Electric Circuits Laboratory & Computational Physics 2 Last compiled August 8, 2017 1
Contents 1 Introduction 3 1.1 Prelab questions................................ 4 2 Breadboard electronics 5 2.1 Background theory.............................. 5 2.1.1 Conventional current and electron flow............... 5 2.1.2 Power supplies............................ 5 2.1.3 Earth or common ground for circuits................ 5 2.1.4 Resistors............................... 6 2.1.5 Capacitors.............................. 7 2.1.6 Diodes and LEDs........................... 7 2.1.7 Inductors............................... 8 2.1.8 Transistors.............................. 8 2.1.9 Operational amplifiers........................ 9 2.1.10 The breadboard and connecting components in circuits....... 10 2.2 Procedure................................... 12 2.2.1 Electronic components........................ 12 2.2.2 Photoresistor (light-dependent resistor)............... 13 2.2.3 Multivibrator............................. 14 2.2.4 Adding the rubbery ruler to your circuit............... 15 3 Experiment 2 - Resonance and noise suppression 16 3.1 Background theory.............................. 16 3.1.1 Inductors and capacitors in circuits.................. 16 3.1.2 Filtering signals............................ 16 3.1.3 Impedance and electronic resonance................. 17 3.1.4 The quality factor, Q......................... 18 3.1.5 Analogue filters............................ 18 3.2 Procedure................................... 21 3.2.1 The Rubbery Ruler in a circuit.................... 21 3.2.2 Building the active noise filter.................... 22 2
1 Introduction Example noise suppression on a signal. This laboratory exercise is split into two parts: learning the operation of a breadboard and some electronics components, then using that knowledge, to build a noise suppressing filter. If you have previous experience with breadboards and electronics, it s likely the first section of this experiment will be somewhat familiar to you. However, there are currently many postgraduate students (and admin staff...) that wouldn t be confident in completing the experiment unaided. This lab is intended to give you the knowledge and confidence to build and understand circuits. In the second section you will be provided with some basic information about electronic filters and build a noise suppression circuit. Aims By the end of this first section of this experiment, you should understand: the physics of resistors/potentiometers the physics of capacitors the physics of inductors the physics of diodes/leds basics of transistors basics of operational amplifiers how to use a breadboard how to create a circuit to light an LED and store charge, and vary the LED brightness 3
1.1 Prelab questions 1. Draw a diagram of a standard parallel plate capacitor in a charging circuit (i.e. connected to a battery and resistor). Capacitance is the ability of an object to store electrical charge. a) Does the capacitance increase or decrease as the plate are moved apart? b) What physical properties determine the amount of charge a capacitor can store? c) What material is commonly used in capacitors as the dielectric layer? 2. Write down the equations for resistors and capacitors in parallel and in series. 3. What component of an LED determines the wavelength of the emitted photons? 4. When should we use the RMS (root mean square) measurement of voltage, and how does its value differ from a direct DC voltage measurement? 5. Calculate the resonant frequency for the circuit shown in Figure 14, with the following values: L = 100 µh C = 0.01 µf R = 1000 Ω (neglect the inductor resistance at this stage). 6. Calculate the Q value (quality factor) for the above circuit, hence the bandwidth, ω. 7. Why do FM stations drop out at shorter distances from their source than AM? 8. What additional components would be required to tune into a digital radio signal rather than a traditional analogue signal? 9. What sort of a noise filter you need to use to suppress the noise from mains electricity in your speakers? Explain. 4
2 Breadboard electronics 2.1 Background theory A lot of this information is most likely not entirely new to you. The idea of current flow, circuits, Ohm s law and Kirchoff s laws, are introduced to you at the latest in first year. We re going to apply that knowledge today in constructing circuits, while discussing the physics of the components as we progress. Electronic components can be divided into three main groups (electrical engineering definition): 1. Passive components. These components are incapable of controlling the flow of current in the circuit. Examples: resistors, capacitors, inductors. 2. Active components. These components can control the current flow in the circuit. Examples: diodes, transistors, integrated circuits. 3. Electromechanical components. These components perform electrical operations by using moving (mechanical) parts. Examples: switches, relays, fans. 2.1.1 Conventional current and electron flow The direction of current flow in circuits might at first confuse you. On the one hand, circuits seem to be drawn with current flowing from positive to negative (or at least, that s the way we begin tracing the circuit). On the other hand, we know it s electrons that are providing the charge for the components of our circuits. Electrons are repelled from the negative terminal of a power source towards the positive terminal. So which convention should we follow? Ultimately, we should train ourselves to use conventional current, where current flows from the positive to negative terminals. Some components only operate when in one direction, and when drawn, these are depicted with current flowing positive to negative. 2.1.2 Power supplies Power supplies are used to provide power to electric circuits. They can provide both direct current (DC) and altering current (AC). Usually abbreviations AC and DC stand for altering and direct and then modified current or voltage, depending on the source type (see figure 1). The common sources of electrical power are batteries, fuel cells, solar batteries. 2.1.3 Earth or common ground for circuits One important consideration in circuits is ensuring all of the components have a common ground. Voltage is relative and only makes sense if we have another level to compare it to. This is encompassed in the term potential difference. Ground is usually considered the lowest potential in a circuit, however it can occasionally be any arbitrary reference point for measuring voltage elsewhere in the circuit. Most of the circuits we will be using will have a balanced signal symmetric about the ground, i.e.: one terminal will be held +1 V and the 5
Figure 1: Circuit symbol for AC (left) and DC (right) power supplies. The longer vertical line indicates the positive terminal. Left DC is a single cell, right DC is batteries in series. other at -1 V. A receiver will then consider the voltage difference between the two and no ground wire will be required. BE CAREFUL WHEN USING MEASURING DEVICES (E.G. OSCILLOSCOPE) POWERED FROM MAINS. SINCE THEY ARE USUALLY CONNECTED TO NEUTRAL ( EARTH GROUND) THEY ARE INAPPROPRIATE FOR MEASURING VIRTUAL GROUND (E.G. IN SWITCHING POWER SUPPLIES OR HV GENERATORS). USE BATTERY POWERED MEASURING TOOLS. 2.1.4 Resistors Two symbols are commonly used to represent resistors in electric circuit diagrams, see Figure 2. Resistors are characterised by their electrical resistance which acts to reduce the flow of current in a circuit. The unit of resistance is Ohms, Ω. Resistance values are indicated by coloured bands on the device. Figure 2: The two commonly used circuit symbols for resistors. Remember that resistors add differently when connected in either series or parallel. Here are the formulas to calculate the total resistance of resistors connected in series (left) and parallel (right): R Σ = n R i R Σ = i=1 n i=1 1 R i (1) 6
Figure 3: Electric circuit of n resistors connected in series (left) or parallel (right). 2.1.5 Capacitors A capacitor is an electrical component which usually consists of two electrical conductors separated by a dielectric material: an electrical insulator that can be polarised to support an electrostatic field. Two different symbols for capacitors in circuits are shown in Figure 4. Figure 4: Circuit symbol for a fixed capacitor (left) and polarised capacitor (right). When a potential difference is applied across the conducting plates an electrostatic field develops across the dielectric 1. In this way capacitors are able to store electrical charge. A capacitor is characterised by a value known as capacitance, which is the ability of an object to store electrical charge. Capacitance is defined as: C = q V (2) with the electric charge q on a plate divided by the potential difference V between the plates. If we consider the size and distance between the plates, capacitance is proportional to the area A divided by the distance d (C A/d). Its unit is the Farad, where 1 Farad = 1 Coulomb/Volt When a battery is connected in series with a resistor and a capacitor, the current acts to charge the capacitor. As the capacitor becomes charged to the battery voltage the charging current asymptotically approaches zero, with the capacitor charging rate described in terms of a time constant, τ = RC (3) Remember that capacitors act the opposite way to resistors when placed in series or parallel. 2.1.6 Diodes and LEDs A diode is an electrical component that allows current to flow in one direction only. A lightemitting diode (LED) is a diode including a material that converts electrical energy into photons. The light-emitting materials are semiconductor materials, with two electrodes, an anode and a cathode. The circuit symbols for diodes and LEDs are shown in Figure 5. 1 Did you know?... That due to the high dielectric constant (ɛ=80.4), deionised water is a good dielectric material for capacitors! 7
Figure 5: Circuit symbols for a diode (left) and a light emitting diode (right). Most components, (wire, resistors, fixed capacitors, inductors) are bi-directional, meaning they can be connected either way in a circuit and still function. On the other hand, diodes, including LEDs and some other components, only operate in one direction. Figure 6 shows an example (different components may have different designators) in which way current needs to flow to function properly 2, with the directional markers indicated. Figure 6: Two diodes. The one on the right is light-emitting. 2.1.7 Inductors An inductor is an electrical component that resists changes in electrical current flowing through it. They are generally made of a conducting wire that has been wound into a coil. When current flows through the coil a magnetic field is created. If the current changes, Faraday s law of induction states that a voltage is induced, and Lenz s law the gives the direction of the induced voltage as opposite to the current which created it. In this way inductors oppose a change in current. Inductors are characterized by a property called inductance, which is the ratio of the voltage to the change in current per second, expressed in units of henry (H) : H = V/(A s) = V s/a. The circuit symbol for inductors is shown in Figure 7. Figure 7: Circuit symbol for an inductor. 2.1.8 Transistors A transistor is another semiconductor device which can be used to amplify or switch electronic signals. There are several kinds of transistors, we will use a BJT (Bi-polar Junction 2 Note that in some cases, the components will break down if connected incorrectly! 8
Transistor) for this experiment. The difference between these types is the sign of the charge carrier used by the junction. There are two types of semiconductors - n and p. If the main charge carriers in the semiconductor are electrons then this semiconductors is called n type. Similarly, if the main carriers are holes (virtual positively-charged particles), then semiconductor is called p type. Bi-polar transistor uses two semiconductor junctions: p-n or n-p and, therefore, there are two types of BJTs called PNP or NPN. We will use an NPN transistor and will focus our discussion on the specifics of this type. Transistors typically have three pins, the emitter (e), base (b) and collector (c), see Figure 8. Figure 8: Circuit symbol for transistor (left) and schematic diagram of an NPN transistor (right). A transistor has three pins: the base, collector and emitter. Understanding the physics of a transistor requires some knowledge of semiconductor physics, which you may not have covered in your studies to date. If you are familiar with the Hall effect you may already be aware of the two different types of charge carriers, holes and electrons, as well as p-type and n-type semiconductors. If not, consider this a peek into semiconductor physics... An NPN transistor consists of p-doped semiconductor base between two n-doped layers that act as the emitter and collector, as in Figure 8. The transistor is consider to be on when current flows between the collector and the emitter. This current is carried in the most part by electrons moving from emitter to collector. To turn a transistor on there must be a positive potential difference measured between the emitter to the base, allowing electrons to pass. 2.1.9 Operational amplifiers Operational amplifiers (OPAMP) are high-gain voltage (or current) amplifiers with differential inputs and single-ended (sometimes differential) output. OPAMPs are amplifying the potential difference between their inputs 3. These amplifiers have extremely high input and low output impedances. This property permits the utilisation of OPAMPs in small-signal circuits. The circuit symbol for operational amplifiers is shown in Figure 9. Typically, OPAMPs have an amplification coefficient called open-loop gain G OL which is in order of 10 4-10 7. This coefficient is usually dependent on many parameters and can vary from sample to sample. Even a small difference between the inputs will result in OPAMP saturation (when output voltage is equal to the supply one). Therefore, open-loop circuits are impractical for analogue signal amplifiers (although they can be used as comparators) so negative feedback (closed-loop) is used to strictly determine the gain. 3 Operational amplifiers were firstly used in analogue computers to perform various mathematical operations such as addition, subtraction, integration, differentiation, exp/log operator and many more. 9
Figure 9: Circuit symbol for an operational amplifier. Power pins are usually omitted. Figure 10: Typical negative feedback applications of OPAMPs: (a) Non-inverting and (b) Inverting amplifiers respectively. ( V OUT = V IN 1 + R ) 2 R 1 V OUT = V IN R 2 R 1 (4) Figure 10 shows typical amplifier circuits utilising OPAMPs. Both circuits utilise negative feedback to set amplification coefficient. Figure 10a. shows a non-inverting amplifier, which means that the output voltage changes in the same direction as the input one. Similarly, Figure 10b. shows an inverting amplifier where input and output voltages change in opposite directions. Since amplification of the circuits depends on values of feedback components, high-precision resistors must be used in order to increase accuracy. 2.1.10 The breadboard and connecting components in circuits We ve now discussed most of the electrical components we ll be using. But how will we connect them in a circuit? One option would be to solder the different devices to wires, and de-solder them if we want to swap any out. Alternatively, we can use a fantastic device called a breadboard which is commonly used for prototyping. A breadboard is essentially a solder-free circuit building board that allows for easy installation, removal and re-arranging of electrical components. Figure 11 shows a section of a breadboard, showing the electrically connected rows and columns. We ll call the vertical lines columns and the horizontal lines rows just for clarity. On the left and right of the breadboard are two columns which are connected across the entire 10-square column. These are typically reserved as power lines as it offers an easy way to ensure common ground. For example, if the positive terminal of a battery is connected to the red column, and the negative terminal connected to the black column, we can then use 10
Figure 11: (a) A diagram of a section of a breadboard. (b) Breadboard clip that sits under plastic holes where wires are inserted. You can see the bottom connects the five clips - these go along the rows or columns as in (a). jumper wires to power circuits on the numbered central rows. The central rows are electrically connected in groups of five, where under each of these rows is a metal clip where the exposed wires of components are inserted. They are not connected vertically, which means components are isolated from each other and only become electrically connected when jumper wires cross the rows. It is very important to note that the two wires from the components should always be in different rows, otherwise they will simply short-circuit. 11
2.2 Procedure 2.2.1 Electronic components You should have a variety of the following items on your desk: Jumper wires (various coloured wires in a plastic tray) Resistors Diodes Light-emitting diodes (LEDs) Capacitors Inductors Transistors Operational amplifiers (OPAMPs) 9V battery All of the components you have on your desk have exposed wires on their ends, allowing them to be easily inserted into the breadboard. Lighting up an LED This is just to get you in to the swing of things. Connect the following components to the breadboard to produce light from one of the LEDs: an LED of your choosing a resistor appropriate for the voltage necessary across the LED a power source. Either two 1.5 V AA batteries, or a bigger 9V battery Remember to orient the LED such that the anode and cathode are in the correct position given the direction of current flow from the power source. The LED you choose will have a certain voltage at which it performs best. It is of course necessary to introduce a load in the circuit along with the LED, as otherwise it would burn too brightly (and die shortly). 1. Using Ohm s law and Kirchoff s voltage law, calculate the resistance needed to produce the correct voltage for your LED given your input voltage. 2. Look up the resistor colour chart and choose an appropriate resistor. 3. Compare your reading of the chart with actual value and its tolerance using an ohmmeter. 4. Once your circuit is working correctly, change the resistance of the load in the circuit, but don t let the LED current exceed 30 ma or it may burn out. Question 1 How brightly does the LED glow? What happens across the LED when you 12
increase or decrease the resistance? Question 2 Does adding unnecessary lengths of wire between the power source and the LED decrease its brightness? That is, is the resistance in the wires noticeable? Question 3 Are all LEDs created equal? Choose a different colour LED and put it in the circuit. Comment on the brightness. 2.2.2 Photoresistor (light-dependent resistor) On your desk should be a photoresistor. Using an ohmmeter, measure the value of the resistor in four different light conditions: with your thumb covering the sensor, with ambient room light, with a strong light source shining on the sensor from roughly 5 cm away (e.g.: the flashlight on your phone), with the strong light source 1 cm from the surface of the resistor. Question 4 How did the resistance change under different light conditions? Draw a diagram to explain this behaviour what you think is happening inside the resistor. Question 5 By what proportion did the resistance change between the two strong light sources? Is this roughly what you expected? You may notice the variable resistor has much higher resistance values than the one you calculated for your earlier circuit. 1. Calculate whether you still expect light to be emitted from the LED in each of your light conditions. 2. Now place the variable resistor in your circuit in place of the static resistor, change the light conditions, and observe the behaviour. Question 6 What do you observe? Compare this with what you noticed in the last section when changing the resistance load in the circuit. Question 7 How could you make the LED brightness act in the opposite way relative to the light hitting the resistor? Question 8 Can you think of two applications of a light-dependent resistor? 13
2.2.3 Multivibrator In this section you build a more complicated circuit using LEDs, transistors, capacitors and resistors to create a multivibrator circuit. You may also use a switch, or simply have a loose wire you plug into the breadboard to complete the circuit. The circuit we are going to build is called an astable multivibrator. It is an electronic circuit which have two unstable states and continually switches between them. It is used to build relaxation oscillators. A period between the states switch is determined by values of components. Question 9 List two examples of equipment that would need an astable multivibrator. What purpose do the multivibrators serve? The circuit diagram is shown in Figure 12. Follow this diagram and construct the circuit on the breadboard yourself. You will need to use jumper wires to connect components that cross the insulating channel, and when connecting the collector link to the LED. Figure 12: Circuit diagram for constructing the astable multivibrator. Once you have built your circuit and it is operating correctly, try to explain the importance or the function of each component within the circuit. It may help to break the circuit down into simpler circuits that you already understand, such as the simple LED circuit and a capacitor charging/discharging circuit. It might some obvious to you, but in the next section, an open circuit is one that is not connected, and a closed circuit is complete and has current flowing. (Some people refer to lights as open when they re on and closed when turned off, so it s important to be clear in your definitions.) Question 10 What happens to the circuit when the switch is closed? Try to explain why does it flip to one of the states? Question 11 What happens to the current flow through the circuit when it flips in one of the unstable states? Why does one of the LEDs turn off? Why does it come back on after a period of time? 14
Question 12 Why do we have four resistors in this circuit? Explain the purpose of these resistors and the need for their values. Question 13 PREDICT how you think the period will change with capacitance. What will happen if you change only one of capacitors? Question 14 Swap in capacitors with greater or less capacitance, what happens? How does this compare with your answer to the previous question? Question 15 Using a stopwatch and whatever capacitors are available to you, plot a graph of capacitance versus LED illumination time. What shape does this graph take? Is the relationship linear? Add another capacitor in series with the first. Note your observations. Does the LED stay lit for a longer or shorter time for the same amount of charging time? Remove the second capacitor and re-arrange the circuit so that the second capacitor is in parallel with the first. Question 16 How did you modify your circuit so the capacitors were in parallel? Draw a circuit diagram and label on the diagram what changes you made. What did you observe? Hypothesize as to the advantages and disadvantages of using two capacitors in parallel, rather than a single capacitor. 2.2.4 Adding the rubbery ruler to your circuit Take measurements of the capacitance of the Ruler across its range directly using a multimeter. Question 17 Produce a plot to show the relationship between ruler extension and capacitance. What is the relationship? Is the change in capacitance between points very large? Question 18 How does the rubbery ruler capacitance compare to the other capacitors you ve been using? From the previous sections, you can probably imagine what will happen if we tried to use the rubbery ruler as our sole capacitor. However, what would happen if we used the rubbery ruler in parallel with another capacitor? Question 19 Write down the formula for capacitance in series, and substitute in your known capacitance values. What might you expect the total capacitance to be at the shortest and longest extensions of the rubbery ruler? 15
Can we successfully vary the time the LED is illuminated by placing the rubbery ruler in series with another capacitor? What is the total possible capacitance of the rubbery ruler and other capacitor in series/parallel? 4 3 Experiment 2 - Resonance and noise suppression 3.1 Background theory This theory will be somewhat more complicated than the previous section. It s designed to teach you particularly about electronics when in alternating current (AC) circuits, rather than DC circuits. We use the term reactive to refer to components that respond differently to AC currents than DC currents. 3.1.1 Inductors and capacitors in circuits In AC circuits, across capacitors and inductors, the voltage is out of phase with the current going through it. This can mean that circuits employing capacitors can seem to have a lower total voltage than their input voltage. If we change the frequency of the input voltage, we may find that the input and total voltage are closer, but still do not add up like in a circuits made only of resistors. We therefore need to be careful when measuring signals in circuits containing combinations of resistors, capacitors and inductors. 3.1.2 Filtering signals When dealing with DC signals, a pair of resistors in series forms a voltage divider. The output voltage then depends on the relative values of the two resistors as: R 2 V OUT = V in (5) R 1 + R 2 This is an ideal circuit, in which no output current is drawn, and holds for both AC and DC circuits. However, when reactive components (inductors or capacitors) are used instead of resistors, we need to start referring to the impedance (symbol Z) of the components, rather than the resistance. Impedance, unlike resistance, will vary with the applied voltage frequency. When working with reactive circuits it is common to define the frequency in terms of ω (omega), where: ω = 2πf (6) with f the applied AC voltage frequency. The impedance, Z C, of a capacitor can be defined as: Z C = 1 jωc (7) 4 The behavior of electric circuits can be simulated using so-called PSpice circuit simulator. You can try to simulate the circuits presented here using a freeware online PSpice simulator such as www.partsim.com/simulator. 16
Figure 13: A simple DC voltage divider. where C is the capacitance, in farads (F), of the capacitor. We use the term j to represent the imaginary component of the current through the capacitor to distinguish it from typical current. We should note that the voltage across a capacitor lags behind the current through the capacitor by a phase of π/2. The impedance, ZL, of an inductor is given by: Z L = jωl (8) where L is the inductance, in henrys (H), of the inductor. We can also note that for an inductor the voltage leads the current by π/2. Imagine we now replace one of the resistors shown in Figure 13 with one of these reactive components. As the impedance of these components changes with frequency, we can make a voltage divider where the output voltage depends on the input frequency. We have now created a filter. Analogue filters will be discussed further in more detail. 3.1.3 Impedance and electronic resonance Imagine a circuit as shown in Figure 14. The total impedance of a series RLC circuit is a complex quantity, and can be expressed in the form: Z = R + jωl + 1 jωc Looking at the total impedance, and recalling that the inductor impedance, Z L = jωl and the capacitor impedance, Z C = 1 vary in exactly opposite ways with frequency, there must jωc be a frequency at which the resultant impedance is simply R - a minimum. This frequency is called the circuit s resonant frequency and is given by: (9) which can be written as: jω 0 L + 1 jω 0 C = 0 (10) ω 0 = 1 LC (11) 17
Figure 14: A simple R (resistor) L (inductor) C (capacitor) circuit. At this frequency, the phase angle will be zero, meaning the two imaginary components have cancelled out. A typical plot of the way the magnitude of the current varies with frequency for various resonant circuits is shown in Figure 14, where it reaches a peak for a given frequency and tails off either side. Note that the plot is normalised to a maximum current of 100 ma. 3.1.4 The quality factor, Q A useful measure of the properties of reactive circuits is the quality factor, Q. This is a measure of how sharp a resonance peak is, or how quickly the current rises to its peak and falls again as you scan through frequency. Looking at Figure 15, Q can be obtained by measuring the width of the resonance peak at half of its maximum height, commonly known as the full width at half maximum, or FWHM. The FWHM can represent a measure of the sharpness of any peak. The quality factor is defined as: Q = ω 0 ω (12) where ω 0 is the frequency of the peak maximum, and ω is the total width, or bandwidth of the peak. The Q factor can be otherwise defined as: Q = 1 L (13) R C As we have seen, the larger the Q value, the sharper the peak. From equation 13, we can see that a larger resistance, R, will result in a broader peak. Remember also that for any inductor, L, there will always be a real resistance associated, because inductors are essentially just wound wires, and all wires have some intrinsic resistance. We must add this this resistance, RL, in series with the inductor when calculating a value for Q. For our applications, a high Q factor allows a circuit to better isolate specific frequencies. Broad peaks with low Q factors will have a large ω and thus a large range of frequencies with none well isolated. 3.1.5 Analogue filters Analogue filters are used to remove unwanted frequency components from the signal. Electronic filters can be divided into two main groups: passive and active. Passive filters utilise 18
Figure 15: A plot of frequency versus current, showing changes in the amplitude and the quality factor Q. only a combination of R,L and C and do not require power supply. They can operate at a very broad frequency range, but usually have relatively small Q factor. Active filters utilise amplifying components such as transistors or operational amplifiers together with passive ones. They permit high Q factors, although their operating frequency is limited. Active filters can also amplify the signal, while passive filters always have amplification coefficient lower than 1 5. Figure 16: Different filter types and their transfer functions. 5 Bandwidth of operational amplifier depends on amplification coefficient. The bigger is amplification the smaller is the bandwidth. In datasheets OPAMPs are characterized using a gain-bandwidth product (GBWP) parameter. 19
Figure 17: Circuit diagrams of low-pass (left) and high-pass (right) filters. There are 4 types of filters implementing different transfer functions. If a low frequency signal experience less attenuation than a signal with higher frequency (Figure 16 upperleft), such filter is called low-pass. Similarly, if a high frequency signal experiences lower attenuation (Figure 16 upper-right) then a filter is called high-pass. There are also two types of filters used either to filter out or pass a certain frequency range. These filters are called band-stop (Figure 16 lower-left) and band-pass (Figure 16 lower-right) respectively. The latest two can be represented as a combination of low- and high-pass filters. Figure 17 shows implementations of simple low-pass and high-pass filters. These filters are usually characterised by parameter called cut-off frequency f C. This parameter can be calculated using following expression: f C = 1 2πRC (14) Utilisation of these filters in conjunction allows us to obtain band-pass and band-stop filters. We can now build an active filter using a non-inverting amplifier discussed in previous section. An active filter shown in Figure 18 can have a non-unity gain, which is defined by resistors Ra and Rb. This permits signal amplitude recovering after inevitable losses associated with passive filter cascades (Figure 18 pink region). Figure 18: Circuit diagram of an active low-pass filter. 20
3.2 Procedure NOTE: DO NOT STRETCH THE RULER BEYOND THE FURTHEST POINT ON THE BOARD. HANDLE THE RULER CAREFULLY. ONCE YOU ARE DONE WITH A MEASUREMENT, RELAX THE RULER TO ITS NATURAL EXTENSION. 3.2.1 The Rubbery Ruler in a circuit Figure 19: Circuit for constructing your I vs V plot. 1. Build the circuit depicted in Figure 19, setting the signal generator to 20 khz. 2. Vary the amplitude of the input voltage (use the amplitude control on the signal generator) and construct a table of the V 1 and V 2 values you measure. 3. V 1 is a measure of the voltage across the whole circuit, which we are assuming to be equal to the voltage across the Rubbery Ruler. 4. V 2 is a measure of the voltage across the resistor. You need to convert your V 2 readings to values of current in the circuit. 5. Obtain at least 5 readings of V 1 and V 2 at this fixed length. Convert your V 2 readings into current, and plot the V vs I graph for this length (capacitance) in Excel. Question 20 Is the V vs I graph Ohmic? Explain. Question 21 Explain how to convert our measurement of voltage at V 2 into a measurement of current in the circuit. 6. Repeat this process for two other extensions of the Rubbery Ruler. 7. Fit a linear trend line to your V vs I graph. For reactive circuits, V = IZ. 8. Use your trendline and the relationship below to determine a capacitance for each rubbery ruler extension. 21
Z = v i = 1 ωc (15) Note that these values only apply at this frequency of f = 20 khz. Question 22 Show how the output voltage V 1 compares to the voltage purely across the capacitor. Use your capacitance values, and compare the impedance of the Rubbery Ruler with the resistance of the resistor, then use the equation for the voltage divider. Measuring the ruler impedance with frequency 1. Choose a single extension, and leave the Rubbery Ruler fixed at that position. 2. Set the signal generator so it produces voltage at a frequency of 5 khz. 3. Produce a V vs I graph as in the section above. 4. Similarly produce graphs for 10 khz and 40 khz input frequencies. 5. Calculate the impedance and capacitance of your capacitor for each input frequency. Question 23 What comments can you make about the change in impedance and capacitance with frequency? Is the relationship similar to the relationship between these values and extension? Question 24 How do your impedance values compare with resistance values? That is, is the impedance of a component quite similar to its resistance? Do you think the impedance drops off relatively quickly or is quite broad with frequency? 3.2.2 Building the active noise filter Figure 20: Circuit diagram of the active low-pass filter. 22
1. Calculate the value of resistor R 1 so the low-pass filter has a cut-off frequency around 10kHz (use the closest available resistor value). 2. Build the circuit shown in Figure 20 using the provided equipment (breadboard, wires, etc) and your choice of conductor. NOTE THAT THE CIRCUIT SHOWN IN FIGURE 20 UTILISES BIPOLAR POWER SUPPLY (POSITIVE AND NEGATIVE 15 V). THIS MEANS THAT IT IS ESSENTIAL TO CONNECT THE CIRCUIT GROUND TO THE GROUND OF THE POWER SOURCE. Question 25 What is the purpose of capacitor C 2? 3. Using a signal generator apply a sine signal with a frequency of 1kHz to the input of the circuit and use oscilloscope to measure the signal after capacitor C 2 (you might want to disconnect the speaker for a while). Question 26 Change the resistance of variable resistor R3 and find the point when the amplitude of the output signal is equal to the amplitude of the input one. Can you calculate the value of this resistor analytically? How is it different to the real value? Try to explain why. 1. Change the source frequency from 1 to 7 khz (1kHz step) and write down the amplitude of the output signal. Build a graph. 2. Rotate potentiometer R 3. How does the output amplitude changes? Disconnect the oscilloscope and connect a speaker to the output of the circuit. How does the sound changes? 3. Connect the oscilloscope back to the output of the circuit and disconnect capacitor C 1. Change the frequency of the signal generator. How does the output signal amplitude change? 4. Substitute resistor R 1 with a variable resistor (potentiometer). Change its resistance and watch how the output amplitude changes. Explain why. Question 27 How can you use the Rubbery Ruler to adjust the cut-off frequency of the circuit? Will you be able to hear the difference in sound? Explain. Question 28 What were the difficulties in building and tuning the circuit? Question 29 How can you modify this circuit to obtain a band-pass filter? Draw a circuit diagram with additional components. Question 30 Where would be the best place to put another filter? Explain. 23