A block diagram of the mystery circuit

Sketches are a handy visual guide that allow you to integrate observations and focus on the components of interest. Make sketches similar to the ones shown, to facilitate the troubleshooting and analysis of your mystery circuit.

The sketch on the left displays a basic schematic of the mystery circuit, grouped into functional blocks.

To the left is the resistor voltage divider ladder with connections to the black box.

To the right is the resistor/capacitor network that was previously analysed. It was determined that transistor Q1, here inside the black box, controls the state of the R4-R5 node and hence the charging and discharging of the capacitor.

Analysis of capacitor voltage

• Connect the scope Ch2 to the R5-C node to monitor the capacitor voltage.
• Use the multimeter to measure the power supply voltage.
• Measure and record the high V_H and low V_L limits of the capacitor voltage. What is the ratio of these two values relative to the power supply voltage?
• Measure and record the voltages at the R1-R2 and R2-R3 nodes of the voltage divider. What is the ratio of these values relative to the power supply voltage?
• What would these voltage ratios be in the ideal case where the values of R1, R2 and R3 are identical?
• Based on the above results, propose a possible functional relationship, realized within the black box, between the RC circuit and the resistor voltage divider.

Analytical derivation of the timing equation

Follow the steps outlined in the next slide to analytically determine the time delays t_c and t_d between V_H and V_L for the capacitor charging and discharging cycles.

The timing equation is then T = t_c + t_d. It should be algebraically equivalent to that previously derived using dimensional analysis.

Recall that the capacitor discharging from V₀ to V₁ is given by

V₁ = V₀ exp ( -t / RC)

and the capacitor charging from V₁ to V₀ is given by

V₁ = V₀ ( 1-exp ( -t / RC ) )

Square wave indicates switch state: high=open, low=closed

Using your measured V_H and V_L:

• use the discharging equation to solve for t = t₁ at V_H;
• use the discharging equation to solve for t = t₂ at V_L;
• evaluate t_d = t₂ - t₁ with R = R_d;
• use the charging equation to solve for t = t₁ at V_L;
• use the charging equation to solve for t = t₂ at V_H;
• evaluate t_c = t₂ - t₁ with R = R_c;
• determine T = t_c + t_d.

Is this T equation algebraically equivalent to that obtained from dimensional analysis? Do the coefficients have values similar to the analytical derivation? If yes, well done!

Derive the timing equation for the mystery circuit using the ideal voltage ratios for V_H = 2V₀/3 and V_L = V₀/3 to determine the theoretical values for the scaling constants k1 and k2.

Manual control of capacitor charge/discharge

Instead of having the mystery circuit control the capacitor charge/discharge you can use a switch to control Q1 and hence the capacitor voltage. To set up the circuit:

• disconnect R6 from pin 5 of IC2
• connect R6 to the centre pin of a three-way switch;
• connect one end pin of the switch to 0V;
• connect the other end pini of the switch to 5V;
• use R4 = R5 = 22kΩ and C = 10μF.
• connect CH1 to the centre pin of the switch;
• connect CH2 to the R5-C node of the circuit;
• set the timebase to a 1s Roll.

Toggle the switch to vary the voltage at the capacitor. Try to reproduce the mystery circuit capacitor charge/discharge curve.

Measuring an RC constant

If the time between two points on a capacitor charge/discharge curve is such that t₁ - t₀ = τ = RC, then the charging/discharging equations can be generalized to

V₁ - V_∞ = ( V₀ - V_∞)/e

where V_∞ is the voltage at the steady-state end of the transient. Displayed is a capacitor discharge curve. To measure τ:

• determine the correct value of V_∞ to use;
• position one cursor at some voltage V₀ along the transient;
• record the time t₀ corresponding to V₀;
• position the second cursor at the calculated voltage V₁;
• record the time t₁ corresponding to V₁.

To validate the general capacitor charging/discharging equation by measuring the RC constant:

• calculate the theoretical τ for this circuit. Use the measured values for R4, R5, C and the +5V supply;
• toggle the switch to generate a full discharge curve;
• freeze the scope display and adjust the gain/timebase to zoom in on the exponential decay curve;
• follow the previously outlined steps;
• save a screenshot showing the cursors placement.

Compare your experimental and theoretical results for τ. Do they agree within experimental error?

Try a couple of other initial V₀ positions along the curve and compare these experimental results with the theoretical τ. Repeat for a capacitor charging curve, shown here.