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(mt+ø), where Z, is the offset, Zpeak is the peak value, o=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 o is defined in radians However we often write it in degrees!)
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(mt+ø), where Z, is the offset, Zpeak is the peak value, o=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 o is defined in radians However we often write it in degrees!)
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I just want the answer to options d and c
![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(t)=Z,+ZpeakCOS(ot+¢),
where Z, is the offset, Zpeak is the peak value, w=27 /T=2tf is the angular frequency (in
rad/s), o 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!)
2.5
1.5
1
Ao
T
0.5
-1
1
2
3
t[s]
a. Express a sine wave (i.e. y(t)=YpeakSin(wt)) in the form given above.
реа
b. Let us given the signal r(t)=-5+10cos(314t-45°). Find the offset, peak,
frequency, period and phase of this signal.
c. The rms (root-mean-square) value Pms (also, nominal value or effective value)
of a periodic function p(t) is defined as P
dt . Note that, the
%3D
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)
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
el0 +e¯j0
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)J](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd9e509b-cb6c-49ea-b64f-0e00fd9a531e%2F9e180926-7e95-48e0-be05-ff4bbbb2cfe4%2Fo4nfb7_processed.jpeg&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(t)=Z,+ZpeakCOS(ot+¢),
where Z, is the offset, Zpeak is the peak value, w=27 /T=2tf is the angular frequency (in
rad/s), o 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!)
2.5
1.5
1
Ao
T
0.5
-1
1
2
3
t[s]
a. Express a sine wave (i.e. y(t)=YpeakSin(wt)) in the form given above.
реа
b. Let us given the signal r(t)=-5+10cos(314t-45°). Find the offset, peak,
frequency, period and phase of this signal.
c. The rms (root-mean-square) value Pms (also, nominal value or effective value)
of a periodic function p(t) is defined as P
dt . Note that, the
%3D
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)
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
el0 +e¯j0
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)J
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