Electronics Lab - Day 4

Today we learned about different types of useful RC circuits. The first of these was thedifferentiator circuit which basically looks like this:

And produces an output voltage Vout=RC (dVin/dt), the derivative of the input voltage as long as the frequency of the input voltage, w, is much smaller than the time constant of the circuit, T=1/RC. (w<<1/RC).

Lab 2-2

After building an RC circuit that would give an output voltage that is the derivative of the input voltage over time, we were asked to drive this circuit with a square wave. We observed the input and output voltages with an oscilloscope:

The output voltage is consistent with what a differentiator would do in that it is zero over the square wave except for at the points when the voltage changes at which there's a positive or negative spike in the output voltage.

When we drove the circuit with a triangle wave:

The output voltage is a square wave.

And when we drove a sine wave:

The output voltage is a cosine wave. Although there is a phase shift that was explained later on in the lab.

For a DC voltage source, this circuit will present an infinite resistance because the capacitor can be thought of having an infinite frequency with a DC circuit. For an infinite frequency voltage source, this capacitor will present almost no impedance and the impedance of the circuit is just the resistance that the parallel resistor presents.

When we change the frequency so that w<<1/RC is no longer satisfied. (So that the input frequency is no longer much smaller than the time constant of the RC circuit). The output frequencies start to look odd because of the capacitor is no longer differentiating the input circuit but the capacitor is allowed to charge and discharge over time (thereby changing the impedance of the circuit overtime).

Phase Shifts - Differentiator Circuit

We learned that capacitors have a complex impedance which causes a phase shift between the input and output voltages. The complex impedance = -i/wC. As we can see, its is obviously dependent on the capacitance but also on the input frequency. From that and the formula to calculate the output voltage for a voltage divider, we find that the output frequency for this type of circuit is:

If we then plug in the w=1/RC for the input frequency we find that we should expect a 45 degree shift in the output frequency which explains the shift in our previous picture.

Note: At the 3dB point, the signal is attentuated by 1/sqrt(2). This point occurs at the cutoff frequency wo=1/RC