R3 R1 A R3R2 R1 + jwL1  =   R + / ( jwC ) 4 I1 C D or, R2 R2R3 + R1 L = R1R4 + jwL1R4 +  jwC4 C4 (7.7) C4 Equating the real and imaginary terms for the Eq (7.7), we get R4 I2 I2 D e2 R2R3 = R1R4 + e4 e wL1R4 = (a) I2 R1 wC L1  C4 (7.8) or  R1 = (w)2 L1R4C4  (7.9)   Substituting for R1 from Eq (7.9) in Eq (7.8), we get e1 = e2 e L1 = I1wL1 R2R3C4 (wR4C4 ) + and I2R4 90° e3 = e4 I1R1 I1 I2/wC4 R1 = (wC4 ) 2R2R3R4 + (wR4 C4 ) These expressions can be used to find R1 and L1 (b) Figure 7.10 |   Hay’s bridge (a) Circuit diagram (b) Phasor diagram 7.3.1.3 Owen’s Bridge The construction of Hay’s bridge is as follows: The circuit for Owen’s bridge is shown in Fig 7.11 The construction of Owen’s bridge is as follows:   1.  One arm of the bridge consists of a capacitor C4 in series with a resistor R4 The resistor R4 and C4 are both adjustable   2.  The second arm consists of an inductor L1 in series with a resistor R1 These are the unknown values   3. The other two arms contain known resistors R2 and R3   1.  One arm consists of a capacitor C2 in series with a resistor R2   2.  Second arm consists of an inductor L1 in series with a resistor R1 These are the values to be determined   3.  Third arm contains a known capacitor C4   4.  The fourth arm just contains known resistor R3.  Chapter Theory.indd 621 3/23/2016 1:22:07 PM 622     Chapter 7:  ELECTRICAL AND ELECTRONIC MEASUREMENTS  Equating real and imaginary parts, we have B e1 e3 I1 L1 = R2R3C4 L1 R3 and R1 I1 A C4R3 C2 R1 = C D 7.3.1.4 Anderson Bridge R2 C4 Anderson bridge is a modified version of Maxwell bridge, used for measurement of unknown value of inductance by comparison with known values of electrical resistance and capacitance The circuit for Andersen bridge is shown in Fig 7.12 The construction of the bridge is as follows: C2 I2 I2 D e2 e4   1.  One arm contains the unknown inductor L1 connected with resistance R1 (purely resistive)   2.  The other three arms consist of resistances R2, R3 and R4 (purely resistive)   3.  A standard capacitor C is connected in series with variable resistance r and this combination is connected in parallel with arm CD e (a) I2 I2/wC2 e B I1wL1 I1 R1 R3 L1 A I1 D IC I3 C e3 = I1R3 = I2/wC4 I1R1 C r (b) I4 R2 Figure 7.11 |   Owen’s bridge (a) Circuit diagram (b) Phasor diagram R4 I2 D When the bridge is balanced, (a) R Z1 = Z2 Z4 e1 e where Z2 is the impedance of C2 and R2 and Z1 is the impedance of the arm containing L1 and R1, and Z4 is the impedance of the arm containing C4.  I1wL1 IC I2 Then Z4 = ICr jwC4 jwC2 = R1 + jwL1 When the bridge is balanced, / j (wC4 ) R2 + / j (wC2 ) = R3 R1 + jwL1 IC r I1 e3 = I1R3 = IC/w C (b) Figure 7.12 |   Andersen bridge (a) Circuit diagram (b) Phasor diagram When the bridge is balanced Chapter Theory.indd 622 I4 I1(R1 + r) Z2 = R2 + Z1 e2 = I2 R2 e4 = I4 R4 I1 = I3 and I2 = IC + I  (7.10) 3/23/2016 1:22:20 PM SET 2     1181 = C2 Therefore, y = xe2x Thus y(1) = e2 = 7.38 Similarly major axis equation is y + x = 1 So,   lies on the minor axis 1 Ans (d) Ans (7.38) ? 33 ?Let 2the? probability density function of a random 31 The line integral of the vector field F = 5xzi + (3x2 + 2y)jvariable, + x zk X, be given as: ? ? ? F = 5xzi + (3x + 2y)j + x zk along a path from (0, 0, 0) to fX (x) = e−3x u(x) + ae4x u(−x) (1, 1, 1) parametrised by (t, t2, t) is       where u(x) is the unit step function Solution:  Given that Then the value of `a’ and Prob {X ≤ 0}, respec? tively, are ? ? E = 5xzi + (3x2 + 2y)j + x2zk 1 1 (a) 2,    (b) 4,    (c) 2,    (d) 4,    2 4 = ∫ F ⋅ dr Solution:  For the given function: C  ae4x x < = ∫ 5t2dt + (3t2 + 2t2 )2t dt + t3dt = 4.41 f (x) =  −3x  e x≥0  Ans (4.41) x 3   Consider the set S of all vectors   32 Let P =  1 3 y  a x 2 such that a + b = where   = P   Then S is b  y  (a) a circle of radius 10 (b) a circle of radius 10 1 (c) an ellipse with major axis along   1 1 (d) an ellipse with minor axis along   1 Solution:  Given that Then  1  x   a     =   1 3  y   b  3x + y = a x + 3y = b a2 + b2 = Therefore, +¥ ∫ Therefore, 1 Length of minor axis =2r =   =   −¥  −1  +   e −3x   ¥ =1 0 P (X ≤ ) = ∫ 2e4x dx = −¥  Ans (a) 34 The driving point input impedance seen from the source VS of the circuit shown below, in W, is 2Ω + V1 − IS + VS − 2Ω 3Ω V1 Ω Solution:  For the given circuit, the equivalent circuits are: 2Ω IS + V1 − 2Ω   a −  x + hy = 2  r  Set 2_Gate_2016.indd 1181 −3 x e dx = a + =1⇒a=2 Equation of minor axis is (10 − 16)x + 6y = y−x = ∫ ae 10x + 10y + 12xy = ⇒ a = 10, b = 10 and h = (ab − h2 )r − (a + b)r + = 1 r = or ¥ ae4x dx + ∫ 4x This represents an ellipse Then length of semi-axis is −¥ (3x + y)2 + (x + 3y)2 = fx (x) =  ¥ + VS − 3Ω V1 4Ω Zeq 3/24/2016 11:49:56 AM 1182     SOLVED GATE(EE) 2016 + V1 − Is 2Ω º≡ + 3Ω Vs − Is Ω V1 Vx C L ≡ º+ Vs − 6Ω 36 In the balanced three-phase, 50 Hz, circuit shown below, the value of inductance (L), is 10 mH The value of the capacitance (C) for which all the line currents are zero, in millifarads, is 2Ω L C LC V1 Solution:  Converting the inductive Δ to Y, we get ZΔ = 3ZY Therefore, V x − VS V x + = 4V1 2 But, V1 = VS − Vx, therefore, V x − VS V + x = 4(VS − Vx ) 2 Vx = VS 10        V1 = 2IS    VS −Vx = 2IS   VS − 0.9VS = 2IS    0.1VS = 2IS Thus, the driving impedance, which is a ratio of voltage to current from the port, is VS = 20 Ω IS Ans (20) 35 The Z-parameters of the two port network shown in the figure are Z11 = 40 W, Z12 = 60 W, Z21 = 80 W and Z22 = 100 W The average power delivered to RL = 20 W, in watts, is 10 Ω I1 I2 + + 20 V − V1 + [Z] − V2 L C RL C C L L For line current i = Zph = XL  XC = Ơ jw L ì jwC = L jw + jwC jwL j = wC C= w 2L = 3.03 mF Ans (3.03) 37 In the circuit shown below, the initial capacitor voltage is V Switch S1 is closed at t = The charge (in μC) lost by the capacitor from t = 25 μs to t = 100 μs is − S1 Solution:  From the given circuit: V2 = −20I2 V1 = 40I1 + 60I2 V2 = 80I1 + 100I2 5Ω µF I2 = −2/3 I1 Therefore, Solution:  Given that VC(0) = 4V Also, when V1 = I1 = A I2 = −4/3 A P = I22RL =   Set 2_Gate_2016.indd 1182 4V Capacitor discharging equation − 16 × 20 = 35.55 W Ans (35.55) VC (t) = V0 e t RC − = 4e t 25 ×10−6 Q = CV 3/24/2016 11:50:01 AM SET 2     1183 −6 Q1 = × 10 = × 10−6  − 25×10−6   −6   4e 25×10  e 0.6 º ≡ j0.2 x Q2 = × 10−6 [4e−4] j0.3 1  Q2 − Q1 = × 10−6 ×  −  C  e e  = × 10−6 × × 0.3498 C = μC Ans (7) 0.3 × 0.3 = 0.75 0.6 + 0.3 + 0.3 0.6 × 0.3 y= = + + × = 1.5 z= 0.6 + 0.3 + 0.3 X2eq = j1.2 pu j1.5 j0.2 The ratio of the maximum real power that can be transferred during the pre-fault condition to the maximum real power that can be transferred under the faulted condition is j0.5 Line j0.2 j0.5 Line j0.1 Solution:  In the given power system with occurrence of fault j0.1 j0.5 F j0.2 pu j0.5 j0.1 1.0 pu X2eq P1 = P2 X1eq Therefore, P1 1.2 = = 2.4 P2 0.5 39 The open loop transfer function of a unity feedback control system is given by K(s + 1) G(s) = , K > 0, T > s(1 + Ts)(1 + 2s)   The closed loop system will be stable if, (a) < T < 4(k + 1) 4(T + 2) T −2 8(K + 1) (d) < T < K −1 (b) < K < k −1 T +2 (c) < K < T −2 Solution:  We know that + G(s)H(s) = Substituting values for the unity feedback system, we have K(s + 1) 1+ =0 s(1 + Ts)(1 + 2s)     2Ts3 + (2 + T)s2 + (1 + K)s + K = Using RH criteria X1eq = j0.2 + [( j.01 + j0.5)  ( j0.1 + j0.5)] = j0.5 Similarly, from the circuit below j0.6 2T ( + T) s2 ( + T )( + K ) − 2TK s1 2+T s0 K s3 (1 + K ) K For stability K > 0, therefore (2 + T) (1 + K) − 2TK > j0.2 j0.3 Set 2_Gate_2016.indd 1183 j1.5 j0.075 Infinite bus j0.1 j0.3 x= 38 The single line diagram of a balanced power system is shown in the figure The voltage magnitude at the generator internal bus is constant and 1.0 pu The pu reactances of different components in the system are also shown in the figure The infinite bus voltage magnitude is 1.0 pu A three phase fault occurs at the middle of line Generator internal bus y z j0.3 ⇒K< T +2 T −2 3/24/2016 11:50:08 AM 1184     SOLVED GATE(EE) 2016 0