Question

1 7. Growth and decay of the voltage in an RC series circuit The aim of this exercise is to study the response of a series RC circuit to a square wave voltage. In figure (1), a capacitor of capacitance C is connected in series with a resistor of resistance R = 10 k across an LFG adjusted to give a square wave output of period T and of maximum value E. An oscilloscope is connected to display the variations of the voltages UDM = Uc and UFM = UG across the capacitor and the generator respectively. Figure (2) is a diagram that shows the voltages over one period. a) Use figure (2) to: i. pick out the period T and the voltage E ii. identify (with justification) the four curves. b) i. Determine the differential equation that governs the variation of the voltage Uc = UDM across the terminals of the capacitor during the charging process. ii. Verify that uc = E (1 - e7) is a solution of the differential equation where t = RC. iii. Use figure (2) to determine the value of t. Deduce the value of C. u(V) 16- 14+ 12+ 10- P=E E UB64 8- 6+ 4+ 2 0 17 0 K +2 w M 4 6 duc dt 8 Ch2, 9 D LFG (2) Fig. 1 Chl R = 10 k c) i. Use the differential equation to determine the expression of the tangent to uc = f(t) at t = 0 in terms of E, T, and t. ii. Deduce that this tangent cuts the curve of equation u₁ = E at point K in figure (2). d) The capacitor starts discharging at t = 12 ms, taken as a new initial time. i. Determine the differential equation that governs the variation of uc. F Ot(ms) 10 12 14 16 18 20 22 24 Fig. 2 at t = 0. Deduce the equation of ii. The solution of the obtained differential equation is uc = Pewhere P and t' are constants. Determine the expressions of P and T'. iii. Use this expression of uc to prove that t' has a time unit.​

Answer 1

Exercise 5-3: Series RC circuit with square-wave input Consider the series RC circuit shown in Figure 5−5 with R=10kΩ,C=0.047μF. Notice this RC circuit is quite similar to the circuit depicted in Figure 5-4 except that the positions of the resistor and the capacitor are swapped. In the present case, an oscilloscope is placed to monitor the transient responses of the capacitor and resistor when the input vs ( t ) is a squarewave voltage signal. Figure 5-5: Series RC circuit driven by square-wave voltage signal E3.1 Capture the schematic of the RC circuit in Multisim and provide a captured image of the circuit schematic in the lab report. E3.2 Set the function generator to generate a square wave with IV amplitude and 0 V DC offset at 200 Hz characterized by 50% duty circle. E3.3 Simulate the capacitor and resistor responses by turning the switch ON and OFF repeatedly. Adjust the settings on the oscilloscope such that the transient capacitor voltage v

C

(t) and resistor voltage v

R

(t) are clearly displayed. Capture the screenshot of the oscilloscope display which shows v

C

(t) and v

R

(t) clearly when the capacitor is charging as well as discharging. The captured image should be included in the lab report. E3.4 Use the cursors to determine the time Δt

R

taken for v

R

(t) to reach the steady state while the capacitor is discharging. Δt

R

= (ms) 5τ= Compare Δt

R

with the expected value of 5τ. E3.5 Explain if KVL can be verified by examining the waveforms v

R

(t) and v

C

(t) only..