TUTORIAL PROBLEMS. E.1 KCL, KVL, Power and Energy. Q.1 Determine the current i in the following circuit. All units in VAΩ,,

Transcription

1 196 E TUTORIAL PROBLEMS E.1 KCL, KVL, Power and Energy Q.1 Determne the current n the followng crcut

2 Appendx E Tutoral Problems 197 Q. Determne the current and the oltage n the followng crcut for R = Ω and R = 50KΩ. 5A Network A R 10 V Network B Q.3 Determne the source oltage and the oltage across the 3Ω resstor n the followng crcut. A Ω 1 Ω Ω Ω Ω 3 Ω Q. The Ammeter AM and oltmeter VM, connected as shown below, measure current and oltage, respectely. The ammeter wll ge a poste readng f the current flowng nto ts "+ " termnal s poste. Electrcal dece A B + AM + VM Determne f the electrcal dece s consumng or supplyng power when (a) both meters ge poste readngs; (b) both meters ge negate readngs; and (c) one meter ges a poste readng whle the other ges a negate readng. (d) one meter or both meters ge zero readngs.

3 Appendx E Tutoral Problems 198 Q.5 Determne the current n the followng crcut. 15 V Ω 1 Ω 5 V How much power s each component consumng or supplyng? Q.6 Determne the oltages and currents n the followng crcut. Ω 10 A 6 Ω 5 A Is the 10A current source consumng or supplyng power? Q.7 In the followng system, determne the effcency of the motor and the torque on the fan shaft? 0 A DC power supply 100 V DC motor 100 re/mn Fan load.5 h.p. How much energy s lost n the motor per mnute? (Note that 1h.p. = 76W.) Q.8 In the followng system, the effcency of the generator s A Steam turbne re/mn Reducton gearng 5 : 1 DC generator 100 V Electrcal load Determne the torque n ts shaft.

4 Appendx E Tutoral Problems 199 Q.9 The followng fgure shows the equalent crcut of an audo amplfer when connected to an nput audo source (CD player) and a load (loudspeaker). R CD Player Audo amplfer Loudspeaker Calculate the oltage gan 1, current gan 11 of the amplfer n db. What s the relatonshp between these gans? Show that these gans wll be equal to one another when the load resstance s equal to nput resstance of the amplfer. Do you expect ths to be the case n practce? 1 and power gan ( ) ( )

5 Appendx E Tutoral Problems 00 E. KCL, KVL and Groundng Q.1 Determne the current n each branch of the followng crcut by nspecton. R If the oltage s 16V, determne R. Q. The followng crcut shows a common oltage dder for obtanng a certan oltage across a load resstor R R In fndng, a noce (= 1st year student) may forget (okay, f t's only for ths tme) to nclude the loadng effects of R. To understand these effects, determne and the current n R when (a) R = (open crcut or no load stuaton); (b) R = 8000Ω; (c) R = 00Ω; and (d) R = 0 (short crcut stuaton).

6 Appendx E Tutoral Problems 01 Q.3 Are the two crcuts shown below equalent? Why? A R 3 3 A R Ω X 10 V 5 A R 5 R 1 Ground 5 A R 1 R 5 X Ω R A R 3 3 A 10 V Ground As s done n the aboe crcuts, oltages n electrcal systems are often measured wth respect to a reference or ground potental. In practce, the ground wll be connected to a large number of components and wll run through the entre system usng thck conductors. It s sometmes connected to the chasss of the system or the earth pn of the 3 pn power plug. Assumng that there are other components (not shown) connected to ground, determne the current flowng through the oltage source and the potental of node X wth respect to ground when = A and = 3A. Why s the ground node seems to be takng/gng current all the tme and stll Krchhoff's current law s not olated?

7 Appendx E Tutoral Problems 0 Q. In the followng crcut, the oltmeter VM s used to measure the oltages of arous ponts wth respect to ground. A Ω D 10 V 5Ω 3Ω B + VM C Assumng that VM does not take any current, determne the current and the oltmeter readng when (a) Pont C s grounded and Pont B s connected to Pont D; (b) there s no connecton for Pont C and Pont B s connected to Pont D; and (c) Pont B s grounded and Pont C s connected to Pont D. Q.5 Determne the current and the potental of node X wth respect to ground n the followng crcut. 1 Ω Ω 3 Ω 6A V R Ω X How wll these be changed f the crcut s not grounded or f the crcut s grounded at Pont X only?

8 Appendx E Tutoral Problems 03 Q.6 Determne the relatonshps between and for the followng three crcuts. Ω 3 3 V 1 A Ω 3 5 V A 1 Ω Ω What can you say about these three crcuts?

9 Appendx E Tutoral Problems 0 E.3 DC Crcut Analyss I Q.1 Fnd the source current n the followng ladder network by reducng the seres and parallel combnatons to a sngle resstor Determne the current by usng current dson. Q. Determne the equalent resstance that would be measured between Ponts A and B of the followng crcut. A B What wll the result be f the outer two resstors are short-crcuted?

10 Appendx E Tutoral Problems 05 Q.3 Formulate the equatons for obtanng the loop currents of the followng crcut by usng mesh analyss. Determne the oltages of nodes A, B and C wth respect to ground n terms of these currents A B C Usng nodal analyss, dere the equatons for obtanng the oltages of the three nodes. Q. Usng nodal analyss, calculate the equalent resstance of the followng - termnal network

11 Appendx E Tutoral Problems 06 Q.5 Determne Norton's and Theenn's equalent crcuts for the followng - termnal network. R 1 R R n 1 n Q.6 In the followng crcut, the ammeter AM reads 1A when the swtch s closed. 1 AM 6 3 Usng Norton's equalent crcut, determne the readng when the swtch s open. Q.7 Usng Theenn's equalent crcut, determne the current 1 In the followng crcut. 1 R R 1 R R R

12 Appendx E Tutoral Problems 07 E. DC Crcut Analyss II Q.1 The followng crcut shows three batteres connected n parallel and supplyng power to a 10Ω load Battery 1 Battery Battery 3 Usng superposton, determne the component of the load current due to each battery. Hence, calculate the load current. Q. Determne the current n the followng crcut by usng superposton Q.3 Determne the maxmum power that can be obtaned from the two termnals of the followng crcut

13 Appendx E Tutoral Problems 08 Q. Determne the resstor that wll draw a current of A when connected across the two termnals of the followng network What resstor wll absorb the maxmum power? Calculate ths power. Q.5 The followng fgure shows the oltage-current cure of an electrcal dece D and a crcut n whch the dece s employed. Current d 1 D Voltage Determne the dece current d f = 16. A. At what alue of wll the power dsspated n D be 06. W?

14 Appendx E Tutoral Problems 09 Q.6 The followng fgure shows the equalent crcut of a transstor amplfer connected to amplfy the magntude of the oltage 1 from an nput oltage source. The man acttes of the transstor are represented by the two dependent sources, whle the substrate and other contact resstances are represented by the two resstors. 1 VmA,,kΩ Transstor amplfer Determne the output oltage of the amplfer n terms of 1. What s the gan of the amplfer n terms of oltage magntudes? Ge your answer n db. Determne the Theenn's equalent crcut of the system as seen from the output termnals.

15 Appendx E Tutoral Problems 10 E.5 AC Crcut Analyss I Q.1 Wrte down the peak and rms alues, frequency, phase, complex and phasor representatons of the followng ac quanttes. (a) 5 sn( ω t ) (b) 5 cosω ( t ) (c) 10 sn( 0t ) (d) 10 cos( 31 t 5 0 ) π (e) 50sn t 3 (f) 0. 5cos( t ) Q. Takng the frequency to be 50Hz, wrte down the snusodal oltages and currents correspondng to the followng phasors. (a) 100 (b) e j 0 3 e jπ V V (c) 01. e jπ A (d) o A Q.3 In the followng ac crcut, ( t) ( t) R = 1 cos V. (t) L(t) R(t) 3 R(t) VA,, Ω,H, F 0.1. Determne ( t) R, ( t) L and t ( ) by usng phasor analyss.

16 Appendx E Tutoral Problems 11 Q. In the followng seres RLC crcut, t ( ) = 50 cos( 150t ) V. Determne the phasors of the oltages across the three components and that of the current t. ( ) (t) 0 VA,, Ω,H, F (t) Draw a phasor dagram showng all these phasors n the complex plane. Q.5 The followng fgure shows an ac crcut wth two parallel branches. Each branch conssts of two seres components: RL (resstor and nductor) or RC (resstor and capactor) or LC (nductor and capactor). From the mpedances gen, fnd the components and ther alues n the crcut. I 50 Hz V 10 j j30 Calculate the admttance (recprocal of mpedance) of the entre crcut and the phase angle between I and V. What s the power factor of the crcut?

17 Appendx E Tutoral Problems 1 E.6 AC Crcut Analyss II Q.1 A Ω resstor s connected n seres wth a col across a 0V 50Hz supply. The oltage across the resstor s 9V and that across the col s 1 V. If the col can be represented as an deal nductor L n seres wth a resstor R (accountng for the wndng resstance and other losses), calculate the alues of L and R, the power absorbed by the col, and the power factor of the whole crcut. Q. A load connected across a 000V 50Hz lne draws 10 kw at a laggng power factor of 05.. Determne the current taken by the load. To mproe the power factor, a capactor C s now connected n parallel wth the load. Determne the alue of C so that the oerall power factor s 0. 9laggng, unty and 0. 8leadng. Q.3 The followng dagram shows the Theenn's equalent crcut of an ac system when connected to a load wth mpedance Z = R + jx. a+ jb V Z = R + jx Electrcal system Show that R and X should be gen by ( ) Z = R+ jx = a+ jb = a jb n order for the load to absorb the maxmum power. Determne ths maxmum power.

18 Appendx E Tutoral Problems 13 Q. Usng Norton's equalent crcut, determne the maxmum power that can be absorbed by the load wth mpedance Z n the followng ac system. 10 j o 5 o 30 j5 Z Q.5 A certan tune crcut conssts of a 100 μ H col wth a resstance of 10Ω (representng losses due to thermal resstance and other effects) connected n seres wth a 100pF capactor. Determne the resonant frequency, Q factor, bandwdth and 3dB cutoff frequences of the crcut. Q.6 A arable tunng capactor n a rado receer has a maxmum capactance of 500pF and a mnmum capactance of 0pF. What nductance s requred so that the lowest frequency to whch the crcut can be tuned s 666kHz? What s the hghest frequency to whch ths crcut can be tuned?

19 Appendx E Tutoral Problems 1 E.7 Perodc Sgnals Q.1 The oltage t ( ) appled to and the current t ( ) resulted from an electrcal dece are perodc and are gen below.,s (t) ( t) t (t) Electrcal dece ( t) t Determne (a) the perod of the oltage and current waeforms; (b) the aerage or mean alue of t ( ) and t ( ) ; (c) the tme when the dece s consumng power and when t s supplyng power; (d) the perod of the nstantaneous power cure; and (e) the aerage power consumed by the dece.

20 Appendx E Tutoral Problems 15 Q. The followng fgure shows a symmetrcal and perodc sawtooth oltage waeform ( ) t from a sgnal generator.,s (t) t (a) Determne the perod and mean alue of ( ) t. (b) Sketch the araton of ( t) as a functon of tme t. Hence, determne the perod and mean alue of ( t). Calculate rms, the square root of ths mean alue. rms s the rms (root mean square) alue of the perodc sgnal t ( ). (c) If t ( ) s appled to a resstor R, show that the aerage power beng consumed s gen by rms R. (From ths, t s easy to see why the rms alue s sometmes called the effecte alue.) Q.3 Wrte down the phasor representatons of ( t) 1 and ( t) and determne ther phase dfference for the followng two stuatons. 0 0 (a) ( t) 1 = 10 cos( π t+ ) and ( ) t = 0 cos( π t+ 3 ) 0 0 (b) ( t) 1 = 10 cos( π t+ ) and ( ) t = 0 cos( π t+ 3 ) What happens when ( t) 1 and ( t) are obsered by an obserer wth a tme orgn of t = 05. so that 1( t+ 05. ) and ( t+ 05. ) are seen nstead? Q. Determne the Fourer seres representaton of the followng symmetrcal perodc square waeform. (t),s t

21 Appendx E Tutoral Problems 16 Q.5 In the followng crcut, ( t) s the exctaton and ( t) o s the response. Dere the frequency response of the crcut. Sketch the magntude response and show that the crcut s lowpass n nature. Deduce ths from the low and hgh frequency characterstcs of the capactor as well. 9 VA,, Ω,H, F (t) 1 1 o(t) Determne ( t) o f ( t) s gen by the non-perodc sgnal () cos( π ) sn( π ) t = t + t and also when t s a perodc square waeform wth Fourer seres ( 6πt) sn( 10πt) sn ( ) t = sn( π t)

22 Appendx E Tutoral Problems 17 E.8 Transents I Q.1 The current t ( ) n the followng fgure flows through a 5H nductor and an ntally uncharged 5 μ F capactor. (t) (ma) t (ms) Plot the oltages across the nductor and capactor as functons of tme t. Q. In the followng crcut, the two swtches hae been n the postons shown for t < 0 and are thrown to the other postons for t t 0 t 0 6 (t) VA,, Ω,H, F 5 1 (t) 1 (t) (a) Assumng that the swtches hae been n ther orgnal postons for a long tme so that the oltages and currents n the crcut hae settled down to constant alues, determne the alues of ( t) 1, ( t) and t ( ) for t < 0. (b) Determne ( t) 1, ( t) and t ( ) just after the swtches were thrown to the other postons at t = 0. (c) Dere and sole the goernng dfferental equaton for t ( ) for t 0. Hence, obtan expressons for ( t) 1 and ( t) for t 0. (d) Sketch ( t) 1, ( t) and t ( ) as functons of t.

23 Appendx E Tutoral Problems 18 Q.3 In the followng crcut, the swtch has been closed for a long tme. At tme t = 0, the swtch s opened. t 0 L R 1 R (a) Determne the oltages and currents n the crcut for t < 0. What s the energy stored n the nductor? (b) Determne the current n the nductor t ( ) just after the swtch s opened at t = 0. (c) Dere an expresson for ( ) t for t 0. (d) By ntegratng ( R R ) ( t) 1 +, determne the energy consumed by the resstors after the swtch s opened. Dere the same result by usng conseraton of energy. Q. In the followng dagram, the nductor s carryng a current of 3A when the swtch on the left s turned from the poston shown to the other poston at tme t = 0. At the end of 3s, the other swtch s closed. 3 t < 0 t 0 t 3 VA,, Ω,H, F 15 5 (t) 5 Dere an expresson for ( ) t and sketch t.

24 Appendx E Tutoral Problems 19 E.9 Transents II Q.1 In the followng crcut, the swtch has been opened for a long tme. At tme t = 0, the swtch s closed. (t) t 0 R 1 R R 3 C L Dere an expresson for t ( ) as a functon of tme. For what alues of R 1, R and R 3 wll ths current be ndependent of tme? Q. In the followng crcut, the swtch s opened at t = 0 after steady state condtons hae prealed VA,, Ω,F (t) t 0 Dere an expresson for ( ) t for t 0.

25 Appendx E Tutoral Problems 0 Q.3 In the followng crcut, the capactor s ntally charged to 10V before the swtch s closed. R C t 0 L (a) Determne the oltages and current n the crcut before the swtch s closed. (b) Determne the oltages and current just after the swtch s closed at t = 0. (c) Dere the goernng dfferental equaton for the current for t 0. Express all the constants noled n terms of the resonant frequency and Q factor of the crcut. What are the ntal condtons that hae to be satsfed? (d) Determne and sketch the current for t 0 under the oerdamped stuaton of Q < 1. (e) Determne and sketch the current for t 0 under the underdamped stuaton of Q > 1. (f) Determne and sketch the current for t 0 under the undamped stuaton when R = 0. Q. In the followng crcut, the capactor s ntally uncharged and there s no current through the nductor when the swtch s closed at tme t = 0. R (t) sn (ωt ) t 0 L C Dere the goernng dfferental equaton and ntal condtons for fndng the crcut current ( ) t for t 0. Determne the steady state current.

26 Appendx E Tutoral Problems 1 E.10 Magnetc Crcuts Q.1 A magnetc crcut comprses an ron core wth a short ar gap. The core crosssecton s 1000mm, ts length s 500mm and the relate permeablty of the ron s 600. The gap length s. 5mm. Estmate the mmf requred to produce a flux of 18. mwb n the gap. Neglect frngng effects. Q. Assumng that the effecte area of each ar gap s 0. 05m, the ron path requres 1 3 of the total mmf and the total flux s kept constant, calculate the force of attracton on the armature of the electromagnet shown n the followng fgure. A 300 turns Armature 1.5 mm What wll the force be f the current s doubled? Wll the force be less than ths n a practcal dece? Why? Q.3 In the followng crcut, an 5Ω load s connected to a 50V 100Ω source through two deal transformers. : 1 1 : n 100 Ω 50 V 5 Ω Gen that maxmum power s beng drawn from the source, determne the turn rato n and all the oltages and currents n the crcut.

27 Appendx E Tutoral Problems Q. A transformer has a ratng of V (6600V prmary, 30V secondary) and 100 kva (maxmum alue whch can be delered under normal operatng condtons). If t s supplyng 1 rated current at 0. 8laggng power factor, calculate the alue of the actual load mpedance. What s the alue of the load mpedance referred to the prmary? Q.5 The flux densty n the core of a 0 0V 50Hz sngle-phase transformer has a maxmum alue of 06. T. If the prmary wndng has 130 turns, determne the number of turns n the secondary wndng. Usng Faraday's law of nducton, dere the relatonshp between the maxmum flux n the core and the rms alue of the oltage across the prmary. Hence, calculate the cross sectonal area of the core. Q.6 There are two man types of power losses n a practcal transformer. Copper loss s caused by the resstances of the copper wre n the wndngs and s proportonal to the square of the currents. Iron loss s caused by the nducton of eddy currents and the repeated changes n magnetsaton n the magnetc core of the transformer. Ths loss ncreases as the frequency ncreases, but s usually ndependent of the load and supply currents. A 50 kva 00 0 V 60Hz transformer has an ron loss of 350W and a copper loss of 630W at full load or rated current. Determne the effcency at full load and unty power factor; at 3 full load and unty power factor; at full load and 0. 8 power factor; and at 3 full load and 0. 8power factor. Determne the all-day effcency f the operatng schedule s as follows hr no load hr full load, unty pf hr full load, unty pf hr full load, 08. pf hr full load, 08. pf hr no load