d. The function exp(t)= Aexp(-t/t)=Ae“ is the (decaying) exponential function, where t=1/a is called the time constant. This function is an important function in mathematics and engineering. Some critical values: Assuming A=1, exp(0)=1, exp(t)=l/e>0.37 and exp(5t)<0. We can relate the (complex) -at exponential function and sinusoidal function as follows: e"=cos(0)+jsin(0), where =-1. Then the sinusoidal function can be expressed as the real part of elo e". We can also use the equation: cos(0)= +e¯jo Derive a similar %3D 2 equation for sin(0) and write the function given in part b. in terms of complex exponentials.
d. The function exp(t)= Aexp(-t/t)=Ae“ is the (decaying) exponential function, where t=1/a is called the time constant. This function is an important function in mathematics and engineering. Some critical values: Assuming A=1, exp(0)=1, exp(t)=l/e>0.37 and exp(5t)<0. We can relate the (complex) -at exponential function and sinusoidal function as follows: e"=cos(0)+jsin(0), where =-1. Then the sinusoidal function can be expressed as the real part of elo e". We can also use the equation: cos(0)= +e¯jo Derive a similar %3D 2 equation for sin(0) and write the function given in part b. in terms of complex exponentials.
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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I just want part d solution
![In Electrical Engineering, the periodic especially sinusoidal signals arise frequently. A
periodic function p(t) with the fundamental period T, satisfies p(t)=p(t+kT) for all t and
all integer k. We define a sinusoidal signal as
z(1)=Z,+ZpeakCOS(@t+$),
where Z, is the offset, Zpeak is the peak value, w=27 /T=2rf is the angular frequency (in
rad/s), ø is the phase (in radians), T is the period and f is the frequency (in Hz=s).
Note that, these are Ao, C1, 0 and T in the figure below. (Note: The phase ø is defined
in radians. However, we often write it in degrees!)
3,
2.5
2
1.5
1
Ao
0.5
2
-1
1
3
4
t[s]
a. Express a sine wave (i.e. y(t)=YpeakSin(@t)) in the form given above.
b. Let us given the signal r(1)=-5+10cos(314t-45°). Find the offset, peak,
frequency, period and phase of this signal.
c. The rms (root-mean-square) value Prms (also, nominal value or effective value)
of a periodic function p(t) is defined as P
T
dt . Note that, the
rms
integration should be on a full-period and the limits can be changed as to and
to+T to get the same result. Find the rms value of a sinusoidal function for both
zero offset (Zo=0) and non-zero offset (Zo=0).
d. The function exp(t)= Aexp(-t/t)=Ae" is the (decaying) exponential function,
where t=1/a is called the time constant. This function is an important function
in mathematics and engineering. Some critical values: Assuming A=1,
exp(0)=1, exp(t)=1/e>0.37 and exp(5t)>0. We can relate the (complex)
-at
exponential function and sinusoidal function as follows: e"=cos(0)+jsin(0),
where j=-1. Then the sinusoidal function can be expressed as the real part of
e". We can also use the equation: cos(0)=
2
Derive a similar
equation for sin(0) and write the function given in part b. in terms of complex
exponentials.
(1)4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd9e509b-cb6c-49ea-b64f-0e00fd9a531e%2F6f57d7c9-7dd3-4bf2-a907-1592ce303ad6%2F058m37_processed.png&w=3840&q=75)
Transcribed Image Text:In Electrical Engineering, the periodic especially sinusoidal signals arise frequently. A
periodic function p(t) with the fundamental period T, satisfies p(t)=p(t+kT) for all t and
all integer k. We define a sinusoidal signal as
z(1)=Z,+ZpeakCOS(@t+$),
where Z, is the offset, Zpeak is the peak value, w=27 /T=2rf is the angular frequency (in
rad/s), ø is the phase (in radians), T is the period and f is the frequency (in Hz=s).
Note that, these are Ao, C1, 0 and T in the figure below. (Note: The phase ø is defined
in radians. However, we often write it in degrees!)
3,
2.5
2
1.5
1
Ao
0.5
2
-1
1
3
4
t[s]
a. Express a sine wave (i.e. y(t)=YpeakSin(@t)) in the form given above.
b. Let us given the signal r(1)=-5+10cos(314t-45°). Find the offset, peak,
frequency, period and phase of this signal.
c. The rms (root-mean-square) value Prms (also, nominal value or effective value)
of a periodic function p(t) is defined as P
T
dt . Note that, the
rms
integration should be on a full-period and the limits can be changed as to and
to+T to get the same result. Find the rms value of a sinusoidal function for both
zero offset (Zo=0) and non-zero offset (Zo=0).
d. The function exp(t)= Aexp(-t/t)=Ae" is the (decaying) exponential function,
where t=1/a is called the time constant. This function is an important function
in mathematics and engineering. Some critical values: Assuming A=1,
exp(0)=1, exp(t)=1/e>0.37 and exp(5t)>0. We can relate the (complex)
-at
exponential function and sinusoidal function as follows: e"=cos(0)+jsin(0),
where j=-1. Then the sinusoidal function can be expressed as the real part of
e". We can also use the equation: cos(0)=
2
Derive a similar
equation for sin(0) and write the function given in part b. in terms of complex
exponentials.
(1)4
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