2/6: RC circuits!

in which Julia learns about capacitors and investigates persistent wiggles

(I'm particularly excited about this because we just talked about modeling viscoelastic materials with an RC circuit in my Structural Biomaterials class!)

Our first task: Lab 2-1 in H&H, in which we verify that a square wave applied to an RC circuit generates an asymmetrically peaked wave over the capacitor:

Image from Electronics pdf.

We found a curve as expected, shown below:

Measuring the square wave (blue) and RC output (yellow).

We measured the time to fall from 100% to 37% (or climb from 0% to 63%), which took about 110 µs.

Next we used the LogoChip to characterize an RC circuit, building a "capacitance meter" wherein the LogoChip could read (and PicoBlocks could interpret) an unknown capacitance.

LogoChip RC circuit diagram (Electronics pdf).

The real-life LogoChip RC circuit.

PicoBlocks code for the LogoChip capacitor identification (Electronics pdf).

 The PicoBlocks code shown above will display (approximately) the amount of time it takes for the unknown capacitor to charge to 63% of its capacity.  From this it is easy to determine the capacity: this Δt (measured in ms) is equal to the resistance of the resistor in series with the capacitor (100 kΩ here) multiplied by the capacitance.  To get the actual capacitance from the displayed value of "Δt":
(Δt) x 10^-6 s = R*C = (100 kΩ)*C = (1 x 10^5 Ω)*C
C = [(Δt) x 10^-3 s]/(1 x 10^5 Ω) = (Δt) x 10^-8 F = (Δt) x 10^-2 µF = C
The "capacitor meter" accurately identified both a 0.1 µF and a 1 µF capacitor by these calculations:

Identifying a capacitor based on the LogoChip/PicoBlocks output reading of 10.

We attempted to program PicoBlocks to display the capacitor's value in µF, as shown above, but ran into difficulties as the programming language (designed for elementary-school kids) could not process fractional values.  We presumed dividing by 100 was something that could be reasonably accomplished in one's head and decided to leave it at that.

In further studies of the capacitance meter, we determined the effective capacitance of identical capacitors in parallel and in series.  Recalling that capacitors in parallel add and capacitors in series add inversely, we were unsurprised to find that two parallel capacitors had double the effective capacitance of one, while two capacitors in series had half the capacitance. (PicoBlocks read 10 ms for one 0.1 µF capacitor, 20 ms for two caps in parallel, 5 ms for series.)

Finally we investigated the integrator circuit in Lab 2-3:

The integrating circuit: RC.

In this circuit, the output voltage (Vout) is proportional to the charge on the capacitor, which in turn is proportional to the integral of the input voltage (Vin).  With a square wave input, we expected the output voltage to appear as a triangle wave in the same phase, increasing when Vin was positive and decreasing when Vin was negative:

Our (correct) prediction for the integrator circuit output voltage (left) compared to the input voltage (right).

Our experimental integrator, at high frequency (100 kHz) in good agreement with the prediction.

A triangle wave input (purple) yields a cycloid output signal (yellow).

We finished a little early and decided to explore the differentiator circuit of Lab 2-2:

The differentiating circuit: CR.

As the name implies, at high frequencies this circuit yields an output voltage signal approximately equal to the differential of the input signal.

As expected, a sine wave input gave a cosine output.

The differentiator also worked as expected for a high-frequency sine wave:

The input square wave (purple) was also correctly differentiated (yellow).

 However, upon increasing the timescale, we noticed a startling aberration in the output signal--it seemed to oscillate very quickly immediately after each peak.

The puzzling wiggle.

Confused, we consulted several professors.  After a good hour of puzzling (thinking about high-pass filters and reflections within the wires), we finally determined that the BNC breakout cables we had been using as oscilloscope leads were not impedence-matched with the scope.  Switching to scope probes seemed to help the issue.

The output wiggle after one calibration (yellow) suggests that we should replace the BNC breakout input cable (purple) with another scope probe.

At last--a clear signal!

Next lab: Impedence and Hi-Pass/Low-Pass Filters!