1. Instructions

Your homework assignment is to complete the following problems and turn them in as hw03. Working together with in a small travelling group, I predict you can get 60%+ of the problems finished within the class time.

The two digits represent the numbered spaces on the left half of the Gather class room where you can meet other folk working on that particular problem. Please avoid straight-up posting the answers, but use the areas and whiteboards to help your discussion.

A great thing to add to the whiteboards are specific references and page numbers of relevant reference material that helped you solve the problem.

The bold digits correspond to the labels of the area on the left half of the Gather class room. Move to the area that matches the problem you are working on. Be aware that the other NN “tables” are for a different ECE class.

2. References

2.1. Hayt book Engineering Circuit Analysis

Chapter 14 is the focus.

p.542 Two- and One-sided transform. It is quite common that our signals only exist (mathematically not zero) for \(t \ge 0\). Any old Laplace transform table you may find will (should) say whether it is one or two-sided.

p.546—​551 Inverse transform \((s \rightarrow t)\)

2.2. Ulaby book Circuit Analysis and Design

Chapter 12.

The text around these pages work through examples using these important tables.

  • p.642 (pdf 656) Table 12-1 — Properties of the Laplace transform

  • p.643 (pdf 657) Table 12-1 — Laplace transform pairs

3. Problems

3.1. 03

Find the Laplace transform of \(x(t) = 5\,\delta(t) - 3\,u(t)\).

(a.k.a. Hayt practice problem 14.5a, so show your work)

Superposition! Perhaps re-write as \(x(t) = 4\,\delta(t) + \left[-3\,u(t)\right]\) to see the addition.


3.2. 04

Find the Laplace transform of \(x(t) = 4\,\delta(t-2) - 3t\,u(t)\).

(a.k.a. Hayt practice problem 14.5b, show your work)


3.3. 05

Find the Laplace transform of \(x(t) = u(t-2)\).

(a.k.a. Hayt practice problem 14.5c, show your work)


3.4. 13

Given the transfer function \(H(s) = \dfrac{2}{s} - \dfrac{4}{s^2} + \dfrac{3.5}{(s+10)(s+10)}\), find the impulse response \(h(t)\).

(a.k.a. Hayt practice problem 14.7, show your work)


3.5. 14

Given the s-domain function \(Q(s) = \dfrac{3\,s^2 - 4}{s^2}\), find \(q(t)\).

Do you believe in superposition? Do you see how this can be rewritten as two terms added together?


3.6. 15

Find the Laplace transform of \(f_1(t) = 2 \left[ 2 - e^{-t} \right] u(t)\).

\(F_1(s)=\)

(Ulaby Exercise 12-5a)


3.7. 23

A capacitor has a time-domain equation of \(i_C(t) = C \dfrac{d}{dt} V_C(t)\), what is this equation in the s-domain?

\(I_C(s)=\)

See Ulaby book section 12-4.


3.8. 24

An inductor has a time-domain equation of \(i_L(t) = \dfrac{1}{L} \int\limits_0^t v_L(\tau) d\tau\), what is this equation in the s-domain?

\(I_L(s)=\)


3.9. 25

Use your results from 23 and 24 to write the Laplace version of “capacitor’s law” and “inductor’s law”. Both are a function of s and the symbolic value of the device (C or L).

\(\dfrac{V_C(s)}{I_C(s)} = \)

\(\dfrac{V_L(s)}{I_L(s)} = \)


4. Submission

Turn in your work for each problem as a PDF to Blackboard. A good strategy is to do one problem per half+ page on (engineering) paper, scan / crop / enhance to multi-page PDF. Or another method you consider reasonable to both create and read.