Electrical Circuit Theory and Technology

Bird J.

Chapter 24 Application of complex numbers to series a.c. circuits - all with Video Answers

Educators

Chapter Questions

Problem 1

Determine the resistance $R$ and series inductance $L$ (or capacitance C) for each of the following impedances, assuming the frequency to be 50 Hz . (a) $(4+j 7) \Omega$ (b) $(3-j 20) \Omega$ (c) $j 10 \Omega$ (d) $-j 3 \mathrm{k} \Omega$ (e) $15 \angle(\pi / 3) \Omega$ (f) $6 \angle-45^{\circ} \mathrm{M} \Omega$
[(a) $R=4 \Omega, L=22.3 \mathrm{mH}$
(b) $R=3 \Omega, C=159.3 \mu \mathrm{~F}$
(c) $R=0, L=31.8 \mathrm{mH}$
(d) $R=0, C=1.061 \mu \mathrm{~F}$
(e) $R=7.5 \Omega, L=41.3 \mathrm{mH}$
(f) $R=4.243 \mathrm{M} \Omega, C=0.750 \mathrm{nF}]$

Varsha Aggarwal

Numerade Educator

01:17

Problem 2

$ \mathrm{~A} 0.4 \mu \mathrm{~F}$ capacitor is connected to a $250 \mathrm{~V}, 2 \mathrm{kHz}$ supply. Determine the current flowing.
[1.257 $\angle 90^{\circ} \mathrm{A}$ or $j 1.247 \mathrm{~A}$ ]

Salamat Ali

Numerade Educator

04:59

Problem 3

Two voltages in a circuit are represented by $(15+j 10) \mathrm{V}$ and $(12-j 4) \mathrm{V}$. Determine the magnitude of the resultant voltage when these voltages are added.
[27.66 V]

Mirza Aslam Beig

Numerade Educator

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Problem 4

A current of $2.5 \angle-90^{\circ}$ A flows in a coil of inductance 314.2 mH and negligible resistance when connected across a 50 Hz supply. Determine the value of the supply p.d.
$\left[246.8 \angle 0^{\circ} \mathrm{V}\right]$

Eduard Sanchez

Numerade Educator

01:18

Problem 5

A voltage $(75+j 90) \mathrm{V}$ is applied across an impedance and a current of $(5+j 12) \mathrm{A}$ flows. Determine (a) the value of the circuit impedance, and (b) the values of the components comprising the circuit if the frequency is 1 kHz .
[(a) $Z=(8.61-j 2.66) \Omega$ or $9.01 \angle-17.19^{\circ} \Omega$
(b) $R=8.61 \Omega, C=59.83 \mu \mathrm{~F}]$

Ze-Han Lee

Numerade Educator

03:15

Problem 6

Determine, in polar form, the complex impedances for the circuits shown in Figure 24.19 if the frequency in each case is 50 Hz .
[(a) $44.53 \angle-63.31^{\circ} \Omega$ (b) $19.77 \angle 52.62^{\circ} \Omega$ (c) $\left.113.5 \angle-58.08^{\circ} \Omega\right]$

Varsha Aggarwal

Numerade Educator

00:51

Problem 7

For the circuit shown in Figure 24.20 determine the impedance $Z$ in polar and rectangular forms.

$$
\left[Z=(1.85+j 6.20) \Omega \text { or } 6.47 \angle 73.39^{\circ} \Omega\right]
$$

Wendi Zhao

Numerade Educator

03:17

Problem 8

A $30 \mu \mathrm{~F}$ capacitor is connected in series with a resistance $R$ at a frequency of 200 Hz . The resulting current leads the voltage by $30^{\circ}$. Determine the magnitude of $R$.
$[45.95 \Omega]$

Vishal Gupta

Numerade Educator

05:28

Problem 9

A coil has a resistance of $40 \Omega$ and an inductive reactance of $75 \Omega$. The current in the coil is $1.70 \angle 0^{\circ} \mathrm{A}$. Determine the value of (a) the supply voltage, (b) the p.d. across the $40 \Omega$ resistance, (c) the p.d. across the inductive part of the coil, and (d) the circuit phase angle. Draw the phasor diagram.
[(a) $(68+j 127.5) \mathrm{V}$ or $144.5 \angle 61.93^{\circ} \mathrm{V}$ (b) $68 \angle 0^{\circ} \mathrm{V}$
(c) $127.5 \angle 90^{\circ} \mathrm{V}$ (d) $61.93^{\circ}$ lagging]

Vishal Gupta

Numerade Educator

03:46

Problem 10

An alternating voltage of $100 \mathrm{~V}, 50 \mathrm{~Hz}$ is applied across an impedance of $(20-j 30) \Omega$. Calculate (a) the resistance, (b) the capacitance, (c) the current, and (d) the phase angle between current and voltage
[(a) $20 \Omega$ (b) $106.1 \mu \mathrm{~F}$ (c) 2.774 A (d) $56.31^{\circ}$ leading]

Varsha Aggarwal

Numerade Educator

02:35

Problem 11

A capacitor $C$ is connected in series with a coil of resistance $R$ and inductance 30 mH . The current flowing in the circuit is $2.5 \angle-40^{\circ} \mathrm{A}$ when the supply p.d. is 200 V at 400 Hz . Determine the value of (a) resistance $R$, (b) capacitance $C$, (c) the p.d. across $C$, and (d) the p.d., across the coil. Draw the phasor diagram.
[(a) $61.28 \Omega$ (b) $16.59 \mu \mathrm{~F}$
(c) $59.95 \angle-130^{\circ} \mathrm{V}$ (d) $242.9 \angle 10.90^{\circ} \mathrm{V}$ ]

Mayukh Banik

Numerade Educator

12:19

Problem 12

A series circuit consists of a $10 \Omega$ resistor, a coil of inductance 0.09 H and negligible resistance, and a $150 \mu \mathrm{~F}$ capacitor, and is connected to a $100 \mathrm{~V}, 50 \mathrm{~Hz}$ supply. Calculate the current flowing and its phase relative to the supply voltage.
[8.17 A lagging $V$ by $35.20^{\circ}$ ]

Yaqub Khan

Numerade Educator

06:08

Problem 13

A $150 \mathrm{mV}, 5 \mathrm{kHz}$ source supplies an ac. circuit consisting of a coil of resistance $25 \Omega$ and inductance 5 mH connected in series with a capacitance of 177 nF . Determine the current flowing and its phase angle relative to the source voltage.
$\left[4.44 \angle 42.31^{\circ} \mathrm{mA}\right]$

Khoobchandra Agrawal

Numerade Educator

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Problem 14

Two impedances, $Z_1=5 \angle 30^{\circ} \Omega$ and $Z_2=10 \angle 45^{\circ} \Omega$ draw a current of 3.36 A when connected in series to a certain a.c. supply. Determine (a) the supply voltage, (b) the phase angle between the voltage and current, (c) the p.d. across $Z_1$, and (d) the p.d. across $Z_2$.
[(a) 50 V
(b) $40.01^{\circ}$ lagging
(c) $16.8 \angle 30^{\circ} \mathrm{V}$
(d) $33.6 \angle 45^{\circ} \mathrm{V}$ ]

Sikandar Baig

Numerade Educator

04:42

Problem 15

A 4500 pF capacitor is connected in series with a $50 \Omega$ resistor across an alternating voltage $v=212.1 \sin \left(\pi 10^6 t+\pi / 4\right)$ volts. Calculate (a) the rms value of the voltage, (b) the circuit impedance, (c) the rms current flowing, (d) the circuit phase angle, (e) the voltage across the resistor, and (f) the voltage across the capacitor.
[(a) $150 \angle 45^{\circ} \mathrm{V}$ (b) $86.63 \angle-54.75^{\circ} \Omega$
(c) $1.73 \angle 99.75^{\circ} \mathrm{A}$ (d) $54.75^{\circ}$ leading
(e) $86.50 \angle 99.75^{\circ} \mathrm{V}$ (f) $122.38 \angle 9.75^{\circ} \mathrm{V}$ ]

Vishal Gupta

Numerade Educator

02:15

Problem 16

If the p.d. across a coil is $(30+j 20) \mathrm{V}$ at 60 Hz and the coil consists of a 50 mH inductance and $10 \Omega$ resistance, determine the value of current flowing (in polar and cartesian forms).

$$
\left[1.69 L-28.36^{\circ} \mathrm{A} ;(1.49-j 0.80) \mathrm{A}\right]
$$

Anurag Kumar

Numerade Educator

05:53

Problem 17

Three impedances are connected in series across a $120 \mathrm{~V}, 10 \mathrm{kHz}$ supply. The impedances are:
(i) $Z_1$, a coil of inductance $200 \mu \mathrm{H}$ and resistance $8 \Omega$
(ii) $Z_2$, a resistance of $12 \Omega$
(iii) $Z_3$, a $0.50 \mu \mathrm{~F}$ capacitor in series with a $15 \Omega$ resistor.

Determine (a) the circuit impedance, (b) the circuit current, (c) the circuit phase angle, and (d) the p.d. across each impedance.
[(a) $39.95 \angle-28.82^{\circ} \Omega$ (b) $3.00 \angle 28.82^{\circ} \mathrm{A}$ (c) $28.82^{\circ}$ leading
(d) $V_1=44.70 \angle 86.35^{\circ} \mathrm{V}, V_2=36.00 \angle 28.82^{\circ} \mathrm{V}$,

$$
\left.V_3=105.56 \angle-35.95^{\circ} \mathrm{V}\right]
$$

Km Neeraj

Numerade Educator

Problem 18

Determine the value of voltages $V_1$ and $V_2$ in the circuit shown in Figure 24.21, if the frequency of the supply is 2.5 kHz . Find also the value of the supply voltage $V$ and the circuit phase angle. Draw the phasor diagram.

$$
\begin{aligned}
& {\left[V_1=164 \angle-12.68^{\circ} \mathrm{V} \text { or }(160-j 36) \mathrm{V}\right.} \\
& V_2=104 \angle 67.38^{\circ} \mathrm{V} \text { or }(40+j 96) \mathrm{V} \\
& V_3=208.8 \angle 16.70^{\circ} \mathrm{V} \text { or }(200+j 60) \mathrm{V} \\
& \text { Phase angle } \left.=16.70^{\circ} \text { lagging }\right]
\end{aligned}
$$

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03:27

Problem 19

A circuit comprises a coil of inductance 40 mH and resistance $20 \Omega$ in series with a variable capacitor. The supply voltage is 120 V at 50 Hz . Determine the value of capacitance needed to cause a current of 2.0 A to flow in the circuit.
[46.04 F]
20 For the circuit shown in Figure 24.22, determine (i) the circuit current I flowing, and (ii) the p.d. across each impedance.

$$
\begin{aligned}
& \text { [(i) } 3.71 \angle-17.35^{\circ} \mathrm{A} \text { (ii) } V_1=55.65 \angle 12.65^{\circ} \mathrm{V}, \\
& V_2=37.10 \angle-77.35^{\circ} \mathrm{V}, V_3=44.52 \angle 32.65^{\circ} \mathrm{V} \text { ] }
\end{aligned}
$$

Khoobchandra Agrawal

Numerade Educator