2.5.3: Sinusoidal Signals and Complex Exponenials Revision: June 11, 2010 215 E Main Suie D Pullman, W 99163 (509) 334 6306 Voice and Fax Overview Sinusoidal signals and complex exponenials are exremely imporan o any engineer who is concerned wih deermining he dynamic response of a sysem. Elecrical circuis, in paricular, are ofen characerized by heir response o sinusoidal inpus. This chaper provides some background relaive o hese signals. Before beginning his chaper, you should be able o: Wrie expressions for sinusoidal funcions Express complex numbers in recangular and polar form fer compleing his chaper, you should be able o: Wrie complex numbers in erms of complex exponenials Express sinusoidal signals in erms of complex exponenials This chaper requires: N/ Doc: XXX-YYY page 1 of 5 Copyrigh Digilen, Inc. ll righs reserved. Oher produc and company names menioned may be rademarks of heir respecive owners.
Sinusoidal Signals: Sinusoidal signals are represened in erms of sine and/or cosine funcions. In general, we will represen sinusoids as cosine funcions. Our general expression for a sinusoidal signal is: v( ) = VP cos( ω + θ ) (1) where V P is he zero-o-peak ampliude of he sinusoid, ω is he radian frequency of he sinusoid (we will always use radians/second as he unis of ω) and θ is he phase angle of he sinusoid (in unis of eiher radians or degrees are used for phase angle recall ha 2π radians = 360 ). represenaive plo of a sinusoidal signal is provided in Figure 1. In Figure 1, he frequency of he sinusoid is indicaed as a period of he signal (he period is defined as he shores ime inerval a which he signal repeas iself). The radian frequency of a sinusoid is relaed o he period by: ω = 2π (2) T Figure 1. rbirary sinusoidal signal. Noe: Complex exponenial signals have boh real and imaginary pars; when we inroduce complex exponenials laer in his chaper, we will see ha he cosine funcion is he real par of a complex exponenial signal. Complex exponenials make dynamic sysems analysis relaively simple hus, we ofen analyze a signals response in erms of complex exponenials. Since any measurable quaniy is real-valued, aking he real par of he analyical resul based on complex exponenials will resul in a cosine funcion. Thus, cosines become a naural way o express signals which vary sinusoidally. www.digileninc.com page 2 of 5 Copyrigh Digilen, Inc. ll righs reserved. Oher produc and company names menioned may be rademarks of heir respecive owners.
The frequency of a sinusoidal signal is alernaely expressed in unis of Herz (abbreviaed Hz). Herz is he number of cycles which he sinusoid goes hrough in one second. Thus, Herz correspond o cycles/second. The frequency of a signal in Herz is relaed o he period of he signal by f 1 = (3) T Radian frequencies relae o frequencies in Herz by: 2π f = ω = 2πf (4) ω lhough frequencies of signals are ofen expressed in Herz, i is no a uni which lends iself o calculaions. Thus, all our calculaions will be performed in radian frequency if given a frequency in Herz, i should be convered o radians/second before any calculaions are performed based on his frequency. Complex Exponenials: In our presenaion of complex exponenials, we firs provide a brief review of complex numbers. complex number conains boh real and imaginary pars. Thus, we may wrie a complex number as: where = a +jb (5) j = 1 (6) The complex number can be represened on orhogonal axes represening he real and imaginary par of he number, as shown in Figure 2. (In Figure 2, we have aken he libery of represening as a vecor, alhough i is really jus a number.) We can also represen he complex number in polar coordinaes, also shown in Figure 2. The polar coordinaes consis of a magniude and phase angle θ, defined as: + 2 2 = a b (7) 1 b θ = an (8) a Noice ha he phase angle is defined counerclockwise from he posiive real axis. Conversely, we can deermine he recangular coordinaes from he polar coordinaes from { } = cos( θ ) a = Re (9) www.digileninc.com page 3 of 5 Copyrigh Digilen, Inc. ll righs reserved. Oher produc and company names menioned may be rademarks of heir respecive owners.
{ } = sin( θ ) b = Im (10) where he noaion Re { } and { } Im denoe he real par of and he imaginary par of, respecively. The polar coordinaes of a complex number are ofen represened in he form: = θ (11) sin( θ ) cos( θ ) Figure 2. Represenaion of a complex number in recangular and polar coordinaes. n alernae mehod of represening complex numbers in polar coordinaes employs complex exponenial noaion. Wihou proof, we claim ha e =1 θ (12) Thus, e is a complex number wih magniude 1 and phase angle θ. From Figure 2, i is easy o see ha his definiion of he complex exponenial agrees wih Euler s equaion: ± θ e j = cosθ ± j sinθ (13) Wih he definiion of equaion (12), we can define any arbirary complex number in erms of complex numbers. For example, our previous complex number can be represened as: = e (14) We can generalize our definiion of he complex exponenial o ime-varying signals. If we define a ime varying signal e ω, we can use equaion (13) o wrie: e ± jω = cosω ± j sinω (15) www.digileninc.com page 4 of 5 Copyrigh Digilen, Inc. ll righs reserved. Oher produc and company names menioned may be rademarks of heir respecive owners.
e ω The signal can be visualized as a uni vecor roaing around he origin in he complex plane; he ip of he vecor scribes a uni circle wih is cener a he origin of he complex plane. This is illusraed in Figure 3. The vecor roaes a a rae defined by he quaniy ω -- he vecor makes one 2π complee revoluion every seconds. The projecion of his roaing vecor on he real axis races ω ou he signal cosω, as shown in Figure 3, while he projecion of he roaing vecor on he imaginary axis races ou he signal sinω, also shown in Figure 3. Thus, we inerpre he complex exponenial funcion as an alernae ype of sinusoidal signal. The real par of his funcion is cosω while he imaginary par of his funcion is sinω. e ω Im sin Re ime cos ime Figure 3. Illusraion of e ω. www.digileninc.com page 5 of 5 Copyrigh Digilen, Inc. ll righs reserved. Oher produc and company names menioned may be rademarks of heir respecive owners.