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Fundamentals of Electric Circuits

Charles K. Alexander, Matthew N.O. Sadiku

Chapter 9

Sinusoids and Phasors - all with Video Answers

Educators

NT

Chapter Questions

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Problem 1

In a linear circuit, the voltage source is $$v_{s}=12 \sin \left(10^{3} t+24^{\circ}\right) \mathrm{V}$$ (a) What is the angular frequency of the voltage?
(b) What is the frequency of the source?
(c) Find the period of the voltage.
(d) Express $v_{s}$ in cosine form.
(e) Determine $v_{s}$ at $t=2.5 \mathrm{ms}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:08

Problem 2

A current source in a linear circuit has
\[
i_{s}=8 \cos \left(500 \pi t-25^{\circ}\right) \mathrm{A}
\]
(a) What is the amplitude of the current?
(b) What is the angular frequency?
(c) Find the frequency of the current.
(d) Calculate $i_{s}$ at $t=2 \mathrm{ms}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:29

Problem 3

Express the following functions in cosine form:
(a) $4 \sin \left(\omega t-30^{\circ}\right)$
(b) $-2 \sin 6 t$
(c) $-10 \sin \left(\omega t+20^{\circ}\right)$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:47

Problem 4

(a) Express $v=8 \cos \left(7 t+15^{\circ}\right)$ in sine form.
(b) Convert $i=-10 \sin \left(3 t-85^{\circ}\right)$ to cosine form.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:29

Problem 5

Given $v_{1}=20 \sin \left(\omega t+60^{\circ}\right)$ and $v_{2}=$
$60 \cos \left(\omega t-10^{\circ}\right),$ determine the phase angle between the two sinusoids and which one lags the other.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:07

Problem 6

For the following pairs of sinusoids, determine which one leads and by how much.
(a) $v(t)=10 \cos \left(4 t-60^{\circ}\right)$ and
$i(t)=4 \sin \left(4 t+50^{\circ}\right)$
(b) $v_{1}(t)=4 \cos \left(377 t+10^{\circ}\right)$ and
$v_{2}(t)=-20 \cos 377 t$
(c) $x(t)=13 \cos 2 t+5 \sin 2 t$ and
$v(t)=15 \cos (2 t-$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:07

Problem 7

$$\text { If } f(\phi)=\cos \phi+j \sin \phi, \text { show that } f(\phi)=e^{j \phi}$$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
06:16

Problem 8

Calculate these complex numbers and express your results in rectangular form:
(a) $\frac{15 \angle 45^{\circ}}{3-j 4}+j 2$
(b) $\frac{8 \angle-20^{\circ}}{(2+j)(3-j 4)}+\frac{10}{-5+j 12}$
(c) $10+\left(8 \angle 50^{\circ}\right)(5-j 12)$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:45

Problem 9

Evaluate the following complex numbers and express your results in rectangular form:
(a) $2+\frac{3+j 4}{5-j 8}$
(b) $4 \angle-10^{\circ}+\frac{1-j 2}{3 \angle 6^{\circ}}$
(c) $\frac{8 \angle 10^{\circ}+6 \angle-20^{\circ}}{9 \angle 80^{\circ}-4 \angle 50^{\circ}}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:35

Problem 10

Given the complex numbers $z_{1}=-3+j 4$ and $z_{2}=12+j 5,$ find:
(a) $z_{1} z_{2}$
(b) $\frac{z_{1}}{z_{2}^{*}}$
(c) $\frac{z_{1}+z_{2}}{z_{1}-z_{2}}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:54

Problem 11

Let $\mathbf{X}=8 \angle 40^{\circ}$ and $\mathbf{Y}=10 \angle-30^{\circ} .$ Evaluate the following quantities and express your results in polar form.
(a) $(\mathbf{X}+\mathbf{Y}) \mathbf{X}^{*}$
(b) $(\mathbf{X}-\mathbf{Y})^{*}$
(c) $(\mathbf{X}+\mathbf{Y}) / \mathbf{X}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:45

Problem 12

Evaluate these determinants:
(a) $\left|\begin{array}{cc}10+j 6 & 2-j 3 \\ -5 & -1+j\end{array}\right|$
(b) $\left|\begin{array}{cc}20 \angle-30^{\circ} & -4 \angle-10^{\circ} \\ 16 \angle 0^{\circ} & 3 \angle 45^{\circ}\end{array}\right|$
(c) $\left|\begin{array}{ccc}1-j & -j & 0 \\ j & 1 & -j \\ 1 & j & 1+j\end{array}\right|$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:22

Problem 13

Transform the following sinusoids to phasors:
(a) $-10 \cos \left(4 t+75^{\circ}\right)$
(b) $5 \sin \left(20 t-10^{\circ}\right)$
(c) $4 \cos 2 t+3 \sin 2 t$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
08:24

Problem 14

Express the sum of the following sinusoidal signals in the form of $A \cos (\omega t+\theta)$ with $A>0$ and $0<\theta<360^{\circ}$
(a) $8 \cos \left(5 t-30^{\circ}\right)+6 \cos 5 t$
(b) $20 \cos \left(120 \pi t+45^{\circ}\right)-30 \sin \left(120 \pi t+20^{\circ}\right)$
(c) $4 \sin 8 t+3 \sin \left(8 t-10^{\circ}\right)$

Ramesh Singh
Ramesh Singh
Numerade Educator
03:08

Problem 15

Obtain the sinusoids corresponding to each of the following phasors:
(a) $\mathbf{V}_{1}=60 \angle 15^{\circ}, \omega=1$
(b) $\mathbf{V}_{2}=6+j 8, \omega=40$
(c) $\mathbf{I}_{1}=2.8 e^{-j \pi / 3}, \omega=377$
(d) $\mathbf{I}_{2}=-0.5-j 1.2, \omega=10^{3}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:21

Problem 16

Using phasors, find:
(a) $3 \cos \left(20 t+10^{\circ}\right)-5 \cos \left(20 t-30^{\circ}\right)$
(b) $40 \sin 50 t+30 \cos \left(50 t-45^{\circ}\right)$
(c) $20 \sin 400 t+10 \cos \left(400 t+60^{\circ}\right)$
$-5 \sin \left(400 t-20^{\circ}\right)$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:03

Problem 17

Find a single sinusoid corresponding to each of these phasors:
(a) $\mathbf{V}=40 \angle-60^{\circ}$
(b) $\mathbf{V}=-30 \angle 10^{\circ}+50 / 60^{\circ}$
(c) $\mathbf{I}=j 6 e^{-j 10^{\circ}}$
(d) $\mathbf{I}=\frac{2}{j}+10 \angle-45^{\circ}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:13

Problem 18

Find $v(t)$ in the following integrodifferential equations using the phasor approach:
(a) $v(t)+\int v d t=10 \cos t$
(b) $\frac{d v}{d t}+5 v(t)+4 \int v d t=20 \sin \left(4 t+10^{\circ}\right)$

Ramesh Singh
Ramesh Singh
Numerade Educator
06:18

Problem 19

Using phasors, determine $i(t)$ in the following equations:
(a) $2 \frac{d i}{d t}+3 i(t)=4 \cos \left(2 t-45^{\circ}\right)$
(b) $10 \int i d t+\frac{d i}{d t}+6 i(t)=5 \cos \left(5 t+22^{\circ}\right)$

Ramesh Singh
Ramesh Singh
Numerade Educator
02:45

Problem 20

The loop equation for a series $R L C$ circuit gives
\[
\frac{d i}{d t}+2 i+\int_{-\infty}^{t} i d t=\cos 2 t
\]
Assuming that the value of the integral at $t=-\infty$ is zero, find $i(t)$ using the phasor method.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
07:45

Problem 21

A parallel $R L C$ circuit has the node equation
\[
\frac{d v}{d t}+50 v+100 \int v d t=110 \cos \left(377 t-10^{\circ}\right)
\]
Determine $v(t)$ using the phasor method. You may assume that the value of the integral at $t=-\infty$ is zero.

Ramesh Singh
Ramesh Singh
Numerade Educator
01:43

Problem 22

Determine the current that flows through an $8-\Omega$ resistor connected to a voltage source $v_{s}=110 \cos 377 t \mathrm{V}$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
02:18

Problem 23

What is the instantaneous voltage across a $2-\mu \mathrm{F}$ capacitor when the current through it is
\[
i=4 \sin \left(10^{6} t+25^{\circ}\right) \mathrm{A} ?
\].

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
01:05

Problem 24

The voltage across a $4-\mathrm{mH}$ inductor is $v=60 \cos \left(500 t-65^{\circ}\right) \mathrm{V} .$ Find the instantaneous
current through it.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:09

Problem 25

A current source of $i(t)=10 \sin \left(377 t+30^{\circ}\right)$ A is applied to a single-element load. The resulting voltage across the element is $v(t)=$ $-65 \cos \left(377 t+120^{\circ}\right) \mathrm{V} .$ What type of element is
this? Calculate its value.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
09:35

Problem 26

Two elements are connected in series as shown in Fig. $9.40 .$ If $i=12 \cos \left(2 t-30^{\circ}\right) \mathrm{A},$ find the
element values.

NT
Nikhil Tiwari
Numerade Educator
01:07

Problem 27

A series $R L$ circuit is connected to a $110-\mathrm{V}$ ac source. If the voltage across the resistor is $85 \mathrm{V}$, find the voltage across the inductor.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
01:40

Problem 28

What value of $\omega$ will cause the forced response $v_{o}$ in Fig. 9.41 to be zero?

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:55

Problem 29

If $v_{s}=5 \cos 2 t \mathrm{V}$ in the circuit of Fig. 9.42 find $v_{v}$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
11:29

Problem 30

Find $i_{x}$ when $i_{s}=2 \sin 5 t$ A is supplied to the circuit in Fig. 9.43

NT
Nikhil Tiwari
Numerade Educator
06:31

Problem 31

Find $i(t)$ and $v(t)$ in each of the circuits of Fig. 9.44

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
14:56

Problem 32

Calculate $i_{1}(t)$ and $i_{2}(t)$ in the circuit of Fig. 9.45 if the source frequency is $60 \mathrm{Hz}$

Linda Winkler
Linda Winkler
Numerade Educator
06:20

Problem 33

In the circuit of Fig. $9.46,$ find $i_{o}$ when:
(a) $\omega=1 \mathrm{rad} / \mathrm{s}$
(b) $\omega=5 \mathrm{rad} / \mathrm{s}$
(c) $\omega=10 \mathrm{rad} / \mathrm{s}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:39

Problem 34

Find $v(t)$ in the $R L C$ circuit of Fig. 9.47

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:39

Problem 35

Calculate $v_{o}(t)$ in the circuit in Fig. 9.48

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
05:57

Problem 36

Determine $i_{o}(t)$ in the $R L C$ circuit of Fig. 9.49

Ramesh Singh
Ramesh Singh
Numerade Educator
10:42

Problem 37

Calculate $i(t)$ in the circuit of Fig. 9.50

Ramesh Singh
Ramesh Singh
Numerade Educator
09:19

Problem 38

Find current $\mathbf{I}_{o}$ in the network of Fig. 9.51

Ramesh Singh
Ramesh Singh
Numerade Educator
04:33

Problem 39

If $i_{s}=5 \cos \left(10 t+40^{\circ}\right)$ A in the circuit in Fig. 9.52 find $i_{o}$

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
04:48

Problem 40

Find $v_{s}(t)$ in the circuit of Fig. 9.53 if the current $i_{x}$ through the $1-\Omega$ resistor is $0.5 \sin 200 t$ A.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:03

Problem 41

If the voltage $v_{o}$ across the $2-\Omega$ resistor in the circuit of Fig. 9.54 is $10 \cos 2 t \mathrm{V},$ obtain $i$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
05:11

Problem 42

If $\mathbf{V}_{o}=8 \angle 30^{\circ} \mathrm{V}$ in the circuit of Fig. 9.55 find $\mathbf{I}_{s}$.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
03:06

Problem 43

In the circuit of Fig. $9.56,$ find $\mathbf{V}_{s}$ if $\mathbf{I}_{o}=2 \angle 0^{\circ}$ A.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
10:27

Problem 44

Find $\mathbf{Z}$ in the network of Fig. 9.57 , given that $\mathbf{v}_{o}=4 \angle 0^{\circ} \mathrm{V}$.

NT
Nikhil Tiwari
Numerade Educator
15:39

Problem 45

At $\omega=50 \mathrm{rad} / \mathrm{s},$ determine $\mathrm{Z}_{\mathrm{in}}$ for each of the circuits in Fig. 9.58.

NT
Nikhil Tiwari
Numerade Educator
05:16

Problem 46

Calculate $\mathbf{Z}_{\mathrm{eq}}$ for the circuit in Fig. 9.59

NT
Nikhil Tiwari
Numerade Educator
05:16

Problem 47

Find $\mathbf{Z}_{\mathrm{eq}}$ in the circuit of Fig. 9.60

NT
Nikhil Tiwari
Numerade Educator
02:26

Problem 48

For the circuit in Fig. 9.61 , find the input impedance $\mathbf{Z}_{\text {in }}$ at $10 \mathrm{krad} / \mathrm{s}$.

Prachita Kush
Prachita Kush
Numerade Educator
09:32

Problem 49

Determine I and $\mathbf{Z}_{T}$ for the circuit in Fig. 9.62

NT
Nikhil Tiwari
Numerade Educator
06:19

Problem 50

For the circuit in Fig. $9.63,$ calculate $\mathbf{Z}_{T}$ and $\mathbf{V}_{a b}$.

Kajal Gautam
Kajal Gautam
Numerade Educator
01:49

Problem 51

At $\omega=10^{3} \mathrm{rad} / \mathrm{s},$ find the input admittance of each of the circuits in Fig. 9.64

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:42

Problem 52

Determine $\mathbf{Y}_{\mathrm{eg}}$ for the circuit in Fig. 9.65.

NT
Nikhil Tiwari
Numerade Educator
02:05

Problem 53

Find the equivalent admittance $Y_{\mathrm{eq}}$ of the circuit in Fig. 9.66

Amit Srivastava
Amit Srivastava
Numerade Educator
02:36

Problem 54

Find the equivalent impedance of the circuit in Fig. 9.67

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:36

Problem 55

Obtain the equivalent impedance of the circuit in Fig. 9.68

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:47

Problem 56

Calculate the value of $\mathbf{Z}_{u b}$ in the network of Fig. 9.69

M Hassan Anwar
M Hassan Anwar
Numerade Educator
02:36

Problem 57

Determine the equivalent impedance of the circuit in Fig. 9.70

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:34

Problem 58

Design an $R L$ circuit to provide a $90^{\circ}$ leading phase shift.

Ramesh Singh
Ramesh Singh
Numerade Educator
04:34

Problem 59

Design a circuit that will transform a sinusoidal input to a cosinusoidal output.

Ramesh Singh
Ramesh Singh
Numerade Educator
08:15

Problem 60

Refer to the $R C$ circuit in Fig. 9.71
(a) Calculate the phase shift at $2 \mathrm{MHz}$
(b) Find the frequency where the phase shift is $45^{\circ}$.

Ramesh Singh
Ramesh Singh
Numerade Educator
16:49

Problem 61

(a) Calculate the phase shift of the circuit in Fig. 9.72
(b) State whether the phase shift is leading or lagging (output with respect to input).
(c) Determine the magnitude of the output when the input is $120 \mathrm{V}$

Ramesh Singh
Ramesh Singh
Numerade Educator
10:09

Problem 62

Consider the phase-shifting circuit in Fig. $9.73 .$ Let $\mathbf{V}_{i}=120 \mathrm{V}$ operating at $60 \mathrm{Hz}$. Find:
(a) $\mathbf{V}_{o}$ when $R$ is maximum
(b) $\mathbf{V}_{o}$ when $R$ is minimum
(c) the value of $R$ that will produce a phase shift of $45^{\circ}$

Ramesh Singh
Ramesh Singh
Numerade Educator
07:42

Problem 63

The ac bridge in Fig. 9.37 is balanced when $R_{1}=400 \Omega, R_{2}=600 \Omega, R_{3}=1.2 \mathrm{k} \Omega,$ and
$C_{2}=0.3 \mu \mathrm{F} .$ Find $R_{x}$ and $C_{x}$

Ramesh Singh
Ramesh Singh
Numerade Educator
04:55

Problem 64

A capacitance bridge balances when $R_{1}=100 \Omega$ $R_{2}=2 \mathrm{k} \Omega,$ and $C_{s}=40 \mu \mathrm{F}$. What is $C_{x},$ the capacitance of the capacitor under test?

Ramesh Singh
Ramesh Singh
Numerade Educator
03:44

Problem 65

An inductive bridge balances when $R_{1}=1.2 \mathrm{k} \Omega$ $R_{2}=500 \Omega,$ and $L_{s}=250 \mathrm{mH} .$ What is the value of $L_{x},$ the inductance of the inductor under test?

Ramesh Singh
Ramesh Singh
Numerade Educator
12:47

Problem 66

The ac bridge shown in Fig. 9.74 is known as a Maxwell bridge and is used for accurate measurement of inductance and resistance of a coil in terms of a standard capacitance $C_{s} .$ Show that when the bridge is balanced,
\[
L_{x}=R_{2} R_{3} C_{s} \quad \text { and } \quad R_{x}=\frac{R_{2}}{R_{1}} R_{3}
\]
Find $L_{x}$ and $R_{x}$ for $R_{1}=40 \mathrm{k} \Omega, R_{2}=1.6 \mathrm{k} \Omega$
$R_{3}=4 \mathrm{k} \Omega,$ and $C_{s}=0.45 \mu \mathrm{F}$

Ramesh Singh
Ramesh Singh
Numerade Educator
11:50

Problem 67

The ac bridge circuit of Fig. 9.75 is called a Wien bridge. It is used for measuring the frequency of a source. Show that when the bridge is balanced,
\[
f=\frac{1}{2 \pi \sqrt{R_{2} R_{4} C_{2} C_{4}}}
\]

Ramesh Singh
Ramesh Singh
Numerade Educator
05:46

Problem 68

The circuit shown in Fig. 9.76 is used in a television receiver. What is the total impedance of this circuit?

Ramesh Singh
Ramesh Singh
Numerade Educator
10:47

Problem 69

The network in Fig. 9.77 is part of the schematic describing an industrial electronic sensing device. What is the total impedance of the circuit at $2 \mathrm{kHz} ?$

Ramesh Singh
Ramesh Singh
Numerade Educator
06:38

Problem 70

A series audio circuit is shown in Fig. 9.78
(a) What is the impedance of the circuit?
(b) If the frequency were halved, what would be the impedance of the circuit?

Ramesh Singh
Ramesh Singh
Numerade Educator
09:48

Problem 71

An industrial load is modeled as a series combination of a capacitance and a resistance as shown in Fig. $9.79 .$ Calculate the value of an inductance $L$ across the series combination so that the net impedance is resistive at a frequency of $5 \mathrm{MHz}$.

Ramesh Singh
Ramesh Singh
Numerade Educator
12:19

Problem 72

An industrial coil is modeled as a series combination of an inductance $L$ and resistance $R,$ as shown in Fig. $9.80 .$ since an ac voltmeter measures only the magnitude of a sinusoid, the following measurements are taken at $60 \mathrm{Hz}$ when the circuit operates in the steady state:
\[
\left|\mathbf{V}_{s}\right|=145 \mathrm{V}, \quad\left|\mathbf{V}_{1}\right|=50 \mathrm{V}, \quad\left|\mathbf{V}_{o}\right|=110 \mathrm{V}
\]
Use these measurements to determine the values of $L$ and $R$

Ramesh Singh
Ramesh Singh
Numerade Educator
08:48

Problem 73

Figure 9.81 shows a parallel combination of an inductance and a resistance. If it is desired to connect a capacitor in series with the parallel combination such that the net impedance is resistive at $10 \mathrm{MHz}$, what is the required value of $C ?$

Ramesh Singh
Ramesh Singh
Numerade Educator
05:19

Problem 74

A power transmission system is modeled as shown in Fig. $9.82 .$ Given the source voltage $\mathbf{V}_{s}=115 / 0^{\circ} \mathrm{V},$ source impedance
$\mathbf{Z}_{s}=1+j 0.5 \Omega,$ line impedance
$\mathbf{Z}_{\ell}=0.4+j 0.3 \Omega,$ and load impedance $\mathbf{Z}_{L}=23.2+j 18.9 \Omega,$ find the load current $\mathbf{I}_{L}$.

Ramesh Singh
Ramesh Singh
Numerade Educator