a) Find b) Check the results of (a) by first integrating

Step functions can be used to define a window function. Thus u(t + 2) - u(t - 3) defines a window 1 unit high and 5 units wide located on the time axis between 2 and 3. A function is defined as follows:60 s t 6 q. = 2.5t - 150 40 s t 60 s = -50, 30 s t 40 s; = -5t + 100, 10 s t 30 s = 5t, 0 t 10 s f a) Sketch over the interval b) Use the concept of the window function to write an expression for

Use step functions to write the expression for each of the functions shown in Fig. P12.3.

Use step functions to write the expression for each function shown in Fig. P12.4.

Use step functions to write the expression for each function shown in Fig. P12.4.

Explain why the following function generates an impulse function as : P 0 f(t) = P>p P2 + t 2, - q t q.

a) Find the area under the function shown in Fig. 12.12(a). b) What is the duration of the function when ? c) What is the magnitude of f(0) when P 0?

The triangular pulses shown in Fig. P12.7 are equivalent to the rectangular pulses in Fig. 12.12(b), because they both enclose the same area (1/P)and they both approach infinity proportional 1/ to as Use this triangular-pulse representation for d(t) to find the Laplace transform of d(t).

Evaluate the following integrals: a) I = L 3 -1 (t 3 + 2)[d(t) + 8d(t - 1)]dt.b I = L 2t2 [d(t) + d(t + 1.5) + d(t - 3)]dt

In Section 12.3, we used the sifting property of the impulse function to show that Show that we can obtain the same result by finding the Laplace transform of the rectangular pulse that exists between in Fig. 12.9 and then finding the limit of this transform as 0

Find f(t) if f(t) = 1 2p L q -q F(v)ejtv dv F(v) = 4 + jv 9 + jv pd(v).

Show that l5d(n) (t)6 = sn.

a) Show that L q -q f(t)d(t - a)dt = -f (a). (Hint: Integrate by parts.) b) Use the formula in (a) to show that l5d(t)6 = s.

Find the Laplace transform of each of the following functions: a) f(t) = 20e-500(t-10)u(t - 10) b)f(t) = (5t + 20)[u(t + 4) - u(t + 2)] - 5t[u(t + 2) - u(t - 2)] + (5t - 20)[u(t - 2) - u(t - 4)].

a) Find the Laplace transform of the function illustrated in Fig. P12.14. b) Find the Laplace transform of the first derivative of the function illustrated in Fig. P12.14. c) Find the Laplace transform of the second derivative of the function illustrated in Fig. P12.14.

Show that l5e-at f(t)6 = F(s + a).

Show that l5f(at)6 = 1a F s a .

a) Find the Laplace transform of te-at b) Use the operational transform given by Eq. 12.23 to find the Laplace transform of d dt (te-at). c) Check your result in part (b) by first differentiating and then transforming the resulting expression.

a) Find b) Check the results of (a) by first integrating and then transforming. l b L t 0- e-ax dx

a) Find the Laplace transform of by first integrating and then transforming. b) Check the result obtained in (a) by using the operational transform given by Eq. 12.33

Find the Laplace transform of each of the following functions: a)f(t) = te-at; b)f(t) = sin vt; c)v d)f(t) = t e)f(t) = cosh(t + u). Hint: See Assessment Problem 12.1c

a) Given that show that dF(s) ds = l5tf(t) b) (-1)n dn F(s) dsn = l5t nf(t)6. Show that c)Use the result of (b) to find l5t l5t sin bt6 and l5te-t cosh t6.

Find a) l b d dt sin vt rb) Find l b d dt cos vt r c) l b d3 dt3 t 2 u(t) r.Find d) Check the results of parts (a), (b), and (c) by first differentiating and then transforming. Check the results of parts (a), (b), and (c) by first differentiating and then transforming

Find the Laplace transform (when P 0 ) of the derivative of the exponential function illustrated in Fig. 12.8, using each of the following two methods: a) First differentiate the function and then find the transform of the resulting function. b) Use the operational transform given by Eq. 12.23.

Find the Laplace transform for (a) and (b). a)f(t) = d dt(e-at cos vt). b)f(t) = Lt0- e-ax sin vx dx. c) Verify the results obtained in (a) and (b) by first carrying out the indicated mathematical operation and then finding the Laplace transform

a) Show that if and is Laplace-transformable, then L q s F(u)du = l b f(t) t r (Hint: Use the defining integral to write L qs F(u)du = L q s a L q 0- f(t)e-utdtb du and then reverse the order of integration.) b) Start with the result obtained in Problem 12.21(c) for and use the operational transform given in (a) of this problem to find l 5sin bt6.

In the circuit shown in Fig. 12.16, the dc current source is replaced with a sinusoidal source that delivers a current of The circuit components are and Find the numerical expression for V(s).

There is no energy stored in the circuit shown in Fig. P12.27 at the time the switch is opened. a) Derive the integrodifferential equation that governs the behavior of the voltage b) Show that Vo(s) = Idc >C s2 + (1>RC)s + (1>LC) c) Show that Io(s) = sIdc s2 + (1>RC)s + (1>LC). Figure P12.27

The switch in the circuit in Fig. P12.28 has been in position a for a long time. At the switch moves instantaneously to position b. a) Derive the integrodifferential equation that governs the behavior of the current for b) Show that Io(s) = Idc[s (1>RC)] [s2 + (1>RC)s + (1>LC)

The switch in the circuit in Fig. P12.29 has been in position a for a long time. At the switch moves instantaneously to position b. a) Derive the integrodifferential equation that governs the behavior of the voltage for b) Show that Vo(s) = Vdc[s + (R>L)] [s2 + (R>L)s + (1>LC)] .

The switch in the circuit in Fig. P12.30 has been open for a long time. At the switch closes. a) Derive the integrodifferential equation that governs the behavior of the voltage for b) Show that Vo(s) = Vdc >RC s2 + (1>RC)s + (1>LC) c) Show that Io(s) = Vdc>RLC s[s2 + (1>RC)s + (1>LC)]

There is no energy stored in the circuit shown in Fig. P12.31 at the time the switch is opened. a) Derive the integrodifferential equations that govern the behavior of the node voltages and b) Show that Figure P12.31

a) Write the two simultaneous differential equations that describe the circuit shown in Fig. P12.32 in terms of the mesh currents and b) Laplace-transform the equations derived in (a). Assume that the initial energy stored in the circuit is zero. c) Solve the equations in (b) for and Figure P12.32

Find in Problem 12.26.

The circuit parameters in the circuit seen in Fig. P12.27 have the following values:R = 20 L = 50 mH, C = 20 mF, and a) v t 0. o(t) Find for b) i t 0. o(t Find for c) Does your solution for io(t) make sense when t = 0? Explain.

The circuit parameters in the circuit in Fig. P12.28 are R = 2.5 k, L = 50 mH and C = 2 nF If find t 0.

The circuit parameters in the circuit in Fig. P12.29 are and If V (t) dc = 120 V t 0

The circuit parameters in the circuit in Fig. P12.30 are and If is 56 V, find a) t 0 for b)for t 0

The circuit parameters in the circuit in Fig. P12.31 R = 4000 L = 40 mH andC = 15.625 nF

Use the results from Problem 12.32 and the circuit shown in Fig P12.32 to a) Find i2 i (t).and b) Find i (q). 1(q and c) Do the solutions i1 for and i2make sense? Explain.

Find for each of the following functions: F(s) = 6(s + 10) (s + 5)(s + 8 F(s) = 20s2 + 141s + 315 s(s2 + 10s + 21) .F(s) = 15s2 + 112s + 228 (s + 2)(s + 4)(s + 6).F(s) = 2s3 + 33s2 + 93s + 54s(s + 1)(s2 + 5s + 6) .

Find f(t) for each of the following functions a) b) c) d) F(s) = 8(s + 1)2 (s2 + 10s + 34)(s2 + 8s + 20). F(s) = 14s2 + 56s + 152 (s + 6)(s2 + 4s + 20). F(s) = -s2 + 52s + 445 s(s2 + 10s + 89) . F(s) = 280 s2 + 14s + 2

Find f(t) for each of the following functions. a) \(F(s)=\frac{320}{s^{2}(s+8)}\) b) \(F(s)=\frac{80(s+3)}{s(s+2)^{2}}\) c) \(F(s)=\frac{60(s+5)}{(s+1)^{2}\left(s^{2}+6 s+25\right)}\) d) \(F(s)=\frac{25(s+4)^{2}}{s^{2}(s+5)^{2}}\) Text Transcription: F(s)=\frac{320}{s^{2}(s+8)} F(s)=\frac{80(s+3)}{s(s+2)^{2}} F(s)=\frac{60(s+5)}{(s+1)^{2}\left(s^{2}+6 s+25\right)} F(s)=\frac{25(s+4)^{2}}{s^{2}(s+5)^{2}}

Find for each of the following functions F(s) = 135 s(s + 3)3 .F(s) = 10(s + 2)2 (s2 + 2s + 2)2 .F(s) = 25s2 + 395s + 1494 s2 + 15s + 54 F(s) = 5s3 + 20s2 - 49s - 108 s2 + 7s + 10

Derive the transform pair given by Eq. 12.64

a) Derive the transform pair given by Eq. 12.83. b) Derive the transform pair given by Eq. 12.84.

a) Use the initial-value theorem to find the initial value of in Problem 12.26. b) Can the final-value theorem be used to find the steady-state value of ? Why?

Use the initial- and final-value theorems to check the initial and final values of the current and voltage in Problem 12.27.

Use the initial- and final-value theorems to check the initial and final values of the current in Problem 12.28.

Use the initial- and final-value theorems to check the initial and final values of the current and voltage in Problem 12.30.

Apply the initial- and final-value theorems to each transform pair in Problem 12.40.

Apply the initial- and final-value theorems to each transform pair in Problem 12.41.

Apply the initial- and final-value theorems to each transform pair in Problem 12.42.

Apply the initial- and final-value theorems to each transform pair in Problem 12.43.

a) Use phasor circuit analysis techniques from Chapter 9 to determine the steady-state expression for the inductor current in Fig. 12.18. b) How does your result in part (a) compare to the complete response as given in Eq. 12.109?

Find the maximum magnitude of the sinusoidal source in Fig. 12.18 such that the complete response of the inductor current does not exceed the 40 mA current rating at t = 1 ms.

Suppose the input to the circuit in Fig 12.18 is a damped ramp of the form -Kte -100t V. Find the largest value of K such that the inductor current does not exceed the 40 mA current rating.