from IPython.core.display import HTML
def set_width(width):
display(HTML(f"""<style>
.container {{ width:{width}% !important;
min-width:800px !important; margin: 0 auto}}
.jp-Cell {{ width:{width}% !important;
min-width:800px !important; margin: 0 auto}} </style>"""))
# Set container width to X% of the fullscreen
set_width(50)
4C: RCCR circuit with PicoPi#
Learning goal: Understand the properties of an RCCR and RLC circuit.
Structure of an experiment:
Anticipate + Simulate (55 min): per person. This is homework and should be finished before you start your 4 hours practicum session
Implement + Investigate (20+40 min): with your partner (group of 2)
Compare + Conclude (15 min): with a group of 4 (per table)
BACKGROUND:#
⏳ Estimated time: 15 min
In this assignment you will predict and measure the behavior of an RCCR circuit, simulate an RLC circuit, and will compare whether they quantitatively behave the same. These are combined filters, and therefore instead of looking at a single high/low pass filter, you will now have to look at the combined behavior.
When combining two low pass filters, the lowest -3dB point will be having most influence on the overall characteristics of the filter. Similar, when combining to high pass filters, the higherst -3dB point will have have most influence.
The most interesting is when combining a low with a high pass filter. Depending on whether the lows pass -3dB is lower, equal or higher than the high pass -3dB, you can get different bandwidths and different max amplitudes. Visually you could see this as overlaying the two filters, and the overall filter is the area between the two filter graphs.
You can calculate characteristics of a bandpass filter, in a similar way as for a single pass filter. Just derive \(\frac{V_{out}}{V_{in}}\) and look at extreme cases for \(\omega\). When measuring the characteristics in the lab, for a single pass filter with sine input you can find the -3dB frequency as the frequency at which \(V_{out}= \frac{V_{in}}{\sqrt{2}}\). For a bandpass filter you will have to locate the resonance frequency (with highest amplitude), which is flanked by two -3dB points; the difference between the two -3dB frequencies is the bandwidth.
Watch the video below on the derivation of the properties of an RLC filter:
ANTICIPATE: RC + RC behaviour#
⏳ Estimated time: 15 min
Below you see a circuit which combines two RC filters, one with 22 k\(\Omega\), the other with 3.3 k\(\Omega\).
You can see this circuit as two RC filters in series.
Derive their filter type (low/high/bandpass).
Calculate the frequency of -3dB point(s).
Predict what the effect of these two parts in combination will be on the output signal.
###TO DO="your answer to the above 3 questions"
in order to get a quick start on the simulation, feel free to watch the precap video of 2022
SIMULATE 1: RLC circuit behaviour#
⏳ Estimated time: 15 min
Build the below circuit in LTSpice, with R=3kΩ , L=5mH , C=7nF. Evaluate the effect of this filter.
Exercise 1: Find the bandwidth of the circuit.
What is the value of (max) Vout you look for at -3dB?
Do you expect the -3dB points to be symmetric around the resonance frequency?
Change the input frequency to find the -3 dB points (accuracy 100 \(\Omega\)) by trial and error.
Also have a look at the phase at these -3 dB points!
ℹ️ Hint
When you have found the -3dB points, try to plot both Vout at one of the -3dB points (change the frequency) and Vin in a single graph. Using the time delay between two peaks and the period (which corresponds to 360 degrees), you can determine the phase delay.
Support your findings with a screendump of the simulation
from ipywidgets import FileUpload
from IPython.display import Image
import os
upload=FileUpload()
upload
file_name="4C_1_bandwidthRLC.jpg"
if upload.value!={}:
with open(file_name,"wb") as f:
f.write(upload.data[-1])
Image(filename=file_name, width="50%")
### TO DO="your bandwidth, discuss symmetry of the band, and phase of the signal"
Optional challenge#
Try to retrieve the BW from Vout (in time) with \(R_{ser}\)=100 \(\Omega\). Have a look at the Vout in time for this \(R_{ser}\). The amplitude of Vout should decrease with \(e^\frac{-i t BW}{2}\).
Derive the BW from Vout for \(R_{ser}\)=100 \(\Omega\)?
why was this approach impossible for \(R_{ser}\)=3 k\(\Omega\).?
### TO DO="your answer (optional)"
SIMULATE 2: RCCR#
⏳ Estimated time: 10 min
Use the file “4_RC_CR_RCCR_Simulate” that you can find in the folder for this week. It will contain an RCCR circuit as shown here:
Run this simulation as you did in 4B to find the -3dB points.
Do they match the predicted values?
How do you think it will affecr the amplitude of Vout?
If you implement this circuit on the Alpaca do you expect to find similar values?
### TO DO="your answers on the above 3 questions"
in order to get a visual introduction to the measurements with the picopi, feel free to watch the following precap movie from 2022
IMPLEMENT & INVESTIGATE 1: Analog Input & Output#
⏳ Estimated time: 20 min
In assignment 4B you already used the analog AinX - ext pins on an area of the Cria called the Multifunction connector.
DAQ#
The code in the cell below allows you to output a continuous, arbitrary voltage between 0 and 3.0V from the ALPACA using the Digital-to-Analog Converter (DAC). Interfacing with the DAC is done using the functiongenerator module, and made to imitate the function generator in the Student Class Room as much as possible. From the functiongenerator module you import FuncGen, which you can use in parallel to other operations using the with environment. See the example below:
from functiongenerator import FuncGen, Sine # type: ignore
print("Starting function generator")
with FuncGen(Sine(Vpp=2, offset=1, freq=100)): # Use the function generator
print("Doing something else in the mean time") # Do something else in parallel
time.sleep(10) # Wait 10 s. After this the function generator turns of automatically
Similar to the limitation of the digital channel, the analog input channel is also restricted to 0-3V. So be careful with connecting signals. In this assignment you will output a sine from the Analog output channel and read it in into Analog in, plus connect it to the Voltmeter of the Cria.
📝 Todo:
In the simple example below, the function generator is turned on while the PicoPi waits for whatever is inside the with to finish (acts as a while loop). Because the stuff on the inside is a delay, the end result is that the function generator is on for 5 seconds
Connect DAC-A output (on ALPACA) to the voltmeter (on the multifunction pin)
Run the code
%serialconnect to --port="COM4"
from functiongenerator import FuncGen, DC # type: ignore
import time
with FuncGen(DC(V=2)):
print("Procedure starts - Duration: 5 seconds\n")
print("Have a look at the Voltmeter LEDs")
time.sleep(5)
print("\nProcedure ended")
📝 Todo:
Now we change to an AC signal, specifically to a Sine with a certain frequency, \(V_{PP}\), and offset.
For example the following function generates a sine with amplitude 1.5V and DC value 1.5V.
with FuncGen(Sine(Vpp=3, offset=1.5, freq=10)):
Connect DAC-A to Ain0 (top-left pin Multifunction connector).
Run the code
ℹ️ Hint
You want to use the breadboard to connect the output of DAC A to both the voltmeter and Ain0. Or run the code once with the Voltmeter connected, and the second time (after checking it is save) Ain0 connected.
from functiongenerator import FuncGen, Sine # type: ignore
from machine import ADC # type: ignore
import time
import numpy as np
import matplotlib.pyplot as plt
# Initialize the output pin: Ain0 has index 26, Ain1=27, Ain2=28.
a0 = ADC(26)
# Set the frequency to 1Hz
frequency = 1
# Initialize an array for the wave that will be plotted
waveform = np.zeros(300)
time_steps = np.zeros(300)
# Set the sampling rate to 5ms (0.005s)
sampling_rate = 0.005
with FuncGen(Sine(Vpp=3, offset=1.5, freq=frequency)):
start_time = time.ticks_ms()
for ii in range(300):
integer = a0.read_u16()
current_time = time.ticks_ms()
elapsed_time = current_time - start_time # Calculate the difference from the start time
time_steps[ii] = elapsed_time # Append the elapsed time to the time_steps list
voltage = (integer / 65535) * 3.0
waveform[ii] = voltage # Stores each measurement of the voltage
time.sleep(sampling_rate) # Time step
#time_steps = np.arange(len(waveform), dtype=np.float)
print(f"Waveform:\n{waveform}\n")
print(f"Time Steps:\n{time_steps}\n")
print(f"Waveform Shape: {waveform.shape}")
print(f"Time Steps Shape: {time_steps.shape}")
plt.plot(time_steps, waveform, label='Frequency: {:.2f} Hz'.format(frequency)) #This plots for last frequency
plt.xlabel('Time [ms]')
plt.ylabel('Voltage [V]')
plt.legend()
# Initialize an array for the wave that will be plotted
waveform = np.zeros((2, 300)) # Use a 2D array to store waveforms for each frequency
# We use 2 frequencies in the range 1-10Hz
freq_range = np.linspace(1, 10, 2)
time_steps = np.zeros((2, 300)) # Use a 2D array to store time steps for each frequency
# Try to understand the for loop over the frequencies
for idx, frequency in enumerate(freq_range): # Use enumerate to get the index
with FuncGen(Sine(Vpp=3, offset=1.5, freq=frequency)):
start_time = time.ticks_ms()
for ii in range(300):
integer = a0.read_u16()
current_time = time.ticks_ms()
elapsed_time = current_time - start_time # Calculate the difference from the start time
time_steps[idx, ii] = elapsed_time # Store the elapsed time in the 2D array
voltage = (integer / 65535) * 3.0
waveform[idx, ii] = voltage # Store each measurement of the voltage in the 2D array
time.sleep(0.005) # Set the sampling rate
print(f"Waveform:\n{waveform}\n")
print(f"Time Steps:\n{time_steps}\n")
print(f"Waveform Shape: {waveform.shape}")
print(f"Time Steps Shape: {time_steps.shape}")
plt.plot(time_steps[0,:], waveform[0,:], label='Frequency: {:.2f} Hz'.format(freq_range[0])) #This plots for last frequency
plt.plot(time_steps[1,:], waveform[1,:], label='Frequency: {:.2f} Hz'.format(freq_range[1])) #This plots for last frequency
plt.xlabel('Time [ms]')
plt.ylabel('Voltage [V]')
plt.legend()
Make sure you understand the concept of looping over all frequencies. This makes it a lot easier for you to compare multiple frequencies and you do not need to change them manually. In it especially useful when you want to plot the results live.
IMPLEMENT & INVESTIGATE 2: RCCR (or CRRC) circuit#
⏳ Estimated time: 40 min
Disconnect the wires from the analog inputs (Ain0,Ain1,Ain2) on the Cria for safety reasons
Adapt your circuit on your Alpaca so that it consists of both filters:
In the next exercises you will look for both cutoff frequencies of this circuit
Caution: Applying a too high voltage (or negative voltage) to the Cria may destroy the chip on the PicoPi (=delay for you). Thus, to ensure that the range of the signal is within safe limits, it is good practice to always use the Voltmeter before connecting any signal to the Cria and PicoPi.
📝 Todo:
Build the circuit as shown in the image below
Connect the circuits Vout to the voltmeter
Connect both USB cables to the Alpaca, and turn the switches for the +12 and -12 voltage supply to the ON position
Make sure you have no jumpers on AMPLIFIER DIRECT TO NANO, remove the one indicated in the blue circle in photo on the right:
Run the code below, which generates a sine wave, and check the LEDs of the voltage meter
ℹ️ Hints for ALPACA breadboard connections
%serialconnect to --port="COM4"
#ADD COM PORT ABOVE, e.g. --port="COM3"
If the measurement and setup was correct, you should have seen that a negative and a positive voltage is generated at the ouput of your circuit. This negative voltage can damage your pico.
Therefore we need to use AMP 0 or 1 to measure our signal. This amplifier can be used as a protective circuit, which forces your signal to be between 0 and 5V, which should not damage the picopi. Make sure to use the amplifier that is present on the ALPACA, to measure the signal when you know it’s going to have voltages exceding the PicoPi’s limits. This way your PicoPi is always protected.
This will be implemented below:
📝 Todo:
Build the circuit as shown in the image below
Connect the output of the DAC, e.g. \(V_{in}\), to Ain0 on the multifunction connector.
Connect the output of your circuit, e.g. \(V_{out}\), to the SIGNAL+ pin at AMP1 (on the ALPACA)
Connect the jumpers as indicated using magenta in the image below. You do need one jumper for AMPLIFIER DIRECT TO NANO (I suggest you use a wire as removing it will be easier in the future).
Similar as before, a sine wave will be used. The code used to generate this sine wave and to perform the measurement is provided below. The frequency of the signal can be set in the code.
from machine import ADC, Pin # type: ignore
import time
import matplotlib.pyplot as plt
import numpy as np
from functiongenerator import FuncGen, Sine # type: ignore
MEASUREMENTS = 300 # number of measurements
v_in = np.zeros(MEASUREMENTS, dtype=np.uint16) # Store ADC output as integer, convert to voltage later
v_out = np.zeros(MEASUREMENTS, dtype=np.uint16)
tt= np.zeros(MEASUREMENTS)
start = time.ticks_us() #Measure in us
Ain0 = ADC(26) # Pin 26
Ain1 = ADC(27) # Pin 27
with FuncGen(Sine(Vpp=3, offset=1.5, freq=110)):
time.sleep(0.5) # Wait half a second for the Function Generator, before continuing to measure
for ii in range(MEASUREMENTS):
v_in[ii] = Ain0.read_u16()
v_out[ii] = Ain1.read_u16()
tt[ii] = time.ticks_us() #
# tt[ii]=ii
time.sleep_ms(1) # Sleep in milliseconds
tt=(tt-tt[0])*1e-3
#print(v_in)
#print(v_out)
plt.plot(tt,v_in * 3.0 / 65535, color='r', label='DAC A (Ain0) [V]') # Convert integer to voltage
plt.plot(tt,v_out * 3.0 / 65535, color='b', label='Vout (Ain1 via AMP1) [V]')
plt.ylabel('Signal [V]')
plt.xlabel('Time [ms]')
plt.legend()
from functiongenerator import FuncGen, Sine # type: ignore
import time
with FuncGen(Sine(Vpp=3, offset=1.5, freq=10)):
time.sleep(10)
### TO DO ="your answer, what does the voltmeter show"
You should notice the shape of Vout and the amplitude of Vout differ from Vin.
Exercise 1: Explain why there is a difference between Vin and Vout, and if this is what you expected for this circuit.
ℹ️ Hint 1 Do you see the full sine? Is the signal shifted vertically? If you have trouble seeing the sine, try changing frequency to 50Hz
ℹ️ Hint 2 DC components have 0Hz, what do you think happens when you pass them through the bandpass filter?
### TO DO="your answer, explaining the difference between Vin and Vout"
Because of one of the RC filters, the average of the output signal is now zero (make sure to understand why this is - Hint2 above). You can also observe the lack of negative voltage. This is thanks to the amplifier.
Despite only a part of the signal being measured, the 3 dB points can still be determined from the plot.
Exercise 2: Set the frequency back of 110 Hz in the code, and calculate the amplitude (the max of the positive part of the sine wave) of the measured signal.
Report the amplitude. Why do you think it is not half of the Vin amplitude? Think back to ANTICIPATE and SIMULATE 2
print(np.max(v_out*3.0/65535)) ## you already did that measurement, just extract the max voltage
### TO DO="your found peak voltage"
Now you found the maximum voltage value. Next, you will compare the -3dB points to the one you found in ANTICIPATE and SIMULATE 2. Think back what -3dB means in terms of voltage drop.
Exercise 3: Calculate the Voltage you should see at -3dB points based on vmax you found
### TO DO="Which voltage you should see at -3dB - V(-3dB)"
Exercise 4: Look for the lower -3 dB point of the signal by first setting the predicted frequency (ANTICIPATE) and then varying it to find the V(-3dB) voltage
You can reuse/rerun the above code, the three cells which do the acquisition, fetch the data and extract the max value
Does the frequency match the predicted value?
ℹ️ Hint Did your SIMULATE 2 match ANTICIPATE? Think about why was that and apply the same logic to Alpaca
### TO DO="your answer for the lower -3dB point"
Exercise 5: Increase the frequency to look for the upper 3 dB point, up to a max of 500 Hz (physical limitation of Alpaca). You might want to change the parameter num_samples to 100 to better visualize the result.
At what frequency is the upper 3 dB point expected - look ANTICIPATE?
Set that frequency and measure the peak voltage at this expected -3dB frequency
Compare the measured cut off frequency with the theoretical value, does this result surprise you? (it should)
find the V(-3dB) voltage by varying the frequency and report where it is
### TO DO="your higher -3 dB point"
Compared to the equipment in the studio classroom, you see that measuring with the Alpaca, has its limitations. In this case on of these limitations is that the frequency of the sine wave is limited to around 500 Hz. Designing a system with cut off frequencies which are further apart is in this case not possible, but would be possible with the equipment located in the studio classroom.
Optional challenge 1: Switch position of the resistors
When we switch only the resistors around we get a circuit, which behaves differently.
Will the output signal of the system be the same, or will it respond differently when we switch around the resistors? (think about Vmax, -3dB points)
Argue what happens for a frequency of 20, 280 and 150 Hz.
### TO DO="answers optional challenge"
Optional challenge 2: In the code that you used above, can you add a for loop over the frequencies to automatically find the -3dB points without having to manually change it?
### TO DO="your code idea (there is a solution at the bottom of the assigment )"
COMPARE & CONCLUDE:#
⏳ Estimated time: 15 min
Wait until all (4) group members finish their observation
Compare your results with your other group members.
If your results agree, and are in line with all predictions, then talk to a TA and get checked off
Otherwise, so if your results do not agree, or your results are not in line with your predictions, then first discuss amongst your group before getting a TA.
to be checked off by a TA:
Why does the output only show half of the sine wave with the RCCR circuit
How well did your predictions agree with the measurements on the -3dB frequencies for the RCCR circuit? Explain the difference.
exit card:
Write a brief abstract on what you learned (conclusion, useful graph),
Which troubleshooting skills do you want to remember for next sessions,
Which code do you copy for use in next sessions,
How do you think this notebook could be improved
### TO DO =" 1. why do you see only half of the sine with the RCCR circuit?"
### TO DO="2. give values and explain the difference between found measured and predicted -3dB frequencies"
### TO DO="3a. abstract"
### TO DO="3b. troubleshooting"
### TO DO="3c. code"
### TO DO="4. what changes would you suggest?"
Optional code: Below is an adapted version of the code you used above and completely optional to do. The concepts used here will be explained in the next octal. As you can see there is a loop over a range of different frequencies (beware, the code takes a long time to run, thus, maybe you have some smart ideas to improve this code):
from machine import ADC, Pin # type: ignore
import time
import matplotlib.pyplot as plt
import numpy as np
from functiongenerator import FuncGen, Square # type: ignore
MEASUREMENTS = 300 # number of measurements
v_in = np.zeros(MEASUREMENTS, dtype=np.uint16) # Store ADC output as integer, convert to voltage later
v_out = np.zeros(MEASUREMENTS, dtype=np.uint16)
tt= np.zeros(MEASUREMENTS)
start = time.ticks_us() #Measure in us
Ain0 = ADC(26) # Pin 26
Ain1 = ADC(27) # Pin 27
FREQ=np.arange(1,500,1)
VOUT=np.zeros(len(FREQ))
for ff in range(len(FREQ)):
with FuncGen(Sine(Vpp=3, offset=1.5, freq=FREQ[ff])):
time.sleep(0.5) # Wait half a second for the Function Generator, before continuing to measure
if ff%10==0: print(FREQ[ff])
for ii in range(MEASUREMENTS):
v_in[ii] = Ain0.read_u16()
v_out[ii] = Ain1.read_u16()
tt[ii] = time.ticks_us() #
# tt[ii]=ii
time.sleep_ms(1) # Sleep in milliseconds
tt=(tt-tt[0])*1e-6
VOUT[ff]=np.max(v_out)
plt.plot(FREQ, VOUT * 3.0 / 65535, color='b', label='Output voltage')
plt.ylabel('Signal [V]')
plt.xlabel('freq[Hz]')
plt.legend()
### TO DO='perhaps you can make a smarter frequency range'
A=(2**np.arange(0,9,0.2))
print(len(A),A[0],A[-1])
A=np.arange(1,500,1)
print(len(A),A[0],A[-1])
If you got stuck during the measurement, at the end of the lab assignment we offer you a movie clip with our recorded efforts in the lab. If you were successfull with measuring, then skip this movie clip