SERIES A.C. CIRCUITS. Learning Objectives. This chapter discusses series AC circuits, and how they function

C H A P T E 3 Learning Objectives A.C. Through esistance and Inductance Power Factor Active and eactive Components of Circuit Current-I Active, eactive and Apparent Power Q-factor of a Coil Power in an Iron-cored Chocking Coil A.C. Through esistance and Capacitance Dielectric Loss and Power Factor of a Capacitor esistance, Inductance and Capacitance in Series esonance in -L-C Circuits Graphical epresentation of esonance esonance Curve Half-power Bandwidth of a esonant Circuit Bandwidth B at any Offresonance Frequency Determination of Upper and Lower Half-Power Frequencies Values of Edge Frequencies Q-Factor of a esonant Series Circuit Circuit Current at Frequencies Other than esonant Frequencies elation Between esonant Power P and Off-resonant Power P SEIES A.C. CICUITS This chapter discusses series AC circuits, and how they function

58 Electrical Technology 3.. A.C. Through esistance and Inductance A pure resistance and a pure inductive coil of inductance L are shown connected in series in Fig. 3.. Let V r.m.s. value of the applied voltage, I r.m.s. value of the resultant current V I voltage drop across (in phase with I), V L I. X L voltage drop across coil (ahead of I by 9 ) Fig. 3. Fig. 3. These voltage drops are shown in voltage triangle OAB in Fig. 3.. Vector OA represents ohmic drop V and AB represents inductive drop V L. The applied voltage V is the vector sum of the two i.e. OB. V V ( V VL) [( I) ( I. XL) ] I XL, I ( X ) The quantity ( + X L ) is known as the impedance (Z) of the circuit. As seen from the impedance triangle ABC (Fig. 3.3) Z + X L. i.e. (Impedance) (resistance) + (reactance) From Fig. 3., it is clear that the applied voltage V leads the current I by an angle φ such that VL I. XL XL ω L reactance XL tan φ φ tan V I. reactance The same fact is illustrated graphically in Fig. 3.4. In other words, current I lags behind the applied voltage V by an angle φ. Hence, if applied voltage is given by ν V m sin ω t, then current equation is i I m sin (ω t φ) where I m V m / Z L Fig. 3.3 Fig. 3.4

Series A.C. Circuits 59 In Fig. 3.5, I has been resolved into its two mutually perpendicular components, I cos φ along the applied voltage V and I sin φ in quadrature (i.e. perpendicular) with V. Fig. 3.5 Fig. 3.6 The mean power consumed by the circuit is given by the product of V and that component of the current I which is in phase with V. So P V I cos φ r.m.s. voltage r.m.s. current cos φ The term cos φ is called the power factor of the circuit. emember that in an a.c. circuit, the product of r.m.s. volts and r.m.s. amperes gives voltamperes (VA) and not true power in watts. True power (W) volt-amperes (VA) power factor. or Watts VA cos φ* It should be noted that power consumed is due to ohmic resistance only because pure inductance does not consume any power. Now P VI cos φ VI (/Z) (V/Z) I. I (ä cos φ /Z) or P I watt Graphical representation of the power consumed is shown in Fig. 4.6. Let us calculate power in terms of instantaneous values. Instantaneous power is v i V m sin ωt I m sin (ωt φ) V m I m sin ω t sin (ωt φ) Vm I m[cos φ cos ( ωt Φ)] Obviously, this power consists of two parts (Fig. 3.7). (i) a constant part V m I m cos φ which contributes to real power. Fig. 3.7 * While dealing with large supplies of electric power, it is convenient to use kilowatt as the unit kw kva cos φ

5 Electrical Technology (ii) a pulsating component V m I m cos (ω t φ) which has a frequency twice that of the voltage and current. It does not contribute to actual power since its average value over a complete cycle is zero. Hence, average power consumed V m I m cos φ Vm. Im cos φ VI cos φ, where V and I represent the r.m.s values. Symbolic Notation. Z + jx L Impedance vector has numerical value of ( + X L ). Its phase angle with the reference axis is φ tan (X L /) It may also be expressed in the polar form as Z Z φ V V V (i) Assuming V V ; I φ (Fig. 3.8) Z Ζ φ Z It shows that current vector is lagging behind the voltage vector by φ. The numerical value of current is V/Z. (ii) However, if we assumed that I I, then V IZ I Z φ IZ φ It shows that voltage vector is φ ahead of current vector in ccw direction as shown in Fig. 3.9. Fig. 3.8 Fig. 3.9 3.. Power Factor It may be defined as (i) cosine of the angle of lead or lag (ii) the ratio resistance Z (...Fig. 3.3) (iii) the ratio true power watts W impedance apparent power volt amperes VA 3.3. Active and eactive Components of Circuit Current I Active component is that which is in phase with the applied voltage V i.e. I cos φ. It is also known as wattful component. eactive component is that which in quadrature with V i.e. I sin φ. It is also known as wattless or idle component. It should be noted that the product of volts and amperes in an a.c. circuit gives voltamperes (VA). Out of this, the actual power is VA cos φ W and reactive power is VA sin φ. Expressing the values in kva, we find that it has two rectangular components : (i) active component which is obtained by multiplying kva by cos φ and this gives power in kw. Fig. 3.

Series A.C. Circuits 5 (ii) the reactive component known as reactive kva and is obtained by multiplying kva by sin φ. It is written as kva (kilovar). The following relations can be easily deduced. kva kw + kva ; kw kva cos φ and kva kva sin φ These relationships can be easily understood by referring to the kva triangle of Fig. 3. where it should be noted that lagging kva has been taken as negative. For example, suppose a circuit draws a current of A at a voltage of, V and has a power factor of.8. Then input,,/, kva; cos φ.8; sin φ.6 Hence kw,.8 6, ; kva,.6, Obviously, 6 +, i.e. kva 3.4. Active, eactive and Apparent Power kw + kva Let a series -L circuit draw a current of I when an alternating voltage of r.m.s. value V is applied to it. Suppose that current lags behind the applied voltage by φ. The three powers drawn by the circuit are as under : (i) apparent power (S) It is given by the product of r.m.s. values of applied voltage and circuit current. S VI (IZ). I I Z volt-amperes (VA) (ii) active power (P or W) It is the power which is actually dissipated in the circuit resistance. P I VI cos φ watts Fig. 3. (iii) reactive power (Q) It is the power developed in the inductive reactance of the circuit. Q I X L I. Z sin φ I. (IZ). sin φ VI sin φ volt-amperes-reactive (VA) These three powers are shown in the power triangle of Fig. 3. from where it can be seen that S P + Q or S 3.5. Q-factor of a Coil P + Q eciprocal of power factor is called the Q-factor of a coil or its figure of merit. It is also known as quality factor of the coil. Q factor Z power factor cos φ If is small as compared to reactance, then Q-factor Z/ ω L/ maximum energy stored Also, Q π in the coil energy dissipated per cycle Example 3.. In a series circuit containing pure resistance and a pure inductance, the current and the voltage are expressed as : i (t) 5 sin (34 t + π/3) and v (t) 5 sin (34 t + 5 π/6) (a) What is the impedance of the circuit? (b) What is the value of the resistance? (c) What is the inductance in henrys? (d) What is the average power drawn by the circuit? (e) What is the power factor? [Elect. Technology, Indore Univ.] Solution. Phase angle of current π/3 8 /3 and phase angle of voltage 5 π/6 5 8 /6 5. Also, Z V m /I m 3 Ω. Hence, current lags behind voltage by 3. It means that it is an -L circuit. Also 34 π f or f 5 Hz. Now, /Z cos 3.866;.6 Ω ; X L /Z sin 3.5

5 Electrical Technology X L.5 Ω 34 L.5, L 4.78 mh (a) Z 3 Ω (b).6 Ω (c) L 4.78 mh (d) P I (5/ ).6 3.5 W (e) p.f. cos 3.866 (lag). Example 3.. In a circuit the equations for instantaneous voltage and current are givn by v 4.4 sin t 3 volt and i 7.7 sin t amp, where ω 34 rad/sec. (i) Sketch a neat phasor diagram for the circuit. (ii) Use polar notation to calculate impedance with phase angle. (iii) Calculate average power & power factor. (iv) Calculate the instantaneous power at the instant t. (F.Y. Engg. Pune Univ.) Solution. (i) From the voltage equation, it is seen that the voltage lags behind the reference quantity by π/3 radian or 8/3. Similarly, current lags behind the reference quantity by π/ radian or 8/ 9. Between themselves, voltage lags behind the current by ( 9) 3 as shwon in Fig. 3. (b). (ii) V V m/ 4.4 V; I I m/ 7.7/ 5 A. V and I 5 9 Z 3 5 9 (iii) Average power VI cos φ 5 cos 3 433 W (iv) At t ; v 4.4 sin ( ).45 V; i 7.7 sin ( 9 ) 7.7 A. instantaneous power at t is given by vi (.45) ( 7.7) 865.7 W. Fig. 3. Example 3.3. The potential difference measured across a coil is 4.5 V, when it carries a direct current of 9 A. The same coil when carries an alternating current of 9 A at 5 Hz, the potential difference is 4 V. Find the current, the power and the power factor when it is supplied by 5 V, 5 Hz supply. (F.Y. Pune Univ.) Solution. Let be the d.c. resistance and L be the inductance of the coil. V/I 4.5/9.5 Ω; With a.c. current of 5 Hz, Z V/I 4/9.66 Ω. X L Z.66.5.6 Ω. Now X L π 5 L ; L.67 Ω At 5 Hz X L.6 5.4 Ω ; Z.5 + 5.4 5.6 Ω Current I 5/5.6 9.5 A ; Power I 9.5.5 45 W.

Series A.C. Circuits 53 Example 3.4. In a particular -L series circuit a voltage of V at 5 Hz produces a current of 7 ma while the same voltage at 75 Hz produces 5 ma. What are the values of and L in the circuit? (Network Analysis A.M.I.E. Sec. B, S 99) Solution. (i) Z 3 + (π 5 L) + 98696 L ; V IZ or 7 ( + 98696 L) ( + 98696 L ) /7 3 /7 or + 98696 L /49...(i) (ii) In the second case Z + (π 75 L) ( + 66 L ) 5 3 ( 66 + L ) i.e. ( + 66 L ) or + 66L 4 (ii) Subtracting Eq. (i) from (ii), we get 66 L 98696 L 4 (/49) or 337 L 96 or L.398 H 4 mh. Substituting this value of L in Eq. (ii), we get, + 66 (.398) 4 6.9 Ω. Example 3.5. A series circuit consists of a resistance of 6 Ω and an inductive reactance of 8 Ω. A potential difference of 4.4 V (r.m.s.) is applied to it. At a certain instant the applied voltage is + V, and is increasing. Calculate at this current, (i) the current (ii) the voltage drop across the resistance and (iii) voltage drop across inductive reactance. (F.E. Pune Univ.) Solution. Z + jx 6 + j8 53. It shows that current lags behind the applied voltage by 53.. Let V be taken as the reference quantity. Then v (4.4 ) sin ωt sin ω t; i (V m /Z sin ωt) 3 sin (ωt 53. ). (i) When the voltage is + Vand increasing; sin ωt; sin ωt.5 ; ω t 3 At this instant, the current is given by i sin (3 53. ) sin 3. 7.847 A. (ii) drop across resistor i 7.847 6 47 V. (iii) Let us first find the equation of the voltage drop V L across the inductive reactance. Maximum value of the voltage drop I m X L 8 6 V. It leads the current by 9. Since current itself lags the applied voltage by 53., the reactive voltage drop across the applied voltage by (9 53. ) 36.9. Hence, the equation of this voltage drop at the instant when ω t 3 is V L 6 sin (3 + 36.9 ) 6 sin 66.9 47. V. Example 3.6. A 6 Hz sinusoidal voltage v 4 sin ω t is applied to a series -L circuit. The values of the resistance and the inductance are 3 Ω and.6 H respectively. (i) Compute the r.m.s. value of the current in the circuit and its phase angle with respect to the voltage. (ii) Write the expression for the instantaneous current in the circuit. (iii) Compute the r.m.s. value and the phase of the voltages appearing across the resistance and the inductance. (iv) Find the average power dissipated by the circuit. (v) Calculate the p.f. of the circuit. (F.E. Pune Univ.) Solution. V m 4 V; V 4/ V V + j X L π 6.6 4 Ω. Z 3 + j 4 5 53. (i) I V/Z /5 53. 53. Since angle is minus, the current lags behind the voltage by 53. (ii) I m 8.8; i 8.8 sin (ω t 53. ) (iii) V I 53. 3 6 53. volt. V L jix L 9 53. 4 8 36.9 (iv) P VI cos φ cos 53. W. (v) p.f. cos φ cos 53..6.

54 Electrical Technology Example 3.7. In a given -L circuit, 3.5 Ω and L. H. Find (i) the current through the circuit and (ii) power factor if a 5-Hz voltage V 3 is applied across the circuit. Solution. The vector diagram is shown in Fig. 3.3. X L π fl π 5. 3.4 Ω Z ( X ) 3.5 3.4 3.6 L Z 3.6 tan (3.4/3.5) 3.6 83.65 (i) I V Z 3 3.6 83.65 6.96 53.65 Fig. 3.3 (ii) Phase angle between voltage and current is 53.65 + 3 83.65º with current lagging. p.f. cos 83.65. (lag). Example 3.8. In an alternating circuit, the impressed voltage is given by V ( j5) volts and the current in the circuit is I (3 j4) A. Determine the real and reactive power in the circuit. (Electrical Engg., Calcutta Univ. 99) Solution. Power will be found by the conjugate method. Using current conjugate, we have P VA ( j 5) (3 + j 4) 3 + j 4 j 5 + 5 + j 5 Hence, real power is 5 W and reactive power of VA is 5. Since the second term in the above expression is positive, the reactive volt-amperes of 5 are inductive.* Example 3.9. In the circuit of Fig. 4.4, applied voltage V is given by ( + j) and the current is (.8 + j.6) A. Determine the values of and X and also indicate if X is inductive or capacitive. (Elect. Technology, Nagpur Univ. 99) Solution. V + j 9 ; I.8 + j.6 36.9 As seen, V leads the reference quantity by 9 whereas I leads by 36.9. In other words, I lags behind the applied voltage by (9 36.9 ) 53. Hence, the circuit of Fig. 3.4 is an -L circuit. Fig. 3.4 Now, Z V/I 9 / 36.9 53. 6 + j8 Hence, 6 Ω and X L 8 Ω. Example 3.. A two-element series circuit is connected across an a.c. source e sin (ωt + ) V. The current in the circuit then is found to be i cos (34 t 5 ) A. Determine the parameters of the circuit. (Electromechanic Allahabad Univ. 99) Solution. The current can be written as i sin (34 t 5 + 9 ) sin (34 t + 65 ). It is seen that applied voltage leads by and current leads by 65 with regards to the reference quantity, their mutual phase difference is 65 ( ) 45. Hence, p.f. cos 45 / (lead). Now, V m and I m Z V m /I m / Ω Z cos φ / Ω 4. Ω ; X c Z sin φ / 4. Ω Now, f 34/π 5 Hz. Also, X c /π fc C /π 5 4. 6 μf Hence, the given circuit is an -C circuit. * If voltage conjugate is used, then capacitive VAs are positive and inductive VAs negative. If current conjugate is used, then capacitive VAs are negative and inductive VAs are positive.

Series A.C. Circuits 55 Example 3.. Transform the following currents to the time domain : (i) 6 j 8 (ii) 6 + j8 (iii) j5. Solution. (i) Now, (6 j8) when expressed in the polar form is 6 + 8 tan 8/6 53.. The time domain representation of this current is i (t) sin (ω t 53. ) (ii) 6 + j8 6 + 8 tan 8/ 6 6.9 i (t) sin (ω t + 6.9 ) (iii) j 5 9 i (t) sin (ω t 9 ) Example 3.. A choke coil takes a current of A lagging 6 behind the applied voltage of V at 5 Hz. Calculate the inductance, resistance and impedance of the coil. Also, determine the power consumed when it is connected across -V, 5-Hz supply. (Elect. Engg. & Electronics, Bangalore Univ.) Solution. (i) Z coil / Ω ; Z cos φ - cos 6 5 Ω X L Z sin φ sin 6 86.6 Ω X L π fl 86.6 L 86.6/π 5.75 H (ii) Now, the coil will have different impedance because the supply frequency is different but its resistance would remain the same i.e. 5 Ω. Since the frequency has been halved, the inductive reactance of the coil is also halved i.e. it becomes 86.6/ 43.3 Ω. Choke coil Z coil 5 + 43.3 66. Ω I /66..5 A, p.f. cos φ 5/66..75 Power consumed by the coil VI cos φ.5.75.5 W Example 3.3. An inductive circuit draws A and kw from a -V, 5 Hz a.c. supply. Determine : (i) the impedance in cartesian from (a + jb) (ii) the impedance in polar from Z θ (iii) the power factor (iv) the reactive power (v) the apparent power. Solution. Z / Ω; P I or ; Ω; X L 7.3Ω. (i) Z + j7.3 (ii) Z ( + 7.3 ) Ω ; tan φ 7.3/.73; φ tan (.73) 6 Z 6. (iii) p.f. cos φ cos 6.5 lag (iv) reactive power VI sin φ.866 73 VA (v) apparent power VI VA. Example 3.4. When a voltage of V at 5 Hz is applied to a choking coil A, the current taken is 8 A and the power is W. When applied to a coil B, the current is A and the power is 5 W. What current and power will be taken when V is applied to the two coils connected in series? (Elements of Elect. Engg., Bangalore Univ.) Solution. Z /8.5 Ω ; P I. or 8 ; 5/8 Ω X Z.5 (5 / 8).36 Ω Z / Ω ; 5 or 5 Ω X 5 8.66 Ω

56 Electrical Technology With Joined in Series + (5/8) + 5 55/9 Ω; X.36 + 8.66. Ω Z (55/8) + (.). Ω, I /. 4.5 A, P I 4.5 55/8 4 W Example 3.5. A coil takes a current of 6 A when connected to a 4-V d.c. supply. To obtain the same current with a 5-Hz a.c. supply, the voltage required was 3 V. Calculate (i) the inductance of the coil (ii) the power factor of the coil. (F.Y. Engg. Pune Univ.) Solution. It should be kept in mind the coil offers only resistance to direct voltage whereas it offers impedance to an alternating voltage. 4/6 4 Ω; Z 3/6 5 Ω (i) X L Z 5 4 3 Ω (ii) p.f. cos φ /Z 4/5.8 (lag) Example 3.6. A resistance of ohm, inductance of. H and capacitance of 5 μf are connected in series and are fed by a 3 V, 5 Hz supply. Find X L, X C, Z, Y, p.f., active power and reactive power. (Elect. Science-I, Allahabad Univ. 99) Solution. X L π fl π 5. 6.8Ω ; X C /π fc 6 π 5 5. Ω ; X (X L X C ) 4.6 Ω ; Z + X + 4.6 46. Ω ; I V/ Z 3/46. 4.98 A Also, Z + jx + j 4.6 46. 64.3 ohm Y /Z /46. 64.3.6 64.3 siemens p.f. cos 64.3.4336 (lag) Active power VI cos φ 3 4.98.4336 497 W eactive power VI sin φ 3 4.98 sin 64.3 3 VA Example 3.7. A -V, 6-W lamp is to be operated on -V, 5-Hz supply mains. Calculate what value of (a) non-inductive resistance (b) pure inductance would be required in order that lamp is run on correct voltage. Which method is preferable and why? Solution. ated current of the bulb 6/.5 A (a) esistor has been shown connected in series with the lamp in Fig. 3.5 (a). P.D. across is V V It is in phase with the applied voltage, /.5 Ω (b) P.D. across bulb V P.D. across L is Fig. 3.5 V L ( ) 84.4 V (emember that V L is in quadrature with V the voltage across the bulb). Now, V L.5 X L or 84.4.5 L π 5 L 84.4/.5 3.4.7 H Method (b) is preferable to (a) because in method (b), there is no loss of power. Ohmic resistance of Ω itself dissipates large power (i.e..5 5 W). Example 3.8. A non-inductive resistor takes 8 A at V. Calculate the inductance of a choke coil of negligible resistance to be connected in series in order that this load may be supplied from -V, 5-Hz mains. What will be the phase angle between the supply voltage and current? (Elements of Elect. Engg.-I, Bangalore Univ. )

Series A.C. Circuits 57 Solution. It is a case of pure resistance in series with pure inductance as shown in Fig. 3.6 (a). Here V V, V L ( ) 96 V Now, V L I. X L or 96 8 π 5 L.78 H Example 3.9. A current of 5 A flows through a non-inductive resistance in series Fig. 3.6 with a choking coil when supplied at 5-V, 5-Hz. If the voltage across the resistance is 5 V and across the coil V, calculate (a) impedance, reactance and resistance of the coil (b) the power absorbed by the coil and (c) the total power. Draw the vector diagram. (Elect. Engg., Madras Univ.) Solution. As seen from the vector diagram of Fig. 3.7 (b). BC + CD...(i) (5 + BC) + CD 5...(ii) Subtracting Eq. (i) from (ii), we get, (5 + BC) BC 5 Fig. 3.7 BC 7.5 V; CD 7.5 98. V (i) Coil impedance /5 4 Ω V I BC or 5 7.5 Ρ 7.5/5 5.5 Ω Also V L I. X L CD 98. X L 98./5 39.6 Ω or X L 4 5.5 39.6 Ω (ii) Power absorbed by the coil is I 5 5.5 37.5 W Also P 5 7.5/ 37.5 W (iii) Total power VI cos φ 5 5 AC/AD 5 5 5.5/5 76.5 W The power may also be calculated by using I formula. Series resistance 5/5 5 Ω Total circuit resistance 5 + 5.5 3.5 Ω Total power 5 3.5 76.5 W Example 3.. Two coils A and B are connected in series across a 4-V, 5-Hz supply. The resistance of A is 5 Ω and the inductance of B is.5 H. If the input from the supply is 3 kw and kva, find the inductance of A and the resistance of B. Calculate the voltage across each coil. (Elect. Technology Hyderabad Univ. 99) Solution. The kva triangle is shown in Fig. 3.8 (b) and the circuit in Fig. 3.8(a). The circuit kva is given by, kva (3 + ) 3.66 or VA 3,66 voltmeters

58 Electrical Technology Fig. 3.8 Circuit current 3,66/4 5.3 A 5.3 ( A + B ) 3, A + B 3,/5.3 3.3 Ω B 3.3 5 8.3 Ω Now, impedance of the whole circuit is given by Z 4/5.3 5.97 Ω 3 X A + X B ( Z ( + ) 5.97 3.3 8.84 Ω A B Now X B π 5.5 4.73 Ω X A 8.843 4.73 4.3 Ω or π 5 L A 4.3 L A.3 H (approx) Now Z A A XA + 5 + 4.3 6.585 Ω P.D. across coil A I. Z A 5.3 6.485 97.5 V; Z b 8.3 + 4.73 9.545 Ω p.d. across coil B I. Z B 5.3 9.545 43.5 V Example 3.. An e.m.f. e 4.4 sin (377 t + 3 ) is impressed on the impedance coil having a resistance of 4 Ω and an inductive reactance of.5 Ω, measured at 5 Hz. What is the equation of the current? Sketch the waves for i, e, e L and e. Solution. The frequency of the applied voltage is f 377/π 6 Hz Since coil reactance is.5 Ω at 5 Hz, its value at 6 Hz.5 6/5 3 Ω Fig. 3.9 Coil impedance, Z 4 + 3, 5Ω ; φ tan (3/4) 36 5 It means that circuit current lags behind the applied voltage by 36 5. Hence, equation of the circuit current is i (4.4/5) sin (377 t + 3 36 5 ) 8.3 sin (377 t 6 5 ) Since, resistance drop is in phase with current, its equation is e i 3. sin (377 t 6 5 ) The inductive voltage drop leads the current by 9, hence its equation is e L ix L 3 8.3 sin (377 t 6 5 + 9 ) 54.9 sin (377 t + 83 8 ) The waves for i, e, e L and e have been drawn in Fig. 3.9.

Series A.C. Circuits 59 Example 3.. A single phase, 7.46 kw motor is supplied from a 4-V, 5-Hz a.c. mains. If its efficiency is 85% and power factor.8 lagging, calculate (a) the kva input (b) the reactive components of input current and (c) kva. output in watts 7.46 Solution. Efficiency.85 7, 46 input in watts VI cos VI.8 VI (a) 746,97 voltamperes.85.8 Input,97/.97 kva (b) Input current I voltamperes,97 7.43 A volts 4 Active component of current I cos φ 7.43.8.94 A eactive component of current I sin φ 7.43.6 6.46 A (ä sin φ.6) (eactive component 7.43.94 6.46 A (c) kva kva sin φ.97.6 6.58 (or kva VI sin φ 3 4 6.46 3 6.58) Example 3.3. Draw the phasor diagram for each of the following combinations : (i) and L in series and combination in parallel with C. (ii), L and C in series with X C > X L when ac voltage source is connected to it. [Nagpur University Summer ] Solution. (i) Fig. 3. (a) Circuit Fig. 3. (b) Phasor diagram Fig. 3. (c) Circuit Fig. 3. (d) Phasor diagram Example 3.4. A voltage v (t) 4.4 sin (34 t + ) is applied to a circuit and the steady current given by i(t) 4.4 sin (34 t ) is found to flow through it.

5 Electrical Technology Determine (i) The p.f. of the circuit (ii) The power delivered to the circuit (iii) Draw the phasor diagram. [Nagpur University Summer ] Solution. v (t) 4.4 sin (34 t + ) This expression indicates a sinusoidally varying alternating voltage at a frequency ω 34 rad/sec, f 5 Hz V MS voltage (Peak voltage)/ volts The expression for the current gives the following data : I MS value 4.4/ amp frequency 5 Hz, naturally. Phase shift between I and V 3, I lags behind V. (i) Power factor of the circuit cos 3.866 lag (ii) P VI cos φ.866 866 watts (iii) Phasor diagram as drawn below, in Fig. 3. (a). Fig. 3. (a) Phasor diagram Example 3.5. A coil of.8 p.f. is connected in series with micro-farad capacitor. Supply frequency is 5 Hz. The potential difference across the coil is found to be equal to that across the capacitor. Calculate the resistance and the inductance of the coil. Calculate the net power factor. [Nagpur University, November 997] Solution. X C /(3.4 C) 8.95 ohms Coil Impedance, Z 8.95 Ω Coil resistance 8.95.8 3.6 Ω Coil reactance 7.37 ohms Coil-inductance 7.37/34 55.3 milli-henrys Total impedance, Z T 3.6 + j 7.37 j 8.95 3.6 j.58 5.9 ohms Net power-factor 3.6/5.9.8943 leading Example 3.6. For the circuit shown in Fig. 3. (c), find the values of and C so that V b 3V a, and V b and V a are in phase quadrature. Find also the phase relationships between V s and V b, and V b and I. [ajiv Gandhi Technical University, Bhopal, Summer ] Fig. 3. (b) Fig. 3. (c)

Series A.C. Circuits 5 Solution. COA φ 53.3 BOE 9 53.3 36.87 DOA 34.7 Angle between V and I Angle between V s and V b 8.43 X L 34.55 8 ohms Z b 6 + j 8 53.3 ohms V b I 3 V a, and hence V a 3.33 I In phasor diagram, I has been taken as reference. V b is in first quadrant. Hence V a must be in the fourth quadrant, since Z a consists of and X c. Angle between V a and I is then 36.87. Since Z a and Z b are in series, V is represented by the phasor OD which is at angle of 34.7. V Va.53 I Thus, the circuit has a total effective impedance of.53 ohms. In the phasor diagram, OA 6 I, AC 8 I, OC I V b 3 V a Hence, V a OE 3.33 I, Since BOE 36.87, OB I OE cos 36.87 3.33.8 I.66 I. Hence,.66 And BE OE sin 36.87 3.33.6 I I Hence X c ohms. For X c ohms, C /(34 ) 59 μf Horizontal component of OD OB + OA 8.66 I Vertical component of OD AC BE 6 I OD.54 I V s Hence, the total impedance.54 ohms 8.66 + j 6 ohms Angle between V s and I DOA tan (6/866) 34.7 Example 3.7. A coil is connected in series with a pure capacitor. The combination is fed from a V supply of, Hz. It was observed that the maximum current of Amp flows in the circuit when the capacitor is of value microfarad. Find the parameters ( and L) of the coil. [Nagpur University April 996] Solution. This is the situation of resonance in A.C. Series circuit, for which X L X C Z V/I / 5 ohms If ω is the angular frequency, at resonance, L and C are related by ω /(LC), which gives L /(ω C).5 4 H.5 mh Example 3.8. Two impedances consist of (resistance of 5 ohms and series-connected inductance of.4 H) and (resistance of ohms, inductance of. H and a capacitance of μf, all in series) are connectd in series and are connected to a 3 V, 5 Hz a.c. source. Find : (i) Current drawn, (ii) Voltage across each impedance, (iii) Individual and total power factor. Draw the phasor diagram. [Nagpur University, Nov. 996] Solution. Let suffix be used for first impedance, and for the second one. At 5 Hz, Z 5 + j (34.4) 5 + j.56 ohms Z 5 +.56 9.56 ohms, Impedance-angle, θ cos (5/9.56) + 4, Z + j (34.) j {/(34 6 6 )} + j 3.4 j 3.85 j.45. ohms, Impedance angle, θ.56, Total Impedance, Z Z Z 5 + j.56 + j.85 5 + j. 7.78 5.85

5 Electrical Technology For this, Phase-angle of + 5.85, the power-factor of the total impedance cos 5.85.9, Lag. Current drawn 3/7.78 8.8 Amp, at 9 lagging p.f. V 8.8 9.56 6 Volts V 8.8. 8.9 Volts Individual Power-factor cos θ cos 4.766 Lagging cos θ cos.56.999 leading Phasor diagram : In case of a series circuit, it is easier to treat the current as a reference. The phasor diagram is drawn as in Fig. 3.. Fig. 3. Example 3.9. esistor ( ), choke-coil (r, L), and a capacitor of 5. μf are connected in series. When supplied from an A.C. source, in takes.4 A. If the voltage across the resistor is V, voltage across the resistor and choke is 45 volts, voltage across the choke is 35 volts, and voltage across the capacitor is 5 V. Find : (a) The values of r, L (b) Applied voltage and its frequency, (c) P.F. of the total circuit and active power consumed. Draw the phasor diagram. [Nagpur Univ. April 998] Solution. Fig. 3.3 (a) Fig. 3.3 (b) (b) Since the current I is.4 amp, and voltage drop across is V, /.4 5 ohms Similarly, Impedance of the coil, Z 35/.4 87.5 ohms Capacitive reactance X c 5/.4 5 ohms With a capacitor of 5.5 μf, and supply angular frequency of ω radians/sec ω X C. C 5 5.5 6, which gives ω 34 ra./sec. The corresponding source frequency, f 5 Hz (c) The phasor diagram is drawn in Fig. 3.3 (b), taking I as the reference. Solving triangle OAB,

45 + 35 cos φ.667, and hence φ 48. 45 Similarly, cos (8 β) 4 + 5 5 6.6 35 This gives Since Series A.C. Circuits 53 β 73.4. From the phasor diagram is Fig. 3.3 (b), OC OA + AC + 35 cos β 3 BC 35 sin β 33.54. The capacitive reactance drop is BD. BD 5, CD 6.46 volts. V s OC + CD 34. volts COD φ cos (OC/OD) 8.75 The power-factor of the total circuit cos φ.877, Leading, Since I Leads V s in Fig. 3.3 (b). (a) For the coil, ACB part of the phasor diagram is to be observed. r AC/I /.4 5 ohms X L BC/I 33.54/.4 83.85 ohms Hence, coil-inductance, L 83.85/34 67 milli-henry. P Active Power Consumed V s I cos φ watts or P (.4) ( + r) watts 3.6. Power in an Iron-cored Choking Coil Total power, P taken by an iron-cored choking coil is used to supply (i) power loss in ohmic resistance i.e. I. (ii) iron-loss in core, P i P I P P + P i or + is known as the effective resistance* of the choke. I I P i P P effective resistance true resistance equivalent resistance I eff + i I I Example 3.3. An iron-cored choking coil takes 5 A when connected to a -V d.c. supply and takes 5 A at V a.c. and consumes 5 W. Determine (a) impedance (b) the power factor (c) the iron loss (d) inductance of the coil. (Elect. Engg. & M.A.S.I. June, 99) Solution. (a) Z /5 Ω (b) P VI cos φ or 5 5 cos φ cos φ 5/5.5 (c) Total loss loss in resistance + iron loss 5 5 + P i P i 5 5 W P 5 (d) Effective resistance of the choke is Ω I 5 X L ( Z ) (4 ) 7.3 Ω Example 3.3. An iron-cored choking coil takes 5 A at a power factor of.6 when supplied at -V, 5 Hz. When the iron core is removed and the supply reduced to 5 V, the current rises to 6 A at power factor of.9. Determine (a) the iron loss in the core (b) the copper loss at 5 A (c) the inductance of the choking coil with core when carrying a current of 5 A. * At higher frequencies like radio frequencies, there is skin-effect loss also.

54 Electrical Technology Solution. When core is removed, then Z 5/6.5 Ω True resistance, Z cos φ.5.9.5 Ω With Iron Core Power input 5.6 3 W Power wasted in the true resistance of the choke when current is 5 A 5.5 56. W (a) Iron loss 3 56. 44 W (approx) (b) Cu loss at 5 A 56. W (c) Z /5 Ω ; X L Z sin φ.8 6Ω π 5 L 6 L.59 N Tutorial Problem No. 3.. The voltage applied to a coil having Ω, L 638 mh is represented by e sin π t. Find a corresponding expression for the current and calculate the average value of the power taken by the coil. [i.77 sin ( πt π/4); 5 W] (I.E.E. London). The coil having a resistance of Ω and an inductance of. H is connected to a -V, 5-Hz supply. Calculate (a) the impedance of the coil (b) the reactance of the coil (c) the current taken and (d) the phase difference between the current and the applied voltage. [(a) 63.5 Ω (b) 6.8 Ω (c).575 A (d) 8 57 ] 3. An inductive coil having a resistance of 5 Ω takes a current of 4 A when connected to a -V, 6 Hz supply. If the coil is connected to a -V, 5 Hz supply, calculate (a) the current (b) the power (c) the power factor. Draw to scale the vector diagram for the 5-Hz conditions, showing the component voltages. [(a) 4.46 A (b) 98 W (c).669] 4. When supplied with current at 4-V, single-phase at 5 Hz, a certain inductive coil takes 3.6 A. If the frequency of supply is changed to 4 Hz, the current increases to 6. A. Calculate the resistance and inductance of the coil. [7. W,.5 H] (London Univ.) 5. A voltage v (t) 4.4 sin (34 t + ) is applied to a circuit and a steady current given by i (t) 4.4 sin (34 t ) is found to flow through it. Determine (i) the p.f. of the circuit and (ii) the power delivered to the circuit. [.866 (lag); 866 W] 6. A circuit takes a current of 8 A at V, the current lagging by 3 behind the applied voltage. Calculate the values of equivalent resistance and reactance of the circuit. [.8 Ω ; 6.5 Ω] 7. Two inductive impedance A and B are connected in series. A has 5 Ω, L. H; B has 3 Ω, L. H. If a sinusoidal voltage of 3 V at 5 Hz is applied to the whole circuit calculate (a) the current (b) the power factor (c) the voltage drops. Draw a complete vector diagram for the circuit. [(a) 8.6 (b).648 (c) V A 9.5 V, V B 9.5 V] (I.E.E. London) 8. A coil has an inductance of. H and a resistance of 3 Ω at C. Calculate (i) the current and (ii) the power taken from -V, 5-Hz mains when the temperature of the coil is 6 C, assuming the temperature coefficient of resistance to be.4% per C from a basic temperature of C. [(i).3 A (ii) 58.5 W] (London Univ.) 9. An air-cored choking coil takes a current of A and dissipates W when connected to a -V, 5- Hz mains. In other coil, the current taken is 3 A and the power 7 W under the same conditions. Calculate the current taken and the total power consumed when the coils are in series and connected to the same supply. [., 5 W] (City and Guilds, London). A circuit consists of a pure resistance and a coil in series. The power dissipated in the resistance is 5 W and the drop across it is V. The power dissipated in the coil is W and the drop across it is 5 V. Find the reactance and resistance of the coil and the supply voltage. [9.68Ω; 4Ω; 8.5V]. A choking coil carries a current of 5 A when supplied from a 5-Hz, 3-V supply. The power in the circuit is measured by a wattmeter and is found to be 3 watt. Estimate the phase difference between the current and p.d. in the circuit. [.3768] (I.E.E. London). An ohmic resistance is connected in series with a coil across 3-V, 5-Hz supply. The current is.8

Series A.C. Circuits 55 A and p.ds. across the resistance and coil are 8 V and 7 V respectively. Calculate the resistance and inductance of the coil and the phase difference between the current and the supply voltage. [6. Ω,.9 H, 34 ] (App. Elect. London Univ.) 3. A coil takes a current of 4 A when 4 V d.c. are applied and for the same power on a 5-Hz a.c. supply, the applied voltage is 4. Explain the reason for the difference in the applied voltage. Determine (a) the reactance (b) the inductance (c) the angle between the applied p.d. and current (d) the power in watts. [(a) 8 Ω (b).55 H (c) 53 7 (d) 96 W] 4. An inductive coil and a non-inductive resistance ohms are connected in series across an a.c. supply. Derive expressions for the power taken by the coil and its power factor in terms of the voltage across the coil, the resistance and the supply respectively. If Ω and the three voltages are in order, V, 8 V and 4 V, calculate the power and the power factor of the coil. [546 W;.33] 5. Two coils are connected in series. With A d.c. through the circuit, the p.ds. across the coils are and 3 V respectively. With A a.c. at 4 Hz, the p.ds. across the coils are 4 and V respectively. If the two coils in series are connected to a 3-V, 5-Hz supply, calculate (a) the current (b) the power (c) the power factor. [(a).55 A (b) 6 W (c).684] 6. It is desired to run a bank of ten -W, -V lamps in parallel from a 3-V, 5-Hz supply by inserting a choke coil in series with the bank of lamps. If the choke coil has a power factor of., find its resistance, reactance and inductance. [ 4.44 Ω, X.35 Ω, L.65 H] (London Univ.) 7. At a frequency for which ω 796, an e.m.f. of 6 V sends a current of ma through a certain circuit. When the frequency is raised so that ω 866, the same voltage sends only 5 ma through the same circuit. Of what does the circuit consist? [ 5 Ω, L.38 H in series] (I.E.E. London) 8. An iron-cored electromagnet has a d.c. resistance of 7.5 Ω and when connected to a 4-V 5-Hz supply, takes A and consumes kw. Calculate for this value of current (a) power loss in iron core (b) the inductance of coil (c) the power factor (d) the value of series resistance which is equivalent to the effect of iron loss. [.5 kw,. H,.5;.5 Ω] (I.E.E. London) 3.7. A.C. Through esistance and Capacitance The circuit is shown in Fig. 3.4 (a). Here V I drop across in phase with I. V C IX C drop across capacitor lagging I by π/ As capacitive reactance X C is taken negative, V C is shown along negative direction of Y-axis in the voltage triangle [Fig. 3.4 (b)] Now V Fig. 3.4 V + ( VC) ( I) + ( IXC) I + XC or I V V Z + XC The denominator is called the impedance of the circuit. So, Z + X C Impedance triangle is shown in Fig. 3.4 (c) From Fig. 3.4 (b) it is found that I leads V by angle φ such that tan φ X C /

56 Electrical Technology Hence, it means that if the equation of the applied alternating voltage is v V m sin ωt, the equation of the resultant current in the -C circuit is i I m sin (ωt + φ) so that current leads the applied voltage by an angle φ. This fact is shown graphically in Fig. 3.5. Example 3.3. An a.c. voltage (8 + j 6) volts is applied to a circuit and the current flowing is ( 4 + j ) amperes. Find (i) impedance of the circuit (ii) power consumed and (iii) phase angle. [Elect. Technology, Indore, Univ., Bombay Univ. 999] Fig. 3.5 Solution. V (8 + j 6) 36.9 ; I 4 + j.77 tan (.5).77 (8 68. ).77.8 (i) Z V/I 36.9 /.77.8 9.8 74.9 9.8 (cos 74.9 j sin 74.9 ).4 j 8.96 Ω Hence.4 Ω and X C 8.96 Ω capacitive (ii) P I.77.4.8 W (iii) Phase angle between voltage and current 74.9 with current leading as shown in Fig. 3.6. Alternative Method for Power The method of conjugates will be used to determine the real power and reactive volt-ampere. It is a convenient way of calculating these quantities when both voltage and current are expressed in cartesian form. If the conjugate of Fig. 3.6 current is multiplied by the voltage in cartesian form, the result is a complex quantity, the real part of which gives the real power j part of which gives the reactive volt-amperes (VA). It should, however, be noted that real power as obtained by this method of conjugates is the same regardless of whether V or I is reversed although sign of voltamperes will depend on the choice of V or I.* Using current conjugate, we get P VA (8 + j 6) ( 4 j ) 8 j 4 Power consumed 8 W Example 3.33. In a circuit, the applied voltage is V and is found to lag the current of A by 3. (i) Is the p.f. lagging or leading? (ii) What is the value of p.f.? (iii) Is the circuit inductive or capacitive? (iv) What is the value of active and reactive power in the circuit? (Basic Electricity, Bombay Univ.) Solution. The applied voltage lags behind the current which, in other words, means that current leads the voltage. (i) p.f. is leading (ii) p.f. cos φ cos 3.866 (lead) (iii) Circuit is capacitive (iv) Active power VI cos φ.866 866 W eactive power VI sin φ.5 5 VA (lead) or VA ( VA) W ( ) 866 5 (lead) * If voltage conjugate is used, then capacitive VAs are positive and inductive VAs negative. If current conjugate is used, then capacitive VAs are negative and inductive VAs are positive.

Series A.C. Circuits 57 Example 3.34. A tungsten filament bulb rated at 5-W, -V is to be connected to series with a capacitance across -V, 5-Hz supply. Calculate : (a) the value of capacitor such that the voltage and power consumed by the bulb are according to the rating of the bulb. (b) the power factor of the current drawn from the supply. (c) draw the phasor diagram of the circuit. (Elect. Technology-, Nagpur Univ. 99) Solution. The rated values for bulb are : voltage V and current I W/V 5/ 5A. Obviously, the bulb has been treated as a pure resistance : (a) V C 96 V Now, IX C 96 or 5 X C 96, X C 39. Ω I/ωC 39. or C /34 39. 8 μf (b) p.f. cos φ V /V /.455 (lead) (c) The phasor diagram is shown in Fig 3.7. Example 3.35. A pure resistance of 5 ohms is in series with a pure capacitance of microfarads. The series combination is connected across -V, 5-Hz supply. Find (a) the impedance (b) current (c) power factor (d) phase angle (e) voltage across resistor (f) voltage across capacitor. Draw the vector diagram. (Elect. Engg.-, JNT Univ. Warrangel) Solution. X C 6 π /π 5 3 Ω ; 5 Ω (a) Z 5 + 3 59.4 Ω (b) I V/Z /59.4.684 A (c) p.f. /Z 5 59.4.84 (lead) (d) φ cos (.84) 3 36 (e) V I 5.684 84. V (f) V C IX C 3.684 53.9 V Example 3.36. A 4-V, 5-Hz series -C circuit takes an r.m.s. current of A. The maximum value of the current occurs /9 second before the maximum value of the voltage. Calculate (i) the power factor (ii) average power (iii) the parameters of the circuit. (Elect. Engg.-I, Calcutta Univ.) Solution. Time-period of the alternating voltage is /5 second. Now a time interval of /5 second corresponds to a phase difference of π radian or 36. Hence, a time interval of /9 second corresponds to a phase difference of 36 5/9. Hence, current leads the voltage by º. (i) power factor cos.9397 (lead) (ii) averge power 4.9397 4,5 W (iii) Z 4/ Ω; Z cos Φ.9397.8 Ω X C Z sin Φ sin.34 4. Ω C 6 /π 5 4. 775 μf Example 3.37. A voltage v sin 34 t is applied to a circuit consisting of a 5 Ω resistor and an 8 μf capacitor in series. Determine : (a) an expression for the value of the current flowing at any instant (b) the power consumed (c) the p.d. across the capacitor at the instant when the current is one-half of its maximum value. Solution. X C /(34 8 6 ) 39.8 Ω, Z I m V m /Z /47.3 A φ tan (39.8/5) 57 5. radian (lead) Fig 3.7 5 + 39.8 47 Ω

58 Electrical Technology (a) Hence, equation for the instantaneous current i.3 sin (34 t +.) (b) Power I (.3/ ) 5 56.7 W (c) The voltage across the capacitor lags the circuit current by π/ radians. Hence, its equation is given by v c Vcm sin ( 34 t +. π) where Vcm Im XC.3 39.8 84.8 V Now, when i is equal to half the maximum current (say, in the positive direction) then i.5.3 A.5.3.3 sin (34 t +.) or 34 t +. sin π 5 (.5) or π radian 6 6 π v C 84.8 sin ( π 6 ) 84.8 sin ( π/3) 73.5 V 5π or v c 34.8 sin ( π 6 ) 84.4 sin π/3 73.5 V Hence, p.d. across the capacitor is 73.5 V Example 3.38. A capacitor and a non-inductive resistance are connected in series to a -V, single-phase supply. When a voltmeter having a non-inductive resistance of 3,5 Ω is connected across the resistor, it reads 3 V and the current then taken from the supply is.35 ma. Indicate on a vector diagram, the voltages across the two components and also the supply current (a) when the voltmeter is connected and (b) when it is disconnected. Solution. The circuit and vector diagrams are shown in Fig. 3.8 (a) and (b) respectively. (a) V C 3 5 V It is seen that φ tan (5/3) 49 in Fig. 3.8 (b). Hence (i) Supply voltage lags behind the current by 49. (ii) V leads supply voltage by 49 (iii) V C lags behind the supply voltage by (9 49 ) 4 The supply current is, as given equal to.35 ma. The value of unknown resistance can be found as follows : Current through voltmeter 3/3,5 9.78 ma Current through.35 9.78.57 ma 3/.57 3,5 Ω ; X C 5/.35 3 6.7 Ω (b) When voltmeter is disconnected, Z Supply current /,46 6. ma Fig. 3.8 In this case, V 6. 3,5 68 V V C 6. 3 67 7.4 V; tan φ 7.4/68 + X L,5 + 6,73,5 Ω

Series A.C. Circuits 59 φ 3.5. In this case, the supply voltage lags the circuit current by 3.5 as shown in Fig. 3.8 (c). Example 3.39. It is desired to operate a -W, -V electric lamp at its current rating from a 4-V, 5-Hz supply. Give details of the simplest manner in which this could be done using (a) a resistor (b) a capacitor and (c) an indicator having resistance of Ω. What power factor would be presented to the supply in each case and which method is the most economical of power. (Principles of Elect. Engg.-I, Jadavpur Univ.) Solution. ated current of the bulb is / 5/6 A The bulb can be run at its correct rating by any one of the three methods shown in Fig. 3.9. (a) With reference to Fig. 3.9 (a), we have P.D. across 4 V /(5/6) 44 Ω Power factor of the circuit is unity. Power consumed 4 5/6 W (b) eferring to Fig. 3.9 (b), we have Fig. 3.9 V C 4 7.5 V; X C 7.5 (5 / 6) 49 Ω /34 C 49 or C.8 μf; p.f. cos φ /4.5 (lead) Power consumed 4 (5/6).5 W (c) The circuit connections are shown in Fig 3.9 (c) 5 V (5/6) 5/3 V V L ( ) 4 + 3V 3 34 L (5/6) 3 L.775 H Total resistive drop + (5/3) 8.3 V; cos φ 8.3/4.535 (lag) Power consumed 4 (5/6).535 7 W Method (b) is most economical because it involves least consumption of power. Example 3.4. A two-element series circuit consumes 7 W and has a p.f..77 leading. If applied voltage is v 4. sin (34 t + 3 ), find the circuit constants. Solution. The maximum value of voltage is 4.4 V and it leads the reference quantity by 3. Hence, the given sinusoidal voltage can be expressed in the phase form as V (4.4/ ) 3 3 now, P VI cos φ 7 I.77; I A. Since p.f..77 (lead); φ cos (.77) 45 (lead). It means that current leads the given voltage by 45 for it leads the common reference quantity

53 Electrical Technology by (3 + 45 ) 75. Hence, it can be expressed as I 75 Z V 3 j7. I 75 Since X C 7. /34 C 7.; C 45 μf 7. Ω 3.8. Dielectric Loss and Power Factor of a Capacitor An ideal capacitor is one in which there are no losses and whose current leads the voltage by 9 as shown in Fig. 3.3 (a). In practice, it is impossible to get such a capacitor although close approximation is achieved by proper design. In every capacitor, there is always some dielectric loss and hence it absorbs some power from the circuit. Due to this loss, the phase angle is somewhat less than 9 [Fig. 3.3 (b)]. In the case of a capacitor with a poor dielectric, the loss can be considerable and the phase angle much less than 9. This dielectric loss appears as heat. By phase difference is meant the difference between the ideal and actual phase angles. As seen from Fig. 3.3 (b), the phase difference ψ is given by ψ 9 φ where φ is the actual phase angle, sin ψ sin (9 φ) cos φ where cos φ is the power factor of the capacitor. Fig. 3.3 Since ψ is generally small, sin ψ ψ (in radians) tan ψ ψ cos φ. It should be noted that dielectric loss increases with the frequency of the applied voltage. Hence phase difference increases with the frequency f. The dielectric loss of an actual capacitor is allowed for by imagining it to consist of a pure capacitor having an equivalent resistance either in series or in parallel with it as shown in Fig. 3.3. These resistances are such that I loss in them is equal to the dielectric loss in the capacitor. Ise se As seen from Fig. (3.7b), tan ψ Cse IX C / C se tan ψ/ω C p.f./ωc I V / sh XC Similarly, as seen from Fig. 3.3 (d), tan ψ I VX / C C sh sh Fig. 3.3 sh ωc.tanψ ω C powerfactor ω C p.f. The power loss in these resistances is P V / sh ωcv tan ψ ω CV p.f. or I se (I p.f.)/ω C where p.f. stands for the power factor of the capacitor. XC Note. (i) In case, ψ is not small, then as seen from Fig. 3.4 (b) tan φ (Ex. 3.4) se X C /tan φ se

From Fig. 3.3 (d), we get tan φ I VX / I V X Series A.C. Circuits 53 C sh / sh X C tan φ tan φ/ωc sh C (ii) It will be seen from above that both se and sh vary inversely as the frequency of the applied voltage. In other words, the resistance of a capacitor decreases in proportion to the increase in frequency. se f se f Example 3.4. A capacitor has a capacitance of μf and a phase difference of. It is inserted in series with a Ω resistor across a -V, 5-Hz line, Find (i) the increase in resistance due to the insertion of this capacitor (ii) power dissipated in the capacitor and (iii) circuit power factor. 6 Solution. X C 38.3 π 5 Ω The equivalent series resistance of the capacitor in Fig. 3.3 is se X C /tan φ Now φ 9 ψ 9 8 tan φ tan 8 5.67 se 38.3/5.67 56. Ω (i) Hence, resistance of the circuit increases by 56. Ω. (ii) Z ( + ) + X 56. + 38.3 354.4 Ω ; I / 354.6 A se C Power dissipated in the capacitor I se.6 56..6 W (iii) Circuit power factor is ( + se )/Z 56./354.4.44 (lead) Example 3.4. Dielectric heating is to be employed to heat a slab of insulating material cm thick and 5 sq. cm in area. The power required is W and a frequency of 3 MHz is to be used. The material has a relative permittivity of 5 and a power factor of.3. Determine the voltage necessary and the current which will flow through the material. If the voltage were to be limited to 6-V, to what would the frequency have to be raised? [Elect. Engg. AMIETE (New Scheme) June 99] Solution. The capacitance of the parallel-plate capacitor formed by the insulating slab is 4 r 8.854 5 5 C A εε 33. F d As shown in Art. 3.8 sh 396 Ω ω C p. f. 6 (π 3 ) 33..5 Now, P V / sh or V P sh 396 8 V Current I V/X C ωcv (π 3 6 ) 33. 8 5 A V Now, as seen from above P V V ω C p.f. or P V f / ω C p. f. 8 3 8 ( ) sh 6 f or f 3 53.3 MHz 6 Tutorial Problem No. 3. Fig. 3.3. A capacitor having a capacitance of μf is connected in series with a non-inductive resistance of Ω across a -V, 5-Hz supply, Calculate (a) voltage (b) the phase difference between the current and the supply voltage (c) the power. Also draw the vector diagram. [(a).5a (b) 5.9 (c) 3. W]

53 Electrical Technology. A capacitor and resistor are connected in series to an a.c. supply of 5 V and 5 Hz. The current is A and the power dissipated in the circuit is 8 W. Calculate the resistance of the resistor and the capacitance of the capacitor. [ Ω ; μf] 3. A voltage of 5 V at 5 Hz is applied to a series combination of non-inductive resistor and a lossless capacitor of 5 μf. The current is.5 A. Find (i) the value of the resistor (ii) power drawn by the network (iii) the power factor of the network. Draw the phasor diagram for the network. [(i) 77.3 Ω (ii) W (iii).773 (lead)] (Electrical Technology-, Osmania Univ.) 4. A black box contains a two-element series circuit. A voltage (4 j3) drives a current of (4 j3) A in the circuit. What are the values of the elements? Supply frequency is 5 Hz. [.5 ; C 475 μf] (Elect. Engg. and Electronics Bangalore Univ.) 5. Following readings were obtained from a series circuit containing resistance and capacitance : V 5 V ; I.5 A; P 37.5 W, f 6 Hz. Calculate (i) Power factor (ii) effective resistance (iii) capacitive reactance and (iv) capacitance. [(i). (ii) 6 Ω (iii) 59.7 Ω (iv) 44.4 μf] 6. An alternating voltage of volt at a frequency of 59 khz is applied across a capacitor of. μf. Calculate the current in the capacitor. If the power dissipated within the dielectric is μw, calculate (a) loss angle (b) the equivalent series resistance (c) the equivalent parallel resistance. [.A (a) -4 radian (b). Ω (c) MΩ] 3.9. esistance, Inductance and Capacitance in Series The three are shown in Fig. 3.33 (a) joined in series across an a.c. supply of r.m.s. voltage V. Fig. 3.33 Let V I voltage drop across in phase with I V L I.X L voltage drop across L leading I by π/ V C I.X C voltage drop across C lagging I by π/ In voltage triangle of Fig. 3.33 (b), OA represents V, AB and AC represent the inductive and capacitive drops respectively. It will be seen that V L and V C are 8 out of phase with each other i.e. they are in direct opposition to each other. Subtracting BD ( AC) from AB, we get the net reactive drop AD I (X L X C ) The applied voltage V is represented by OD and is the vector sum of OA and AD OD OA AD or V ( I) ( IXL IXC) I ( XL XC) or I V V V + ( X X ) + X Z L C The term + ( X X ) is known as the impedance of the circuit. Obviously, L C (impedance) (resistance) + (net reactance) or Z + (X L X C ) + X