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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) ! = !!

This can be further generalized to become

(2) ! = !!

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) ! = !(!"(!)!")

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, ! = −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) !! =!!"#

= !!!"

(impedance of a capacitor)

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and

(5) !! = !"#. (impedance of an inductor)

where

(6) ! = 2!"

The impedances are purely imaginary and depend on frequency, indicating that they are only relevant in the time domain. In contrast, the impedance of a resistor is purely real, where

(7) !! = !. (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) !! = !! + !! + !! +⋯

while in parallel:

(9) !! =!

!!!! !!!! !!!!⋯

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) ! = !!"!!"!#$

(Think about it: where is I measured? Does it matter?)

(11) !!"!#$ = !! + !!

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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

(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:

(13) ! = !!"!!"!#$

= !!"!!! !"

= !!" !!! !"!!! !

!!!!

(14) !!"# = !!! = !" = !!" !!! !"!!! !

!!!!!

(15) !!"#!!"

= !!"#!!"

∗ !!"#!!"

= !

!!!! !!!!

Note: the * indicates the complex conjugate.

(16) !!"# =!!"#$

!! !!"#$ ! !!"

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) !!"# =!

!! !!"#$ ! !!"

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

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(18) !!"#!!"

= !!≈ .707

The cutoff frequency in either case is therefore

(19) !c =!

!!"#

If we were to construct a log-log scale plot for this ratio as a function of frequency (using ! = 4700 Ω and ! = 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.

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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 create a simple high pass filter and plot attenuation (the ratio of Vin to Vout) vs. frequency over a broad range. Begin by constructing the circuit drawn in figure 2. Measure and record the capacitance and resistance of your components, and determine the expected cutoff frequency (fc). You will use this calculated cutoff frequency in your measurements. The values should be about 33 nF and 4.7 kΩ, but you should always measure component values for yourself at the start of an experiment. Typical capacitors have a +20% tolerance, standard resistors around +5%. Make sure you have kept the ground tabs on your banana cables consistent, and connected the tabs on between your input and resistor to ensure a common ground. The orientation of the capacitor doesn’t matter. 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). Use a banana cable connector with a BNC attachment to split the ground and circuit input (ground goes to the ground tab on the connector). Make sure all grounded components are connected to the ground tab of the banana connector.

When hooking the circuit up to your AC source, the signal generator, use a BNC splitter to split the input. Keep one cable for your circuit input and attach the other to channel 1 of your scope. 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. Making sure channel 1 is onscreen on your oscilloscope, start up the function generator and set the input to 4 V sine waves. Measure your input voltage peak to peak, making sure to trigger by adjusting the trigger level knob until the signal stops moving horizontally on your screen. 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.

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Figure 3: Wiring Setup (Figure 2 gives the circuit diagram)

Plug the other input into your circuit and your circuit Vout into channel 2. Any time you adjust the input frequency you will need to check and adjust your input voltage back to 4 V peak to peak. 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 your signal 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.). 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.

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Part 3. Signal Amp

Prelab Questions

1) What is the function in Mathematica used to create a log-log scale plot of discrete data points? (1 point)

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