Problem 2. Cascaded first-order circuits Waves in the EEG cover a range from about 1 to 40 Hz. We need digital filters to separate the closely spaced 8, 0, α, ẞ, and y wave ranges, but before sampling we should apply an analog band-pass filter that attenuates low-frequency drift and high-frequency noise. Using the following process, design the filter to have cutoff frequencies near 1.5 Hz and 300 Hz and a gain near (but not equal to) one in the pass band. 1. For this problem, let's use a pair of cascaded first-order circuits that includes one operational amplifier (op-amp), two resistors R1 and R2, one inductor, and one capacitor. Your four components can produce two pairs of cutoff frequencies. a. Write these frequencies in terms of L, C, R1 and R2. b. Assume that we have a 3.3 µF (microfarad) capacitor and 0.68 H inductor. Calculate the resistor value needed to produce the desired cutoff frequencies with the inductor and with the capacitor. Choose the pairs that allow you to use resistances between 1 kQ and 1 MQ. c. If we had to choose resistor values from those listed below, how much error would there be between our desired cutoff frequency and the cutoff frequency calculated using the real resistors? Common 5% resistor values 100, 220, 330, 390, 470, 510, 680, and 820 1, 1.1, 2.0, 2.2, 3.3, 3.9, 4.7, 5.1, 6.8, and 8.2 k 10, 11, 20, 22, 33, 39, 47, 51, 68, and 82 k 100, 110, 200, 220, 330, 390, 470, 510, 680, 820 kn, and 1 M 2. Draw the cascaded first-order filter circuit, using component locations that will give you a usable pass band. The op-amp feedback does not need to be wired correctly but the amp does need to be in the correct location. Label the input Vs(w) and each component, indicate which stage is high-pass and which is low-pass, and indicate where you will measure the output voltage. 3. Write an expression for the complex gain of each stage, G1(jw) and G2(jw), and for the complex gain G(jw) of the whole filter, in terms of L, C, R1 and R2. 4. Sketch the following items on semilog axes. Some grids are provided at the end of this handout. a. Magnitude spectrum of total G(jw) as dB vs. for w (log scale). Label the axes with values for for w and dB; indicate the cutoff frequencies and the dB value(s) there; indicate the slope of at the middle and each end of the magnitude plot. b. Phase spectrum of G(jw). Label the axes with values for for w (horizontal) and degrees or radians (vertical). Indicate the phase value(s) at the cutoff frequencies.

Problem 2. Cascaded first-order circuits Waves in the EEG cover a range from about 1 to 40 Hz. We need digital filters to separate the closely spaced 8, 0, α, ẞ, and y wave ranges, but before sampling we should apply an analog band-pass filter that attenuates low-frequency drift and high-frequency noise. Using the following process, design the filter to have cutoff frequencies near 1.5 Hz and 300 Hz and a gain near (but not equal to) one in the pass band. 1. For this problem, let's use a pair of cascaded first-order circuits that includes one operational amplifier (op-amp), two resistors R1 and R2, one inductor, and one capacitor. Your four components can produce two pairs of cutoff frequencies. a. Write these frequencies in terms of L, C, R1 and R2. b. Assume that we have a 3.3 µF (microfarad) capacitor and 0.68 H inductor. Calculate the resistor value needed to produce the desired cutoff frequencies with the inductor and with the capacitor. Choose the pairs that allow you to use resistances between 1 kQ and 1 MQ. c. If we had to choose resistor values from those listed below, how much error would there be between our desired cutoff frequency and the cutoff frequency calculated using the real resistors? Common 5% resistor values 100, 220, 330, 390, 470, 510, 680, and 820 1, 1.1, 2.0, 2.2, 3.3, 3.9, 4.7, 5.1, 6.8, and 8.2 k 10, 11, 20, 22, 33, 39, 47, 51, 68, and 82 k 100, 110, 200, 220, 330, 390, 470, 510, 680, 820 kn, and 1 M 2. Draw the cascaded first-order filter circuit, using component locations that will give you a usable pass band. The op-amp feedback does not need to be wired correctly but the amp does need to be in the correct location. Label the input Vs(w) and each component, indicate which stage is high-pass and which is low-pass, and indicate where you will measure the output voltage. 3. Write an expression for the complex gain of each stage, G1(jw) and G2(jw), and for the complex gain G(jw) of the whole filter, in terms of L, C, R1 and R2. 4. Sketch the following items on semilog axes. Some grids are provided at the end of this handout. a. Magnitude spectrum of total G(jw) as dB vs. for w (log scale). Label the axes with values for for w and dB; indicate the cutoff frequencies and the dB value(s) there; indicate the slope of at the middle and each end of the magnitude plot. b. Phase spectrum of G(jw). Label the axes with values for for w (horizontal) and degrees or radians (vertical). Indicate the phase value(s) at the cutoff frequencies.