E5.1 Lab E5: Filters and Complex Impedance
Note: It is strongly recommended that you complete lab E4: Capacitors and the RC Circuit
before performing this experiment.
Introduction
Ohm’s law, a well known experimental fact, states that the current through a circuit is
related to the overall voltage drop and the resistance of the components by the relation
(1)
!
𝐼=!
This can be further generalized to become
(2)
!
𝐼=!
where Z represents the complex impedance, which has both real and imaginary components.
Impedance is analogous to direct current (DC) resistance, but in addition to the real “resistance”
also includes an imaginary “reactance” term due to the oscillatory effects of, for instance,
charging and discharging a capacitor that come into play when we switch to using an oscillating
alternating current (AC) source to power our circuit. (Recall that the simplest form of a capacitor
is two parallel plates with a gap in-between them, creating time dependence as observed in the
E4: Capacitors lab. In that lab we looked at a single discharging cycle of the capacitor to
determine the time constant of the circuit. Here we apply an oscillating voltage that causes the
capacitor to constantly charge and discharge.) The imaginary nature of the reactance gives phase
to the impedance, indicating that the current is out of phase with the voltage across that
component. Another common passive component, the inductor, creates a time dependence of its
own. An inductor is essentially just a coil of wire; when current flows through, a magnetic field
is created in the coil. If the input voltage changes, the inductor creates a voltage across itself
opposing that change (you may remember Faraday’s Law of Inductance and Lenz’ Law from
Physics 1120). The voltage drop across the inductor is given by
(3)
𝑉 = 𝐿(
!"(!)
!"
)
where L, known as the inductance, is determined by the geometry and number of coils in the
loop, and i(t) is the time dependent current through the inductor. We have switched to the lower
case i in order to indicate time dependence, a common convention. However, from this point
forward in the lab i will denote the imaginary number, 𝑖 = −1. Notice that there must be a time
dependence in the current for any voltage drop to occur – for this reason inductors don’t factor
into DC circuit analysis, where the input voltage is constant in time. The same is true for
capacitors. The complex impedances of the capacitor and inductor are given by
(4)
Chadly, Wynne 2013 !
!!
𝑍! = !"# = !"
(impedance of a capacitor)
E5.2 and
(5)
𝑍! = 𝑖𝜔𝐿.
(impedance of an inductor)
where
(6)
𝜔 = 2𝜋𝑓
The impedances are purely imaginary and depend on frequency, indicating that they are only
relevant when the voltage changes with time. In contrast, the impedance of a resistor is purely
real, where
(7)
𝑍! = 𝑅.
(impedance of a resistor)
This is to be expected, since the resistor creates a voltage drop even in a DC circuit with a time
independent voltage. These complex impedances add in the same way the resistances add; recall
that in series this means that
(8)
𝑍! = 𝑍! + 𝑍! + 𝑍! + ⋯
while in parallel:
(9)
𝑍! =
!
!
!
!
! ! !⋯
!! !! !!
A common method of decreasing the voltage sent to a given part of your circuit is to
create a voltage divider (pictured below).
Figure 1: Voltage Divider
We can analyze the divider by using Ohm’s law:
(10)
𝐼=
!!"
!!"!#$
(Think about it: where is I measured? Does it matter?)
(11)
Chadly, Wynne 2013 𝑍!"!#$ = 𝑍! + 𝑍!
E5.3 Vout, the voltage we’re trying to find, is the voltage dropped across the second impedance, so we
use Ohm’s law once again to find that
𝑉!"# = 𝐼𝑍! = 𝑉!" !
(12)
!!
! !!!
.
When we use resistors for Z1 and Z2, the divider simply reduces the voltage output independent
of the signal frequency, even if we are using an alternating current (AC) input.
Figure 2: CR high-pass filter
When we use an inductor or a capacitor, however, we find (through a little algebra) interesting
frequency dependence. Given the circuit above, we see that:
!!"
𝐼=!
(14)
𝑉!"# = 𝐼𝑍! = 𝐼𝑅 =
(15)
!!"#
!!"
!"!#$
=
=
!!"
!!! !"
(13)
!!" !!! !"
!!! ! ! !
! !
!
!!" !! !"
𝑅
!!! ! ! !
! !
=
!!"# ∗
!!"#
!!"
!!"
=
!
! ! !!
!! ! !
Note: the * indicates the complex conjugate.
(16)
𝑉!"# =
!!"#$
𝑉!"
!! !!"#$ !
This circuit is called a CR high-pass filter. You can see that, for large frequencies f, Equation
(15) approaches unity, whereas for small f it approaches 0. If we switch the positions of the
resistor and capacitor above we get a similar result,
(17)
𝑉!"# =
!
𝑉!"
!! !!"#$ !
which is known as an RC low-pass filter.
For these circuits we define the cutoff frequency to be the point at which the ratio
Chadly, Wynne 2013 E5.4 (18)
!!"#
!!"
=
!
!
≈ .707
The cutoff frequency in either case is therefore
(19)
!
𝑓c = !!"#
If we were to construct a log-log scale plot for this ratio as a function of frequency (using
𝑅 = 4700 Ω and 𝐶 = 33 nF for example) we would get
for the first circuit and
for the second. The cutoff frequency (~1026 Hz) is marked for each case. As we can see the two
circuits block out low and high frequency inputs respectively.
Chadly, Wynne 2013 E5.5 Similar filters can be constructed using resistors and inductors, although with slightly different
time dependence (see prelab questions).
Procedure
Part 1. CR High Pass Filter
In this part we will analyze a simple high pass filter and plot attenuation (the ratio of Vin
to Vout) vs. frequency over a broad range. The circuit is built for you at the lab station and the
components are labeled with their actual values. Determine the cutoff frequency fc for the CR
circuit (figure 2) at your table. You will use this calculated cutoff frequency in your
measurements. The values should be about 33 nF and 4.7 kΩ, but the actual values can vary so
use the values labeled on the circuit box.
To connect the signal generator to your circuit you will use a BNC cable. These cables
carry your signal internally but also have a grounded shell (not connected to the inner wire).
Connect a BNC cable to the “Input” terminal on the circuit box. Connect the other end to the
function generator using a BNC T-splitter. Use another BNC cable to connect the other end of
the T-splitter and Channel 1 on the Oscilloscope.
Take another BNC cable and attach it from the output TTL on the function generator to
the EXT Trig slot on your scope. This will be used to trigger the measurement. Triggering tells
the scope when to take measurements in order to get a consistent signal, as opposed to taking a
different signal with each sweep which shows up as a signal that appears to move across your
screen. Turn on the function generator making sure channel 1 is onscreen on your oscilloscope
and that the oscilloscope is set to trigger off of the channel connected to “Output TTL” on the
function generator.
Chadly, Wynne 2013 E5.6 Figure 3: Wiring Setup (Figure 2 gives the circuit diagram)
Connect another BNC to the circuit “Output” and into channel 2 on the oscilloscope.
Write down the input voltage, with error, as measured on your scope, using the vertical scale
divisions knob to make sure the entire signal is on screen while also filling it as much as
possible. This allows you to take the most accurate measurements possible- when collecting data
for the remainder of the lab always be sure to use this technique. Any time you adjust the input
frequency you will need to check your waveforms so that they fill the screen on the oscilloscope
again. Make voltage measurements at values of .001*(fc), .01*(fc), .1*(fc), .5*(fc), fc, 2*(fc),
10*(fc), 100*(fc), and 1000*(fc). Where fc is the cutoff frequency that you calculated earlier.
Record the exact frequency off of your function generator. For very low frequencies, such as
.001 and .01 fc, you may not be able to trigger your signal. If this is the case, use the Run/Stop
button on your oscilloscope, allowing you to look at just a single measurement. Divide your
measurements by Vin, and create a data table including error for each result. Plot your data and
the theory curve on a log-log scale plot in Mathematica, and comment on any discrepancy.
Part 2. Integrator and Differentiator Circuits
Using the high pass filter you constructed in the previous part of this lab, look at the input
and output signals for various input waveforms (i.e. square wave, triangle wave, sine wave, etc.)
found on the right side of the function generator. Keep the frequency of the signals well below
your calculated fc (in the 0.01 fc - 0.1 fc range). What do you notice about the waveform of the
output signal as compared to the input (Hint: see section title)?
Switch the resistor and capacitor to create a low pass filter, and change your input
frequency to well above the circuit’s fc. What do you notice now? Which of these two circuits
would you call an integrator? Which a differentiator? Explain. State what the integral and
derivative waveforms of each of the three inputs mentioned above are. Do the circuits make a
good approximation of differentiation and integration?
Hint: think carefully about what the integral of a triangle waveform should look like. It is
NOT a sine wave.
Part 3. Signal Processing and the ECG
One of the most important applications of filters is to reduce electronic noise in signals
from measurement devices. When measuring a signal of a certain frequency you are nearly
certain to have your measurement distorted by a multitude of other signals over a range of other
frequencies. If your signal amplitude is low enough your measurement can be drowned out
completely! The goal of signal processing is to reduce this noise as much as possible so that you
only look at the data you are interested in. By attenuating signals far from the cutoff frequency in
one direction while allowing signals on the opposite side to pass through, RC and RL low and
high pass filters used in combination are good candidates for this task.
The electrocardiogram (ECG) is commonly used in hospitals to monitor patients’ heart
beats and detect heart problems as a diagnostic tool. It works by amplifying tiny voltages present
in the skin caused by polarization and depolarization of the heart. The human heart has four
chambers: two atria, where blood enters the heart, and two ventricles, which pump the blood out.
During the heart’s cycle the atria and then the ventricles contract then relax in succession,
pushing blood through the body. These contractions are caused by an electrical signal
proliferated by the sinoatrial node, otherwise known as the pacemaker due to the fact that the
Chadly, Wynne 2013 E5.7 cells spontaneously and rhythmically depolarize of their own accord. This electrical signal
creates a very small net voltage difference across the body due to the asymmetric positioning of
the transduction cells on the heart. This tiny signal can be read off the skin on opposite sides of
the body and amplified to give doctors an idea of how a heart is functioning.
The largest peak in an ECG readout, known as the QRS complex, represents ventricular
contraction. Depending on the heart of the person in question, the conductivity of their skin, and
the positions at which the signals are measured, the voltage difference across the body at this
point will be around 1-3 mV. This presents a challenge, since the largest source of noise in most
of these circuits, which comes from power lines, is large enough to completely obscure the signal
and will be amplified during the amplification phase of the circuits. Luckily, power lines operate
at 60 Hz, whereas the human heart (which beats only around 100 times per minute) operates at a
frequency between 1 and 2 Hz. Almost all of the circuitry that goes into an ECG is aimed at
eliminating this noise by applying various filters to allow visualization of the heart cycle.
In this part of the lab you will compute the actual maximum voltage difference created at
your hands by your heart. ECG circuits first amplify the signal before filtering it, since otherwise
peaks would be too small to resolve. To determine your heart voltage you will thus need to know
the degree to which the ECG you are using amplifies the signal. Make sure the power cord on the
ECG box is plugged in, then connect the OUTPUT jack to channel 1 of your oscilloscope using a
BNC cable. Turn on the circuit, set the mode to CALIBRATE, and switch three to Vin. Using a
DMM (digital multimeter) with a BNC cable attachment, measure the input voltage to the
calibration pulse and it’s error by connecting the DMM to the LEFT/Vin socket. Pressing the
CALIBRATING PULSE button will send a signal into the circuit at this voltage. Remove the
DMM and flip switch three to OPERATE. On the oscilloscope, adjust the time scale to around
250 ms/division and, using the calibrate button, adjust the voltage divisions until the pulses take
up most of the oscilloscope screen. Measure the output calibration voltage on the oscilloscope
with error (the error on any oscilloscope measurement is + one small division- note that making
the large divisions as small as possible minimizes error, which is why it is best to take up as
much of the screen with the pulses as you can without clipping the signal). Use the input and
output calibration voltages to determine the amplification factor of the circuit and its error.
Next attach the handle bar BNC cables to the LEFT/Vin and RIGHT sockets on the box,
and switch the mode to OPERATE. In this mode the box will pick up signals from the
handlebars, amplify and filter the signal, then output what remains. To get an accurate reading,
rest your hands gently on the bars and stay as still and relaxed as you can- muscle contractions
also create transient voltages that will distort your signal. If you don’t see your heartbeat right off
the bat don’t worry- sometimes it can take 30-40 seconds to resolve to a steady pattern. When
you see your heartbeat, pick a typical cycle and measure the peak to peak voltage of the QRS
complex with error.
Using the amplification factor and the measured output voltage for your heartbeat,
calculate the voltage running across your hands whenever your ventricles contract. Neat!
Prelab Questions
1) What is the function in Mathematica used to create a log-log scale plot of discrete data
points? (1 point)
Chadly, Wynne 2013 E5.8 2) When plotting the theory curve on a log-log scale, be careful not to begin your plot range
at 0. Why is this necessary? (1 point)
3) For an EKG monitor, one problem in signal amplification is power line noise which
comes in at around 60 Hz (the human heart beats at around 1 Hz). If you wanted to
eliminate this noise using an RC filter with a 1 µF capacitor, what valued resistor would
you choose? Draw a schematic of the filter. (2 points)
!
4) Derive the ratio !!"# for the RL filter shown below, and determine the cutoff frequency.
!"
Hint: see equations 13-19. Your final answer should be
!!"#
!!"
=
!
.
!! !/!!"# !
(3 points)
Figure 4: RL low-pass filter
5) Show that the units on the right hand side of the ratio above cancel to give a unit-less
quantity. (1 point)
6) Filters have many real world applications. Write down a couple ideas you have for
possible uses. (1 point)
7) By combining a resistor, an inductor, and a capacitor all into one RLC circuit, as shown
below, it is possible to filter out signals above and below a certain frequency. This is
known as a band-pass filter, since only a narrow band of signals is allowed through. The
width of the band is determined by the Q (for quality) factor of the circuit, which we
won’t discuss in depth. Give the values for Z1 and Z2 in the circuit below. (1 point)
Figure 5: RLC band-pass filter
Chadly, Wynne 2013