Lab E5: Filters and Complex Impedance

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1 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) Z = = "# " (impedance of a capacitor)

2 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). We can analyze the divider by using Ohm s law: Figure 1: Voltage Divider (10) I = " "#\$ (Think about it: where is I measured? Does it matter?) (11) Z "#\$ = Z + Z

3 E5.3 V out, 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) V "# = IZ = V ". When we use resistors for Z 1 and Z 2, 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) I = " = " "#\$ = " " " (14) V "# = IZ = IR = " " R (15) "# " = "# " "# " = 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

4 E5.4 (18) "# " =.707 The cutoff frequency in either case is therefore (19) f c = 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.

5 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 V in to V out ) 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 f c 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.

9 E5.9 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 Z 1 and Z 2 in the circuit below. (1 point) Figure 5: RLC band-pass filter

11 E5.11 Remember, all plots you make in a prelab or lab report should include the following: I. Plot your theory or expectation as a line. (In Mathematica, you should define a function and plot it using the command Plot ) Reference the Mathematica tutorial to do this. II. Label your axes, in English (not just symbols) and with units. III. Include a brief caption (namely, a text statement of what the plot shows.) IV. Set the x and y range of the plot to be close to what you expect for your data. For example: in M1 your longest length pendulum should be no more than 130 cm in length. 5. Write a Discussion of Uncertainty. What are the major sources of uncertainty in the experiment and how will you account for them? 6. Turn in a printout of your Mathematica document that includes 2-5 above. This document should be no more than 2 pages long.

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