Dorf,R C, Wan, Z, Paul, C.R., Cogdell, J. R "Voltage and Current Sources The electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Dorf, R.C., Wan, Z., Paul, C.R., Cogdell, J.R. “Voltage and Current Sources” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
2 oltage an Current sources Richard C. dorf 2.1 Step, Impulse, Ramp, Sinusoidal, Exponential, and Zhen Wan Step Function. The Impulse. Ramp Function. Sinusoidal Function· SIGnal University of California, Davis 2.2 Ideal and Practical Sources Clayton R. Paul Ideal Sources. Practical Sources University of Kentucky, Lexington 2.3 Controlled Sources What Are Controlled Sources?. What Is the Significance of J. R. Cogdell Controlled Sources?. How does the presence of Controlled Sources University of Texas at Austin Affect Circuit Analysis 2.1 Step, Impulse, Ramp, Sinusoidal, Exponential, and dc signals Richard C. Dorf and Zhen Wan The important signals for circuits include the step, impulse, ramp, sinusoid, and dc signals. These signals are widely used and are described here in the time domain. All of these signals have a Laplace transform. Step Function The unit-step function u(r) is defined mathematically by t≥0 0)=1. If A is an arbitrary nonzero number, Au( t) is the step function with 0. Neverth wing the convention Here unit step means that the amplitude of ud(n)is equal to l for t20. Note that we are folle that u(0)=1. From a strict mathematical standpoint, u( t) is not defined at t ess, we usually take amplitude A for t20. The unit step function is plotted in Fig. 2.1 The Impulse The unit impulse 8(r), also called the delta function or the Dirac distribution, is defined by c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 2 Voltage and Current Sources 2.1 Step, Impulse, Ramp, Sinusoidal, Exponential, and DC Signals Step Function • The Impulse • Ramp Function • Sinusoidal Function • DCSignal 2.2 Ideal and Practical Sources Ideal Sources • Practical Sources 2.3 Controlled Sources What Are Controlled Sources? • What Is the Significance of Controlled Sources? • How Does the Presence of Controlled Sources Affect Circuit Analysis? 2.1 Step, Impulse, Ramp, Sinusoidal, Exponential, and DC Signals Richard C. Dorf and Zhen Wan The important signals for circuits include the step, impulse, ramp, sinusoid, and dc signals. These signals are widely used and are described here in the time domain. All of these signals have a Laplace transform. Step Function The unit-step function u(t) is defined mathematically by Here unit step means that the amplitude of u(t) is equal to 1 for t ³ 0. Note that we are following the convention that u(0) = 1. From a strict mathematical standpoint, u(t) is not defined at t = 0. Nevertheless, we usually take u(0) = 1. If A is an arbitrary nonzero number, Au(t) is the step function with amplitude A for t ³ 0. The unit step function is plotted in Fig. 2.1. The Impulse The unit impulse d(t), also called the delta function or the Dirac distribution, is defined by u t t t ( ) , , = ³ < Ï Ì Ô Ó Ô 1 0 0 0 Richard C. Dorf University of California, Davis Zhen Wan University of California, Davis Clayton R. Paul University of Kentucky, Lexington J. R. Cogdell University of Texas at Austin
FIGURE 2.1 Unit-step function. FIGURE 2.2 Graphical representation of the impulse Ko(n) δ(t)=0, t≠0 6O)dX=1, for any real numbere>0 The first condition states that d(n is zero for all nonzero values of t, while the second condition states that the area under the impulse is 1, so 8(n) has unit area. It is important to point out that the value 8(0)of &(r)at t 0 is not defined; in particular, 8(0)is not equal to infinity. For any real number k, ka(t) is the impulse with area K. It is defined by K6(t)=0, t≠0 Kδ(^λ)d^=K, for any real numbere The graphical representation of K8(n is shown in Fig. 2.2. The notation K in the figure refers to the area of the impulse Ko(n The unit-step function u(t) is equal to the integral of the unit impulse 8(t); more precisely, we have (t) δOλ)d入, ll t except t =0 Conversely, the first derivative of u(n), with respect to t, is equal to 8(n), except at t=0, where the derivative of u( t) is not defined Ramp function The unit-ramp function r( r)is defined mathematically by t≥0 r(t) Note that for t20, the slope of r(r) is 1. Thus, r(t) has unit slope, which is the reason r( n)is called the unit-ramp function. If K is an arbitrary nonzero scalar(rea ber), the ramp function Kr( n)has slope K for t20. The FIGURE 2.3 Unit-ramp function unit-ramp function is plotted in Fig. 2.3 The unit-ramp function r(t) is equal to the integral of the unit-step function id r); that is (t) u(a)dn e 2000 by CRC Press LLC
© 2000 by CRC Press LLC The first condition states that d(t) is zero for all nonzero values of t, while the second condition states that the area under the impulse is 1, so d(t) has unit area. It is important to point out that the value d(0) of d(t) at t = 0 is not defined; in particular, d(0) is not equal to infinity. For any real number K, Kd(t) is the impulse with area K. It is defined by The graphical representation of Kd(t) is shown in Fig. 2.2. The notation K in the figure refers to the area of the impulse Kd(t). The unit-step function u(t) is equal to the integral of the unit impulse d(t); more precisely, we have Conversely, the first derivative of u(t), with respect to t, is equal to d(t), except at t = 0, where the derivative of u(t) is not defined. Ramp Function The unit-ramp function r(t) is defined mathematically by Note that for t ³ 0, the slope of r(t) is 1. Thus, r(t) has unit slope, which is the reason r(t) is called the unit-ramp function. If K is an arbitrary nonzero scalar (real number), the ramp function Kr(t) has slope K for t ³ 0. The unit-ramp function is plotted in Fig. 2.3. The unit-ramp function r(t) is equal to the integral of the unit-step function u(t); that is, FIGURE 2.1 Unit-step function. FIGURE 2.2 Graphical representation of the impulse Kd(t) u (t) t 123 1 0 Kd (t) t 0 (K) d d l l e e e ( ) , ( ) , t t d = ¹ = -Ú 0 0 1 for any real number > 0 K t t K d K d d l l e e e ( ) , ( ) , = ¹ = -Ú 0 0 for any real number > 0 u t d t t t ( ) = ( ) , -• Ú d l l all except = 0 FIGURE 2.3 Unit-ramp function r(t) t 123 1 0 r t t t t ( ) , , = ³ < Ï Ì Ó 0 0 0 r t u d t ( ) = ( ) -• Ú l l
Aco π+26 FIGURE 2.4 The sinusoid A cos(ot 0)with -rt/2<0<0 Conversely, the first derivative of r(t) with respect to t is equal to u( t), except at t=0, where the derivative of r(t is not defined. Sinusoidal function The sinusoid is a continuous-time signal: A cos(ot 8) Here A is the amplitude, o is the frequency in radians per second (rad/s), and e is the phase in radians. The frequency f in cycles per second, or hertz(Hz), is f=(/2. The sinusoid is a periodic signal with period 2T/o he sinusoid is plotted in Fig. 2.4 Decaying Exponential In general, an exponentially decaying quantity(Fig. 2.5) can be expressed as where a instantaneous value A= amplitude or maximum value e= base of natural logarithms = 2.718 0.368 t= time constant in second t time in seconds The current of a discharging capacitor can be approxi mated by a decaying exponential function of time. Time Constant FIGURE 2.5 The decaying exponential Since the exponential factor only approaches zero as t increases without limit, such functions theoretically last forever. In the same sense, all radioactive disintegrations last forever. In the case of an exponentially decaying current, it is convenient to use the value of time that makes the exponent -l. When t=t= the time constant, e value In other words, after a time equal to the time constant, the exponential factor is reduced to approximatly 37% of its initial value e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Conversely, the first derivative of r(t) with respect to t is equal to u(t), except at t = 0, where the derivative of r(t) is not defined. Sinusoidal Function The sinusoid is a continuous-time signal: A cos(wt + q). Here A is the amplitude, w is the frequency in radians per second (rad/s), and q is the phase in radians. The frequency f in cycles per second, or hertz (Hz), is f = w/2p. The sinusoid is a periodic signal with period 2p/w. The sinusoid is plotted in Fig. 2.4. Decaying Exponential In general, an exponentially decaying quantity (Fig. 2.5) can be expressed as a = A e –t/t where a = instantaneous value A = amplitude or maximum value e = base of natural logarithms = 2.718 … t = time constant in seconds t = time in seconds The current of a discharging capacitor can be approximated by a decaying exponential function of time. Time Constant Since the exponential factor only approaches zero as t increases without limit, such functions theoretically last forever. In the same sense, all radioactive disintegrations last forever. In the case of an exponentially decaying current, it is convenient to use the value of time that makes the exponent –1. When t = t = the time constant, the value of the exponential factor is In other words, after a time equal to the time constant, the exponential factor is reduced to approximatly 37% of its initial value. FIGURE 2.4 The sinusoid A cos(wt + q) with –p/2 < q < 0. p + 2q 2w p - 2q 2w 3p - 2q 2w 3p + 2q 2w q w A cos(wt + q) 0 –A A t FIGURE 2.5 The decaying exponential. e e e - - t = = = = t 1 1 1 2 718 0 368 .
FIGURE 2.6 The dc signal with amplitude K. DC Signal The direct current signal (dc signal) can be defined mathematically by (0=K ∞<t<+c Here, K is any nonzero number. The dc signal remains a constant value of K for any -oo< t< oo. The dc signal is plotted in Fig. 2.6 Defining Terms Ramp: A continually growing signal such that its value is zero for tso and proportional to time t for t>0 Sinusoid: A periodic signal x(r)=A cos(or 0)where @= 2nf with frequency in hertz. Unit impulse: A very short pulse such that its value is zero for t*0 and the integral of the pulse is 1 Unit step: Function of time that is zero for t< to and unity for t>to. At t= to the magnitude changes from zero to one. The unit step is dimensionl Related Topic 11.1 Introduction References R.C. Dorf, Introduction to Electric Circuits, 3rd ed, New York: Wiley, 1996 R.E. Ziemer, Signals and Systems, 2nd ed, New York: Macmillan, 1989. Further Information IEEE Transactions on Circuits and Systems IEEE Transactions on Education 2.2 Ideal and Practical Sources Clayton R. paul A mathematical model of an electric circuit contains ideal models of physical circuit elements. Some of these ideal circuit elements(e.g, the resistor, capacitor, inductor, and transformer)were discussed previously. Here we will define and examine both ideal and practical voltage and current sources. The terminal characteristics of these models will be compared to those of actual sources. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC DC Signal The direct current signal (dc signal) can be defined mathematically by i(t) = K –• < t < +• Here, K is any nonzero number. The dc signal remains a constant value of K for any –• < t < •. The dc signal is plotted in Fig. 2.6. Defining Terms Ramp: A continually growing signal such that its value is zero for t £ 0 and proportional to time t for t > 0. Sinusoid: A periodic signal x(t) = A cos(wt + q) where w = 2pf with frequency in hertz. Unit impulse: A very short pulse such that its value is zero for t ¹ 0 and the integral of the pulse is 1. Unit step: Function of time that is zero for t < t0 and unity for t > t0. At t = t0 the magnitude changes from zero to one. The unit step is dimensionless. Related Topic 11.1 Introduction References R.C. Dorf, Introduction to Electric Circuits, 3rd ed., New York: Wiley, 1996. R.E. Ziemer, Signals and Systems, 2nd ed., New York: Macmillan, 1989. Further Information IEEE Transactions on Circuits and Systems IEEE Transactions on Education 2.2 Ideal and Practical Sources Clayton R. Paul A mathematical model of an electric circuit contains ideal models of physical circuit elements. Some of these ideal circuit elements (e.g., the resistor, capacitor, inductor, and transformer) were discussed previously. Here we will define and examine both ideal and practical voltage and current sources. The terminal characteristics of these models will be compared to those of actual sources. FIGURE 2.6 The dc signal with amplitude K. i(t) t 0 K