Having completed the previous lab, you should have a functioning mystery circuit that flashes its LED on/off at a rate of a couple of times per second. In this lab you will use the oscilloscope to describe and measure the time-dependent voltages, or waveforms, that can be seen at various key points in the circuit. Then you can derive the timing equation that controls this flashing cycle.

A dimensional analysis approach to determine the mystery circuit timing equation

The flashing LED and periodically repeating waveforms viewed on the scope, in particular the varying voltage at the capacitor (the node between $R5$ and $C$), provide a hint that the capacitor is undergoing a repeated charging and discharging cycle.

Since this flow of charge (current) must be limited somehow, by looking at the circuit diagram, we note that there is some resistance, $R4$ and $R5$ in series, between the capacitor and the (conventional) source of the electric current, in our case, the +5V power supply.

Ignoring the other connections to these components, we make an educated guess that the period of the flashing LED is related to values of the capacitor and the two resistors.

We then use the method of dimensional analysis to determine a tentative relationship, if there is one, between the period $T$ of the waveforms and the components $C, R4, R5$ by matching their physical units (dimensions) so that a function $T(C,R4,R5)$ is derived.

• Given $T$ and $t$ in seconds, $q$ in Coulombs, $V$ in Volts and that $C=q/V$, $i=dq/dt$, $R=V/i$, show that the following relationship between the period $T$ and $C, R4$ and $R5$ is valid: $$ T(C,R4,R5) = k1*C*(R4+k2*R5) $$ where $k1$ and $k2$ are two dimensionless scaling constants to be experimentally determined.

Typically, each variable $C,R4,R5$ is multiplied by a scaling constant that determines it's contribution to the overall $T$ equation. In this case, the $R4$ coefficient has been divided into the other two, resulting in the above equation that, in hindsight, best describes the timing of the mystery circuit.

To explore how $T$ is related to $R_1 , R_2$ and $C$, we change one at a time the values of the three circuit variables, carefully measure the resulting $T_H, T_L$, calculate $T$ and tabulate the results. A review of this table can provide insights into the relationship between the wave timings and the component values used, for example, how a change in $T_H$ might be related to a change in $R4, R5$ or $C$.

Solving for any two simultaneous equations from this table should yield values for the unknowns $k_1$ and $k_2$, although in practice, due to measurement errors, the results may vary somewhat with the choice of data sets.

The master plan is then to collect a table of $T_L , T_H , T$ values, varying $R4$, $R5$ and $C$ one at a time. Then select two trials from the table and solve the two simultaneous equations for the unknowns $k1$ and $k2$ to obtain an explicit relationship between the period $T$ and the components $R4, R5$ and $C$. Determine $k1$ and $k2$ using different pairs of trials to estimate their errors.

Prelab preparation

To prepare for this lab session, include in your "lab book" a validation of the timing equation based on dimensional analysis and a table for the variables that will be set and measure during the session, as outlined in the slides, to determine the scaling constants $k1$ and $k2$.

Consider how you intend to tackle the task of solving two simultaneous equations.

Review also the information pertaining to the oscilloscope summarized in 0.Introduction You will use the scope extensively during this lab session and the ones that follow.