**Building a frequency filter**The non-linear charging of capacitors can be used in series with a resistor (RC circuit) to filter high and low frequency signals. The frequency threshold can be selected as it depends on the rate at which the capacitor charges and discharges. This in turn is related to the time constant $\tau$ (tau): the time taken for the charge stored to reach $\frac{1}{e}$ of the original value. As $\tau = RC$, we can tune discrete electrical components to set the threshold for our filter. Relating $\tau$ to frequency gives the following equation:

$f_c = \frac{1}{2 \pi R C}$The order in which the resistor and capacitor are set up and the point at which voltage is recorded determines whether our filter outputs only high or low frequency components (i.e. whether it’s a high or low pass filter). By placing the capacitor before the resistor and taking our output from between these two components, the capacitor will rapidly charge and then not allow any more current to flow to our output, meaning it will filter for high frequency components of the input. Conversely, when the capacitor is placed after the resistor and output, high frequency input will fail to fully charge the capacitor, preventing the high frequency signal to flow into the output. However, low frequency input will lead to the capacitor being fully charged, preventing current flow to the output, and therefore act as a high pass filter. Thus, by adjusting the configuration of the circuit, we can either produce a low or high pass filter. By combining our low and high pass filters in series, we also aimed to produce a band pass filter, where only a range of signals between two frequencies is able to pass. With the right combination of resistors and capacitors, this will allow us to filter out all non-relevant signals above and below the frequency range for (neuro)physiological signals.

**Operational amplifiers**The physiological signals we aim to measure have amplitudes on the order of $10-100\mu V$ (EEG) or 20 mV (EMG). Furthermore, we observed that the filters we built attenuate the amplitude of the signals even more, which means that amplification will be necessary. Therefore, the second aim of this week was to build an

*operational amplifier*(op-amp). The basic principle of electronic amplification is a circuit that makes clever use of the properties of resistors and transistors. In the circuit shown in the figure, a positive voltage is applied at $V_{CC}$, and a negative voltage is applied at $V_{EE}$. Since the resistor at the tail of the circuit is much larger than the two parallel resistors at the top, the current flowing through this circuit is determined by the potential difference and this large resistor (following Ohm's law). If the input voltages ($ V^+_{IN}$ and $V^-_{IN}$) are equal, this means the current flowing through both arms will be equal, and the voltage measured at $V_{OUT}$ will be 0. When either one of the input voltages surpasses the other, more current will flow through that transistor, causing the measured voltage on the contralateral side to rise. The measured signal $V_{OUT}$ thus reflects the amplified difference between the input signals: $V_{OUT}=A(V^+_{IN}-V^-_{IN}) $, where $A$ is the gain of the amplifier. It should be noted that the circuit shown in the left figure is the simplest, original example of a differential amplifier, and it is dependent on the two transistors having exactly the same properties. Therefore, modern amplifiers use more sophisticated circuits. This gain, which is typically on the order of 100,000, can be regulated with a negative feedback loop as shown in the right figure above. When (a proportion of) the output voltage is applied back into $V^-_{IN}$, the amplifier will drive the output voltage to whatever level necessary to keep the differential voltage between the inputs to zero. Thus, when a voltage divider is used to apply a proportion of the output voltage to the $V^-_{IN}$ port of the amplifier, one can use the relative sizes of the resistors to adjust the gain of the amplifier: $A = \frac{R_1}{R_2} $

**Outstanding difficulties**At the end of the week, we tried to apply the information mentioned above to record some muscle activity using an EMG electrode and an op-amp with negative feedback. We displayed the output voltage on an oscilloscope, but observed mainly noise. Next week, we will attempt to build a more sophisticated circuit, using pre-amplification and filtering.

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