For the network represented in Fig. 15.68, determine the critical frequencies of Zin(s)

Determine the current i(t) in the circuit of Fig. 15.4.

Repeat Example 15.2 using the voltage-source-based capacitor model.

Employ nodal analysis to calculate \(v_x (t)\) for the circuit of Fig. 15.11.

Use nodal analysis to determine the voltages \(v_1\), \(v_2\), and \(v_3\) in the circuit of Fig. 15.13. Assume there is zero energy stored in the inductors at \(t = 0^-\).

Using the method of source transformation, reduce the circuit of Fig. 15.16 to a single s-domain current source in parallel with a single impedance.

Working in the s-domain, find the Norton equivalent connected to the \(1\ \Omega\) resistor in the circuit of Fig. 15.18.

Repeat Example 15.8 using the second integral of Eq. [11].

The impulse response of a network is given by \(h(t) = 5u(t ? 1)\). If an input signal \(x(t) = 2[u(t) ? u(t ? 3)]\) is applied, determine the output y(t) at t equal to (a) ?0.5; (b) 0.5; (c) 2.5; (d) 3.5.

Repeat Example 15.8, performing the convolution in the s-domain.

Referring to the circuit of Fig. 15.26a, use convolution to obtain \(v_o(t)\) if \(v_{in} = tu(t)\ V\).

Sketch the magnitude of the impedance \(Z(s) = 2 + 5s\) as a function of \(\sigma\) and \(j \omega\).

The parallel combination of 0.25 mH and \(5\ \Omega\) is in series with the parallel combination of \(40\ \mu F\) and \(5\ \Omega\). (a) Find \(Z_{in} (s)\), the input impedance of the series combination. (b) Specify all the zeros of \(Z_{in} (s)\). (c) Specify all the poles of \(Z_{in} (s)\). (d) Draw the pole-zero configuration.

If a current source \(i_1(t) = u(t)\ A\) is present at a-b in Fig. 15.37 with the arrow entering a, find \(\mathbf{H}(\mathbf{s})=\mathbf{V}_{c d} / \mathbf{I}_{1}\), and specify the natural frequencies present in \(v_{cd} (t)\).

Specify suitable element values for \(\mathbf{Z}_{1}\) and \(\mathbf{Z}_{f}\) in each of three cascaded stages to realize the transfer function \(\mathbf{H}(\mathbf{s})=-20 s^{2} /(s+1000)\).

Draw an s-domain equivalent of the circuit depicted in Fig. 15.45, if the only quantity of interest is v(t). (Hint: omit the source, but don’t ignore it.)

For the circuit of Fig. 15.46, the only quantity of interest is the voltage v(t). Draw an appropriate s-domain equivalent circuit. (Hint: omit the source, but don’t ignore it.)

For the circuit represented in Fig. 15.47, draw an s-domain equivalent and analyze it to obtain a value for i(t) if i(0) is equal to (a) 0; (b) -2 A.

For the circuit of Fig. 15.47, draw an s-domain equivalent and analyze it to obtain a value for v(t) if i(0) is equal to (a) 0; (b) 3 A.

With respect to the s-domain circuit drawn in Fig. 15.48, (a) calculate \(\mathbf{V}_{C}(\mathbf{s})\); (b) determine \(v_C (t)\), t > 0; (c) draw the time-domain representation of the circuit.

Draw all possible s-domain equivalents (t > 0) of the circuit shown in Fig. 15.49.

With respect to the network of Fig. 15.51, obtain an expression for the input admittance Y(s) as labeled. Express your answer as a ratio of two s-polynomials.

For the circuit of Fig. 15.52, (a) draw both s-domain equivalent circuits; (b) choose one and solve for V(s); (c) determine v(t).

Determine the input impedance 1/Y(s) of the network represented in Fig. 15.51 if the \(1.5\ \Omega\) resistor is replaced with the parallel combination of a 100 mF capacitor and a \(1\ \Omega\) resistor, and the initial current through the inductor (defined as flowing downward) is 540 mA.

For the circuit given in Fig. 15.53, (a) draw the s-domain equivalent; (b) write the three s-domain mesh equations; (c) determine \(i_1\), \(i_2\), and \(i_3\).

Replace the ?4u(t) label in the circuit of Fig. 15.53 with \(4e^{?t}\ u(t)\ V\). Calculate \(i_1\), \(i_2\), and \(i_3\) if it transpires that the initial current through the inductor, \(i_2?i_3\), is equal to 50 mA.

The 2u(t) A source in Fig. 15.55 is replaced with a \(4e^{?t} u(t)\ A\) source. Employ s-domain analysis to determine the power dissipated by the \(1\ \Omega\) resistor.

Calculate the power dissipated in the \(3\ \Omega\) resistor of Fig. 15.56, if \(v_1 (0^- ) = 2\ V\).

For the circuit shown in Fig. 15.57, let \(i_{s1} = 3u(t)\ A\) and \(i_{s2} = 5\ sin\ 2t\ A\). Working initially in the s-domain, obtain an expression for \(v_x (t)\).

For the circuit of Fig. 15.58, (a) draw the corresponding s-domain circuit; (b) solve for \(v_1 (t)\), \(v_2 (t)\), and \(v_3 (t)\); (c) verify your solution with an appropriate PSpice simulation.

Determine the mesh currents \(i_1 (t)\) and \(i_2 (t)\) in Fig. 15.59 if the current through the 1 mH inductor \((i_2 ? i_4)\) is 1 A at \(t = 0^- \). Verify that your answer approaches the answer obtained using phasor analysis as the circuit response eventually reaches steady state.

Assuming no energy initially stored in the circuit of Fig. 15.60, determine the value of \(v_2\) at t equal to (a) 1 ms; (b) 100 ms; (c) 10 s.

Using repeated source transformations, obtain an s-domain expression for the Thévenin equivalent seen by the element labeled Z in the circuit of Fig. 15.61.

Calculate I(s) as labeled in the circuit of Fig. 15.61 if the element Z has impedance of (a) \(2\ \Omega\); (b) \(\frac{1}{2s}\ \Omega\); (c) \(s + \frac{1}{2s} + 3\ \Omega\).

For the circuit shown in Fig. 15.62, determine the s-domain Thévenin equivalent seen by the (a) \(2\ \Omega\) resistor; (b) \(4\ \Omega\) resistor; (c) 1.2 F capacitor; (d) current source.

Calculate both currents labeled in the circuit of Fig. 15.62.

For the circuit of Fig. 15.63, take \(i_s (t) = 5u(t)\ A\) and determine (a) the Thévenin equivalent impedance seen by the \(10\ \Omega\) resistor; (b) the inductor current \(i_L (t)\).

If the current source of Fig. 15.63 is \(1.5e^{-2t}u(t)\ \mathrm{A}\), and \(i_L (0^- ) = 1\ A\), determine \(i_x (t)\).

(a) Use superposition in the s-domain to find an expression for \(\mathbf{V}_{1}(\mathbf{s})\) as labeled in Fig. 15.65. (b) Find \(v_1 (t)\).

If the top right voltage source of Fig. 15.65 is open-circuited, determine the Thévenin equivalent seen looking into the terminals marked a and b.

If the bottom left voltage source of Fig. 15.65 is open-circuited, determine the Thévenin equivalent seen looking into the terminals marked c and d.

Determine the poles and zeros of the following s-domain functions: (a) \(\frac{s}{s+12.5}\); (b) \(\frac{s(s+1)}{(s+5)(s+3)}\); (c) \(\frac{s+4}{s^2 +8s+7}\); (d) \(\frac{s^2 -s-2}{3s^3 + 24s^2 +21s}\).

Use appropriate means to ascertain the poles and zeros of (a) s + 4; (b) \(\frac{2s}{s^2 -8s+16}\); (c) \(\frac{4}{s^3 +8s +7}\); (d) \(\frac{s-5}{s^3 -7s+6}\).

Consider the following expressions and determine the critical frequencies of each: (a) \(5 + s^{-1}\); (b) \(\frac{s(s+1)(s+4)}{(s+5)(s+3)^2}\); (c) \(\frac{1}{s^2 +4}\); (d) \(\frac{0.5s^2 -18}{s^2 +1}\).

For the network represented schematically in Fig. 15.66, (a) write the transfer function \(\mathbf{H}(\mathrm{s}) \equiv \mathbf{V}_{\text {out }}(\mathrm{s}) / \mathbf{V}_{\text {in }}(\mathrm{s})\); (b) determine the poles and zeros of H(s).

For each of the two networks represented schematically in Fig. 15.67, (a) write the transfer function \(\mathbf{H}(\mathbf{s}) \equiv \mathbf{V}_{\text {out }}(\mathbf{s}) / \mathbf{V}_{\text {in }}(\mathbf{s})\); (b) determine the poles and zeros of H(s).

Determine the critical frequencies of \(\mathbf{Z}_{\text {in }}\) as defined in Fig. 15.50.

Specify the poles and zeros of Y(s) as defined by Fig. 15.51.

A particular network is known to be characterized by the transfer function \(\mathbf{H}(\mathbf{s})=\mathbf{s}+1 /\left(\mathbf{s}^{2}+23 \mathbf{s}+60\right)\). Determine the critical frequencies of the output if the input is (a) \(2 u(t)+4 \delta(t)\); (b) \(?5e^{?t}\ u(t)\); (c) \(4te^{?2t}\ u(t)\); (d) \(5 \sqrt{2}e^{-10t}\ cos\ 5t\ u(t)\ V\).

For the network represented in Fig. 15.68, determine the critical frequencies of \(\mathbf{Z}_{\mathrm{in}}(s)\).

Referring to Fig. 15.69, employ Eq. [11] to obtain x(t) ? y(t).

With respect to the functions x(t) and y(t) as plotted in Fig. 15.69, use Eq. [11] to obtain (a) \(x(t) ? x(t)\); (b) \(y(t) ? \delta (t)\).

Employ graphical convolution techniques to determine \(f ? g\) if \(f (t) = 5u (t)\) and \(g(t) = 2u(t) ? 2u(t ? 2) + 2u(t ? 4) ? 2u(t ? 6)\).

Let \(h(t) = 2e^{?3t} u(t)\) and \(x(t) = u(t) ? \delta (t)\). Find \(y(t) = h(t) ? x(t)\) by (a) using convolution in the time domain; (b) finding H(s) and X(s) and then obtaining \(\mathscr{L}^{-1}\{\mathbf{H}(\mathbf{s}) \mathbf{X}(\mathbf{s})\}\).

A \(2\ \Omega\) resistor is placed in series with a 250 mF capacitor. Sketch the magnitude of the equivalent impedance as a function of (a) \(\sigma\) ; (b) \(\omega\); (c) \(\sigma\) and \(\omega\), using an elastic-sheet type approach. (d) Verify your solutions using MATLAB.

Sketch the magnitude of \(\mathbf{Z}(\mathbf{s})=\mathbf{s}^{2}+\mathbf{s}\) as a function of (a) \(\sigma\) ; (b) \(\omega\); (c) \(\sigma\), and \(\omega\), using an elastic-sheet type approach. (d) Verify your solutions using MATLAB.

Sketch the pole-zero constellation of each of the following: (a) \(\frac{\mathbf{s}(\mathbf{s}+4)}{(\mathbf{s}+5)(\mathbf{s}+2)}\); (b) \(\frac{\mathbf{s}-1}{\mathbf{s}^{2}+8 \mathbf{s}+7}\); (c) \(\frac{\mathbf{s}^{2}+1}{\mathbf{s}\left(\mathbf{s}^{2}+10 \mathbf{s}+16\right.}\); (d) \(\frac{5}{\mathbf{s}^{2}+2 \mathbf{s}+5}\).

The partially labeled pole-zero constellation of a particular transfer function H(s) is shown in Fig. 15.71. Obtain an expression for H(s) if H(0) is equal to (a) 1; (b) ?5. (c) Is the system H(s) represents expected to be stable or unstable? Explain.

The three-element network shown in Fig. 15.72 has an input impedance \(\mathbf{Z}_{A}(\mathbf{s})\) that has a zero at \(s = ?10 + j0\). If a \(20\ \Omega\) resistor is placed in series with the network, the zero of the new impedance shifts to \(s = ?3.6 + j0\). Calculate R and C.

Let \(\mathbf{H}(\mathbf{s})=100(\mathbf{s}+2) /\left(\mathbf{s}^{2}+2 \mathbf{s}+5\right)\) and (a) show the pole-zero plot for H(s); (b) find \(\mathbf{H}(j \omega)\); (c) find \(|\mathbf{H}(j \omega)|\); (d) sketch \(|\mathbf{H}(j \omega)|\) versus \(\omega\); (e) find \(\omega_{\max }\), the frequency at which \(|\mathbf{H}(j \omega)|\) is a maximum.

Determine expressions for \(i_1 (t)\) and \(i_2 (t)\) for the circuit of Fig. 15.73, assuming \(v_1 (0^- ) = 2\ V\) and \(v_2 (0^- ) = 0\ V\).

The 250 mF capacitor in the circuit of Fig. 15.73 is replaced with a 2 H inductor. If \(v_1 (t) = 0\ V\) and \(i_1 (0^- ) - i_2 (0^- ) = 1\ A\), obtain an expression for \(i_2 (t)\).

In the network of Fig. 15.74, a current source \(i_x (t) = 2u(t)\ A\) is connected between terminals c and d such that the arrow of the source points upward. Determine the natural frequencies present in the voltage \(v_{ab} (t)\) which results.

With regard to the circuit shown in Fig. 15.75, let \(i_1 (0^- ) = 1\ A\) and \(i_2 (0^- ) = 0\). (a) Determine the poles of \(\mathbf{I}_{\mathrm{in}}(\mathbf{s}) / \mathbf{V}_{\mathrm{in}}(\mathbf{s})\); (b) use this information to obtain expressions for \(i_1 (t)\) and \(i_2 (t)\).

Design a circuit which produces the transfer function \(\mathbf{H}(\mathbf{s})=\mathbf{V}_{\text {out }} / \mathbf{V}_{\text {in }}\) equal to (a) 5(s + 1); (b) \(\frac{5}{(s+1)}\); (c) \(5 \frac{s+1}{s+2}\).

Design a circuit which produces the transfer function \(\mathbf{H}(\mathbf{s})=\mathbf{V}_{\text {out }} / \mathbf{V}_{\text {in }}\) equal to (a) \(2(s + 1)^2\); (b) \(\frac{3}{(s+500)(s+100)}\).

Design a circuit which produces the transfer function \(\mathbf{H}(\mathbf{s})=\frac{\mathbf{V}_{\text {out }}}{\mathbf{V}_{\text {in }}}=5 \frac{\mathbf{s}+10^{4}}{\mathbf{s}+2 \times 10^{5}}\).

Design a circuit which produces the transfer function \(\mathbf{H}(\mathbf{s})=\frac{\mathbf{V}_{\text {out }}}{\mathbf{V}_{\text {in }}}=3 \frac{\mathbf{s}+50}{(\mathbf{s}+75)^{2}}\).

Find \(\mathbf{H}(\mathbf{s})=\mathbf{V}_{\text {out }} / \mathbf{V}_{\text {in }}\) as a ratio of polynomials in s for the op-amp circuit of Fig. 15.42, given the impedance values (in \(\omega\)): (a) \(\mathbf{Z}_{1}(\mathbf{s})=10^{3}+\left(10^{8} / \mathbf{s}\right)\), \(\mathbf{Z}_{f}(\mathbf{s})=5000\); (b) \(\mathbf{Z}_{1}(\mathbf{s})=5000\), \(\mathbf{Z}_{f}(\mathbf{s})=10^{3}+\left(10^{8} / \mathbf{s}\right)\); (c) \(\mathbf{Z}_{1}(\mathbf{s})=10^{3}+\left(10^{8} / \mathrm{s}\right)\), \(\mathbf{Z}_{f}(\mathbf{s})=10^{4}+\left(10^{8} / s\right)\).

Design a circuit that provides a frequency of 16 Hz, which is near the lower end of the human hearing range. Verify your design with an appropriate simulation.

Design a circuit which provides a dual-tone multifrequency (DTMF) signal corresponding to the number 9, which is a voltage output composed of a 1477 Hz signal and an 852 Hz signal.

(a) Design a circuit which provides a signal at 261.6 Hz, which is approximately middle “C”. Use only standard 5% tolerance resistance values. (b) Estimate the likely frequency range of your signal generator based on the range of possible resistor values which can be used in construction.

(a) Many people with partial hearing loss, especially the elderly, have difficulty in detecting standard smoke detectors. An alternative is to lower the frequency to approximately 500 Hz. Design a circuit which provides such a signal, using only standard 10% tolerance resistor and capacitor values. (b) Estimate the actual frequency range expected from your design if it is manufactured based on the possible range of component values.

Find the mesh currents \(i_1\) and \(i_2\) in the circuit of Fig. 15.8. You may assume no energy is stored in the circuit at \(t = 0^-\)

Determine the input impedance \(Z_{in} (s)\) seen looking into the terminals of the network depicted in Fig. 15.50. Express your answer as a ratio of two s-polynomials.

For the circuit shown in Fig. 15.54, (a) write an s-domain nodal equation for \(V_x (s)\); (b) solve for vx (t).

Determine \(v_1\) and \(v_2\) for the circuit of Fig. 15.55 using nodal analysis in the s-domain.

For the s-domain circuit of Fig. 15.64, determine the Thévenin equivalent seen looking into the terminals marked a and b.

If a network is found to have the transfer function \(H(s) = \frac{s}{s^2 +8s+7}\), determine the s-domain output voltage for \(v_{in} (t)\) equal to (a) 3u(t) V; (b) \(25e^{-2t} u(t)\ V\); (c) 4u(t + 1) V; (d) 2 Sin 5t u(t) V.

(a) Determine the impulse response h(t) of the network shown in Fig. 15.70. (b) Use convolution to determine \(v_o (t)\) if \(v_{in} (t) = 8 u(t)\ V\).

Design a circuit which provides either a 200 Hz signal or a 400 Hz signal by closing appropriate switches.