1. Redraw the following schematics with the impedance of each of the element shown in Laplace domain. Then determine the overall impedance of the entire circuit between the two ends of the shown circuit and express it in Laplace domain as a ratio of two polynomials in s, with the coefficients of the highest power if s in the numerator and denominator are made unity. (Follow the method outlined in the lecture to determine the impedances of elements in Laplace domain and then use the formulas for combining impedances in series and parallel.)
2. (a) Apply Laplace transform to the following differential equation and express it as an algebraic equation in s
(b) Given that all initial conditions are zero,
3. An RC circuit with an initial condition is shown below. The toggle switch is closed at t = 0. Assuming that a current i(t) flowsclockwise in the circuit, Write the integral equation that governs the behavior of the circuit current and solve it for the current in the circuit i(t) and voltage across the capacitor as a function of time using Laplace transforms. Note the polarity of the initial condition as marked in the figure. (Take help from the document “Solving RC, RLC, and RL Circuits Using Laplace Transforms” (located in Doc Sharing) and the Week 2 Lecture to see how initial conditions are entered in Laplace domain.)
4. The voltage in a circuit, expressed in Laplace domain, is given by the questions below
5. An RLC circuit is shown below. There is an initial voltage of 5 V on the capacitor, with polarity as marked in the circuit. The switch is closed at t = 0 and a current i(t) is assumed to flow clockwise. Write the integral-differential equation of this circuit using Kirchoff’s method (sum of all voltages around a loop is zero). Apply Laplace transform as outlined in the lecture for Week 2 and in the document “Solving RC, RLC, and RL Circuits Using Laplace Transforms” (located in Doc Sharing) and write i(s) in Laplace transform notation. Express the denominator with the coefficient of the highest power of s unity. Then invert to obtain the current in time domain, i(t).