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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)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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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+$), where Z, is the offset, Zpeak is the peak value, w=27 /T=2rf is the angular frequency (in rad/s), ø is the phase (in radians), T is the period and f is the frequency (in Hz=s). Note that, these are Ao, C1, 0 and T in the figure below. (Note: The phase ø is defined in radians. However, we often write it in degrees!) 3, 2.5 2 1.5 1 Ao 0.5 2 -1 1 3 4 t[s] a. Express a sine wave (i.e. y(t)=YpeakSin(@t)) in the form given above. b. Let us given the signal r(1)=-5+10cos(314t-45°). Find the offset, peak, frequency, period and phase of this signal. c. The rms (root-mean-square) value Prms (also, nominal value or effective value) of a periodic function p(t) is defined as P T dt . Note that, the rms integration should be on a full-period and the limits can be changed as to and to+T to get the same result. Find the rms value of a sinusoidal function for both zero offset (Zo=0) and non-zero offset (Zo=0). d. The function exp(t)= Aexp(-t/t)=Ae" is the (decaying) exponential function, where t=1/a is called the time constant. This function is an important function in mathematics and engineering. Some critical values: Assuming A=1, exp(0)=1, exp(t)=1/e>0.37 and exp(5t)>0. We can relate the (complex) -at exponential function and sinusoidal function as follows: e"=cos(0)+jsin(0), where j=-1. Then the sinusoidal function can be expressed as the real part of e". We can also use the equation: cos(0)= 2 Derive a similar equation for sin(0) and write the function given in part b. in terms of complex exponentials. (1)4
Transcribed Image Text:In Electrical Engineering, the periodic especially sinusoidal signals arise frequently. A periodic function p(t) with the fundamental period T, satisfies p(t)=p(t+kT) for all t and all integer k. We define a sinusoidal signal as z(1)=Z,+ZpeakCOS(@t+$), 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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