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Basic of Electronics What is an electronic circuit?'' A circuit is a structure that directs and controls electric currents, presumably to perform some useful function. Circuit Implies, that the structure is closed or something like a loop. Is a line or a structure enclosing an area? Charge Charge is measured in units of Coulombs, abbreviated (Charles Augustin Coulomb (1736-1806) CHARGE COMES IN TWO STYLE We call the two styles positive charge, +, and (you guessed it) negative charge, - . Charge also comes in lumps of 1.6 ×10-19C, which are about two ten-million- trillionths of a Coulomb. The discrete nature of charge is not important for this discussion, but it does serve to indicate that a Coulomb is a LOT of charge. Charge is conserved. You cannot create it and you cannot annihilate it. You can, however, neutralize it. Early workers observed experimentally that if they took equal amounts of positive and negative charge and combined them on some object, then that object neither exerted nor responded to electrical forces; effectively it had zero net charge. This experiment suggests that it might be possible to take uncharged, or neutral, material and to separate somehow the latent positive and negative charges. If you have ever rubbed a balloon on wool to make it stick to the wall, you have separated charges using mechanical action. What is an electric current?" Electric current is the flow of electrons in a circuit. Charge is mobile and can flow freely in certain materials, called conductors. Metals and a few other elements and compounds are conductors. Materials that charge cannot flow through are called insulators. Air, glass, most plastics, and rubber are insulators, for example. And then there are some materials called semiconductors Historically seemed to be good conductors sometimes but much less so other times. Silicon and germanium are two such materials. Today, we know that the difference in electrical behavior of different samples of these materials is due to extremely small amounts of impurities of different kinds, which could not be measured earlier. This recognition and the ability to precisely control the "impurities" has led to the massive semiconductor electronics industry and the near-magical devices it produces, including those on your RoboBoard. We will discuss semiconductor devices later; now let us return to conductors and charges. Figure 4.3: Two spheres with opposite charges are connected by a conductor, allowing charge to flow. There is a force between them, the potential for work, and thus a voltage. Now we connect a conductor between them, a metal wire. On the positively charged sphere, positive charges rush along the wire to the other sphere, repelled by the nearby similar charges and attracted to the distant opposite charges. The same thing occurs on the other sphere and negative charge flows out on the wire. Positive and negative charges combine to neutralize each other, and the flow continues until there are no charge differences between any points of the entire connected system. There may be a net residual charge if the amounts of original positive and negative charge were not equal, but that charge will be distributed evenly so all the forces are balanced. If they were not, more charge would flow. The charge flow is driven by voltage or potential differences. After things have quieted down, there is no voltage difference between any two points of the system and no potential for work. All the work has been done by the moving charges heating up the wire. The flow of charge is called electrical current. Current is measured in amperes (a), amps for short (named after another French scientist) An ampere is defined as a flow of one Coulomb of charge in one second past some point. While a Coulomb is a lot of charge to have in one place, an ampere is a common amount of current; about one ampere flows through a 100 watt incandescent light bulb, and a stove burner or a large motor would require ten or more amperes. On the other hand low power digital circuits use only a fraction of an ampere, and so we often use units of 1/1000 of an ampere, a milliamp, abbreviated as ma, and even 1/1000 of a milliamp, or a micro amp, µa . The currents on the RoboBoard are generally in the milliamp range, except for the motors, which can require a full ampere under heavy load. Current has a direction, and we define a positive current from point A to B as the flow of positive charges in the same direction. Negative charges can flow as well; in fact, most current is actually the result of negative charges moving. Negative charges flowing from A to B would be a negative current, but, and here is the tricky part, negative charges flowing from B to A would represent a positive current from A to B . The net effect is the same: positive charges flowing to neutralize negative charge or negative charges flowing to neutralize positive charge; in both cases the voltage is reduced and by the same amount. Voltage First we return to the basic assumption that forces are the result of charges. Specifically, bodies with opposite charges attract, they exert a force on each other pulling them together. The magnitude of the force is proportional to the product of the charge on each mass. This is just like gravity, where we use the term "mass" to represent the quality of bodies that results in the attractive force that pulls them together (see Fig. 4.1). Electrical force, like gravity, also depends inversely on the distance squared between the two bodies; short separation means big forces. Thus it takes an opposing force to keep two charges of opposite sign apart, just like it takes force to keep an apple from falling to earth. It also takes work and the expenditure of energy to pull positive and negative charges apart, just like it takes work to raise a big mass against gravity, or to stretch a spring. This stored or potential energy can be recovered and put to work to do some useful task. A falling mass can raise a bucket of water; a retracting spring can pull a door shut or run a clock. It requires some imagination to devise ways one might hook on to charges of opposite sign to get some useful work done, but it should be possible. The potential that separated opposite charges have for doing work if they are released to fly together is called voltage, measured in units of volts (V). (Sadly, the unit volt is not named for Voltaire, but rather for Volta, an Italian scientist.) The greater the amount of charge and the greater the physical separation, the greater the voltage or stored energy. The greater the voltage, the greater the force that is driving the charges together. Voltage is always measured between two points, in this case, the positive and negative charges. If you want to compare the voltage of several charged bodies, the relative force driving the various charges, it makes sense to keep one point constant for the measurements. Traditionally, that common point is called "ground." Early workers, like Coulomb, also observed that two bodies with charges of the same type, either both positive or both negative, repelled each other (Fig. 4.2). They experience a force pushing Figure 4.1: Opposite charges exert an attractive force on each other, just like two masses attract. External force is required to hold them apart, and work is required to move them farther apart. Electrical force, like gravity, also depends inversely on the distance squared between the two bodies; short separation means big forces. Thus it takes an opposing force to keep two charges of opposite sign apart, just like it takes force to keep an apple from falling to earth. It also takes work and the expenditure of energy to pull positive and negative charges apart, just like it takes work to raise a big mass against gravity, or to stretch a spring. This stored or potential energy can be recovered and put to work to do some useful task. A falling mass can raise a bucket of water; a retracting spring can pull a door shut or run a clock. It requires some imagination to devise ways one might hook on to charges of opposite sign to get some useful work done, but it should be possible. The potential that separated opposite charges have for doing work if they are released to fly together is called voltage, measured in units of volts (V). (Sadly, the unit volt is not named for Voltaire, but rather for Volta, an Italian scientist.) The greater the amount of charge and the greater the physical separation, the greater the voltage or stored energy. The greater the voltage, the greater the force that is driving the charges together. Voltage is always measured between two points, in this case, the positive and negative charges. If you want to compare the voltage of several charged bodies, the relative force driving the various charges, it makes sense to keep one point constant for the measurements. Traditionally, that common point is called "ground." Early workers, like Coulomb, also observed that two bodies with charges of the same type, either both positive or both negative, repelled each other (Fig. 4.2). They experience a force pushing Figure 4.2: Like charges exert a repulsive force on each other. External force is required to hold them together, and work is required to push them closer. Them apart, and an opposing force is necessary to hold them together, like holding a compressed spring. Work can potentially be done by letting the charges fly apart, just like releasing the spring. Our analogy with gravity must end here: no one has observed negative mass, negative gravity or uncharged bodies flying apart unaided. Too bad, it would be a great way to launch a space probe. The voltage between two separated like charges is negative; they have already done their work by running apart, and it will take external energy and work to force them back together. So how do you tell if a particular bunch of charge is positive or negative? You can't in isolation. Even with two charges, you can only tell if they are the same (they repel) or opposite (they attract). The names are relative; someone has to define which one is "positive." Similarly, the voltage between two points A and B , VAB , is relative. If VAB is positive you know the two points are oppositely charged, but you cannot tell if point A has positive charge and point B negative, or visa versa. However, if you make a second measurement between A and another point C , you can at least tell if B and C have the same charge by the relative sign of the two voltages, VAB and VAC to your common point A . You can even determine the voltage between B and C without measuring it: VBC = VAC - VAB . This is the advantage of defining a common point, like A , as ground and making all voltage measurements with respect to it. If one further defines the charge at point A to be negative charge, then a positive VAB means point B is positively charged, by definition. Circuit Elements Ohm's Law Ohm's law describes the relationship between voltage, V , which is trying to force charge to flow, resistance, R , which is resisting that flow, and the actual resulting current I . The relationship is simple and very basic: . Power Current flowing through a poor conductor produces heat by an effect similar to mechanical friction. That heat represents energy that comes from the charge traveling across the voltage difference. Remember that separated charges have the potential to do work and provide energy. The work involved in heating a resistor is not very useful, unless we are making a hotplate; rather it is a byproduct of restricting the current flow. Power is measured in units of watts (W), named after James Watt, the Englishman who invented the steam engine, a device for producing lots of useful power. The power that is released into the resistor as heat can be calculated as P=VI , where I is the current flowing through the resistor and V is the voltage across it. Ohm's law relates these two quantities, so we can also calculate the power as The power produced in a resistor raises its temperature and can change its value or destroy it. Most resistors are air-cooled and they are made with different power handling capacity. The most common values are 1/8, 1/4, 1, and 2 watt resistors, and the bigger the wattage rating, the bigger the resistor physically. Some high power applications use special water cooled resistors. Most of the resistors on the RoboBoard are 1/8 watt. Kirchhoff's Current Law This fundamental law results from the conservation of charge. It applies to a junction or node in a circuit -- a point in the circuit where charge has several possible paths to travel. IB + IC + ID = IA Bringing everything to the left side of the above equation, we get (IB + IC + ID) - IA = 0 Figure 1 Possible node (or junction) in a circuit Then, the sum of all the currents is zero. This can be generalized as follows Note the convention we have chosen here: current flowing into the node are taken to be negative, and currents flowing out of the node are positive. It should not really matter which you choose to be the positive or negative current, as long as you stay consistent. However, it may be a good idea to find out the convention used in your class. Kirchhoff's Voltage Law Kirchhoff's Voltage Law (or Kirchhoff's Loop Rule) is a result of the electrostatic field being conservative. It states that the total voltage around a closed loop must be zero. If this were not the case, then when we travel around a closed loop, the voltages would be indefinite. So In Figure 1 the total voltage around loop 1 should sum to zero, as does the total voltage in loop2. Furthermore, the loop which consists of the outer part of the circuit (the Figure 1 Around a closed loop, the path ABCD) should also sum to zero. total voltage should be zero We can adopt the convention that potential gains (i.e. going from lower to higher potential, such as with an emf source) is taken to be positive. Potential losses (such as across a resistor) will then be negative. However, as long as you are consistent in doing your problems, you should be able to choose whichever convention you like. It is a good idea to adopt the convention used in your class. Here are some examples of complicated circuits which cannot be reduced to a series circuit or a parallel circuit. One cannot find equivalent resistances using the rules from resistors in series or in parallel. Instead, Kirchhoff's Current and Voltage Laws are used to solve these circuits. Figure 1 This is an example of a Wheatstone Bridge circuit, where the component labelled 'G' is a galvanometer. This type of circuit is used to calculate the resistance of an unknown resistor, RX. The other three resistors are variable. Kirchhoff's Laws are not the only method of solving such circuits. Different methods have arisen to solve complicated circuits, such as the Superposition Theorem. Some of these methods are easier to use than others, and their simplicity is dependent on the specific circuit to be solved. The Wheatstone Bridge is a fundamental circuit originally invented to measure resistance. To see how this works look at the following simulated Wheatstone Bridge Experiment. Use the "back" button to return to this point. Figure 2 This circuit can be thought of as a 'T-circuit'. It cannot be reduced to series or parallel combinations of resistors because there is more than one emf source. Kirchhoff's Laws are not the only method of solving such circuits. Different methods have arisen to solve complicated circuits, such as the Superposition Theorem. Some of these methods are easier to use than others, and their simplicity is dependent on the specific circuit to be solved. The Wheatstone Bridge is a fundamental circuit originally invented to measure resistance. To see how this works look at the following simulated Wheatstone Bridge Experiment. Use the "back" button to return to this point. Resistors Resistors they resist the flow of charge; they are poor conductors. The value of a resistor is measured in ohms and represented (by the Greek letter capital omega). There are many different ways to make a resistor. ➢ Some are just a coil of wire made of a material that is a poor conductor. The most common and inexpensive type is made from powdered carbon and a glue-like binder. Such carbon composition resistors usually have a brown cylindrical body with a wire lead on each end, and colored bands that indicate the value of the resistor. ➢ The potentiometer is a variable resistor. When the knob of a potentiometer is turned, a slider moves along the resistance element. Potentiometers generally have three terminals, a common slider terminal, and one that exhibits increasing resistance and one that has decreasing resistance relative to the slider as the shaft is turned in one direction. The resistance between the two stationary contacts is, of course, fixed, and is the value specified for the potentiometer. The photo resistor or photocell is composed of a light sensitive material. When the photocell is exposed to more light, the resistance decreases. This type of resistor makes an excellent light sensor. Type of resistors RESISTORS IN SERIES Resistors can be connected in series; that is, the current flows through them one after another. The circuit in Figure 1 shows three resistors connected in series, and the direction of current is indicated by the arrow. Note ➢ That since there is only one path for the current to travel; the current through each of the resistors is the same. ➢ Also, the voltage drops across the resistors must add up to the total voltage supplied by the battery: ➢ Since V = I R, then ➢ But Ohm's Law must also be satisfied for the complete circuit: Setting equations [3] and [4] equal, we get: We know what the current through each resistor (from equation [1]) is just I. So the currents cancel on both sides, and we arrive at an expression for equivalent resistance for resistors connected in series. In general, the equivalent resistance of resistors connected in series is the sum of their resistances. That is, This can also be written in terms of conductance, since conductance is just the reciprocal of resistance: RESISTORS IN PARALLEL Resistors can be connected such that they branch out from a single point (known as a node), and join up again somewhere else in the circuit. This is known as a parallel connection. Each of the three resistors in Figure 1 is another path for current to travel between points A and B. Note That the node does not have to physically be a single point; as long as the current has several alternate paths to follow, then that part of the circuit is considered to be parallel. Figures 1 and 2 are identical circuits, but with different appearances. 1. At A the potential must be the same for each resistor. Similarly, at B the potential must also be the same for each resistor. So, between points A and B, the potential difference is the same. That is, each of the three resistors in the parallel circuit must have the same voltage. 2. Also, the current splits as it travels from A to B. So, the sum of the currents through the three branches is the same as the current at A and at B (where the currents from the branch reunite). 3. By Ohm's Law, equation [2] is equivalent to: 4. By equation [1], we see that all the voltages are equal. So the V's cancel out, and we are left with 5. This result can be generalized to any number of resistors connected in parallel. 6. Since resistance is the reciprocal of conductance, equation [5] can be expressed in terms of conductance. RESISTORS IN COMBINATIONCIRCUITS Here, we will combine series circuits and parallel circuits. These are known as combination circuits. No new equations will be learned here. We can imagine a branch in a parallel circuit, but which contains two resistors in series. For example, between points A and B in Figure 1. In this situation, we could calculate the equivalent resistance of branch AB using our rules for series circuits. So, Now, we can replace the two resistors with a single, equivalent resistor with no effective change to the circuit. As can be seen in Figure 2, the circuit is now a parallel circuit, with resistors RAB and R3 in parallel. This circuit can be solved using the same rules as any other parallel circuit Another combination circuit can occur with parallel circuits connected in series. Figure 3 shows a typical example of two parallel circuits (AB and CD) connected in series with another resistor, R3. Here, the resistors in the parallel circuit AB can be replaced by an equivalent resistance. Again, we will use the equivalence rule for resistors connected in parallel: Figure 3 Combination Circuit 2 This gives: So, the equivalent resistance between points A and B is RAB. Replacing the parallel circuit between these two points with RAB gives the following circuit. Similarly, we can replace the parallel circuit containing R4 and R5 (between points C and D) with its equivalent resistance, RCD, where Figure 4 Circuit 2, partially simplified. Figure 5 Circuit 2, simplified Similarly, we can replace the parallel circuit containing R4 and R5 (between points C and D) with its equivalent resistance, RCD, where Replacing the parallel circuit between CD with its equivalent resistance yields the circuit in Figure 5 (above). Now, you can see that we have simplified Circuit 2 to one which contains resistors connected in series only. That is, this circuit now contains RAB, R3, and RCD in series. The equivalent resistance for this circuit would be found using: Or Rt = RAB + R3 + RCD Capacitors Capacitors are another element used to control the flow of charge in a circuit. The name derives from their capacity to store charge, rather like a small battery. Capacitors consist of two conducting surfaces separated by an insulator; a wire lead is connected to each surface. You can imagine a capacitor as two large metal plates separated by air, although in reality they usually consist of thin metal foils or films separated by plastic film or another solid insulator, and rolled up in a compact package. Figure 4.6: A simple capacitor connected to a battery through a resistor. As soon as the connection is made charge flows from the battery terminals, along the wire and onto the plates, positive charge on one plate, negative charge on the other. Why? The like-sign charges on each terminal want to get away from each other. In addition to that repulsion, there is an attraction to the opposite-sign charge on the other nearby plate. Initially the current is large, because in a sense the charges can not tell immediately that the wire does not really go anywhere, that there is no complete circuit of wire. The initial current is limited by the resistance of the wires, or perhaps by a real resistor, Eventually, the repulsive force from charge on the plate is strong enough to balance the force from charge on the battery terminal, and all current stops. The time dependence of the current in the circuit of Fig. 4.6 For two values of resistance. time for two different values of resistors. For a large resistor, the whole process is slowed because the current is less, but in the end, the same amount of charge must exist on the capacitor plates in both cases. The magnitude of the charge on each plate is equal. The existence of the separated charges on the plates means there must be a voltage between the plates, and this voltage be equal to the battery voltage when all current stops. After all, since the points are connected by conductors, they should have the same voltage; even if there is a resistor in the circuit, there is no voltage across the resistor if the current is zero, according to Ohm's law. The amount of charge that collects on the plates to produce the voltage is a measure of the value of the capacitor, its capacitance, measured in farads (f). The relationship is C = Q/V , microfarads ( µf ) or picofarads pf where Q is the charge in Coulombs. V is the voltage in volts Combinations of Capacitors • Thus, the formula for total capacitance in a parallel circuit is: CT=C1+C2...+Cn , (The same form of equation for resistors in series, which can be confusing unless you think about the physics of what is happening). • The capacitance of a series connection is lower than any capacitor because for a given voltage across the entire group, there will be less charge on each plate. The total capacitance in a series circuit is CT={1{1C1}+{1C2}...+{1Cn}}. Semiconductor Devices The Truth About Charge Our statements above about charge are not wrong, but they are simple and incomplete. In order to understand how semiconductor devices work one needs a more complete description of the nature of charge in the real world. Charge does not exist independently; it is carried by subatomic particles. For this discussion we will be concerned primarily with electrons, which carry a negative charge of 1.6 × 10-19 C , the minimum amount of charge that can exist in isolation. At least, no one has found any smaller amount than this fundamental quantum of charge. Electrons are one component of atoms and molecules. Atoms are the building blocks out of which all matter is constructed. Atoms bond with each other to form substances. Substances composed of just one type of atom are called elements. For example, copper, gold and silver are all elements; that is, each of them consists of only one type of atom. More complex substances are made up of more than one atom and are known as compounds. Water, which has both hydrogen and oxygen atoms, is such a compound. The smallest unit of a compound is a molecule. A water molecule, for example, contains two hydrogen atoms and one oxygen atom. Atoms themselves are made up of even smaller components: protons, neutrons and electrons. Protons and neutrons form the nucleus of an atom, while the electrons orbit the nucleus. Protons carry positive charge and electrons carry negative charge; the magnitude of the charge for both particles is the same, one quantum charge, 1.6 ×10-19 C . Neutrons are not charged. Normally, atoms have the same number of protons and electrons and have no net electrical charge. Figure 4.8: Structure of an Atom Electrons that are far from the nucleus are relatively free to move around under the influence of external fields because the force of attraction from the positive charge in the nucleus is weak at large distances. In fact, it takes little force in many cases to completely remove an outer electron from an atom, leaving an ion with a net positive charge. Once free, electrons can move at speeds approaching the speed of light (roughly 670 million miles per hour) through metals, gases and vacuum. They can also become attached to another atom, forming an ion with net negative charge. Electric current in metal conductors consists of a flow of free electrons. Because electrons have negative charge, the flow of electrons is in a direction opposite to the positive current. Free electrons traveling through a conductor drift until they hit other electrons attached to atoms. These electrons are then dislodged from their orbits and replaced by the formerly free electrons. The newly freed electrons then start the process anew. At the microscopic level, electron flow through a conductor is not a steady stream, like water flowing from a faucet, but rather a series of short bursts. Figure 4.9: A Simple Model of Electron Flow Silicon Semiconductor devices are made primarily of silicon (silicon's element symbol is "Si"). Pure silicon forms rigid crystals because of its four valence (outermost) electron structure -- one Si atom bonds to four other Si atoms forming a very regularly shaped diamond pattern. Figure 4.10: A Silicon Crystal Structure Pure silicon is not a conductor because there are no free electrons; all the electrons are tightly bound to neighboring atoms. To make silicon conducting, producers combine or "dope" pure silicon with very small amounts of other elements like boron or phosphorus. Phosphorus has five outer valence electrons. When three silicon atoms and one phosphorus atom bind together in the basic silicon crystal cell of four atoms, there is an extra electron and a net negative charge. Figure 4.11: Silicon Doped with Phosphorus This type of material is called n-type silicon. The extra electron in the crystal cell is not strongly attached and can be released by normal thermal energy to carry current; the conductivity depends on the amount of phosphorus added to the silicon. Boron has only three valance electrons. When three silicon atoms and one boron atom bind with each other there is a "hole" where another electron would be if the boron atom were silicon; see Fig. 4.12. This gives the crystal cell a positive net charge (referred to as p-type silicon), and the ability to pick up an electron easily from a neighboring cell. Figure 4.12: Silicon Doped with Boron The resulting migration of electron vacancies or holes acts like a flow of positive charge through the crystal and can support a current. It is sometimes convenient to refer to this current as a flow of positive holes, but in fact the current is really the result of electrons moving in the opposite direction from vacancy to vacancy. Diodes Both p-type and n-type silicon will conduct electricity just like any conductor; however, if a piece of silicon is doped p-type in one section and n-type in an adjacent section, current will flow in only one direction across the junction between the two regions. This device is called a diode and is one of the most basic semiconductor devices. A diode is called forward biased if it has a positive voltage across it from the p- to n-type material. In this condition, the diode acts rather like a good conductor, and current can flow Figure 4.13: A Forward Biased Diode There will be a small voltage across the diode, about 0.6 volts for Si, and this voltage will be largely independent of the current, very different from a resistor. If the polarity of the applied voltage is reversed, then the diode will be reversing biased and will appear non-conducting (Fig. 4.14). Almost no current will flow and there will be a large voltage across the device. Figure 4.14: A Reverse Biased Diode The non-symmetric behavior is due to the detailed properties of the pn- junction. The diode acts like a one-way valve for current and this is a very useful characteristic. One application is to convert alternating current (AC), which changes polarity periodically, into direct current (DC), which always has the same polarity. Normal household power is AC while batteries provide DC, and converting from AC to DC is called rectification. Diodes are used so commonly for this purpose that they are sometimes called rectifiers, although there are other types of rectifying devices. Figure 4.15 shows the input and output current for a simple half-wave Figure 4.15: A Half-Wave Rectifier Rectifier the circuits get its name from the fact that the output is just the positive half of the input waveform. A full-wave rectifier circuit (shown in Figure 4.16) uses four diodes arranged so that both polarities of the input waveform can be used at the output. Figure 4.16: A Full-Wave Rectifier The full-wave circuit is more efficient than the half-wave one. Inductors Inductors are the third and final type of basic circuit component. An inductor is a coil of wire with many windings, often wound around a core made of a magnetic material, like iron. The properties of inductors derive from a different type of force than the one we invented charge to explain: magnetic force rather than electric force. When current flows through a coil (or any wire) it produces a magnetic field in the space outside the wire, and the coil acts just like any natural, permanent magnet, attracting iron and other magnets. If you move a wire through a magnetic field, a current will be generated in the wire and will flow through the associated circuit. It takes energy to move the wire through the field, and that mechanical energy is transformed to electrical energy. This is how an electrical generator works. If the current through a coil is stopped, the magnetic field must also disappear, but it cannot do so immediately. The field represents stored energy and that energy must go somewhere. The field contracts toward the coil, and the effect of the field moving through the wire of the coil is the same as moving a wire through a stationary field: a current is generated in the coil. This induced current acts to keep the current flowing in the coil; the induced current opposes any change, an increase or a decrease, in the current through the inductor. Inductors are used in circuits to smooth the flow of current and prevent any rapid changes. The current in an inductor is analogous to the voltage across a capacitor. It takes time to change the voltage across a capacitor, and if you try, a large current flows initially. Similarly, it takes time to change the current through an inductor, and if you insist, say by opening a switch, a large voltage will be produced across the inductor as it tries to force current to flow. Such induced voltages can be very large and can damage other circuit components, so it is common to connect some element, like a resistor or even a capacitor across the inductor to provide a current path and absorb the induced voltage. (Often, a diode, which we will discuss later, is used.) Inductors are measured in henrys (h), another very big unit, so you are more likely to see millihenries, and microhenries. There are almost no inductors on the RoboBoard, but you will be using some indirectly: the motors act like inductors in many ways. In a sense an electric motor is the opposite of an electrical generator. If current flows through a wire that is in a magnetic field (produced either by a permanent magnet or current flowing through a coil), a mechanical force will be generated on the wire. That force can do work. In a motor, the wire that moves through the field and experiences the force is also in the form of a coil of wire, connected mechanically to the shaft of the motor. This coil looks like and acts like an inductor; if you turn off the current (to stop the motor), the coil will still be moving through the magnetic field, and the motor now looks like a generator and can produce a large voltage. The resulting inductive voltage spike can damage components, such as the circuit that controls the motor current. In the past this effect destroyed a lot of motor controller chips and other RoboBoard components. The present board design contains special diodes that will withstand and safely dissipate the induced voltages -- we hope. AC circuits: alternating current electricity Alternating current (AC) circuits, impedance, phase relations, resonance and RMS quantities. AC electricity is ubiquitous not only in the supply of power, but in electronics and signal processing. t first, why study AC circuits? You probably live in a house or appartment with sockets that deliver AC. Your radio, television and portable phone receive it, using (among others) circuits like those below. As for the computer you're using to read this, its signals are not ordinary sinusoidal AC, but, thanks to Fourier's theorem, any varying signal may be analysed in terms of its sinusoidal components. So AC signals are almost everywhere. And you can't escape them, because even the electrical circuits in your brain use capacitors and resistors. Some terminology For brevity, we shall refer to electrical potential difference as voltage. Throughout this page, we shall consider voltages and currents that vary sinusoidally with time. We shall use lower case v and i for the voltage and current when we are considering their variation with time explicitly. The amplitude or peak value of the sinusoidal variation we shall represent by Vm and Im, and we shall use V = Vm/√2 and I = Im/√2 without subscripts to refer to the RMS values. For an explanation of RMS values, see Power and RMS values. For the origin of the sinusoidally varying voltage in the mains supply, see Motors and generators. So for instance, we shall write: v = v(t) = Vm sin (ωt + φ) i = i(t) = Im sin (ωt). where ω is the angular frequency. ω = 2πf, where f is the ordinary or cyclic frequency. f is the number of complete oscillations per second. φ is the phase difference between the voltage and current. We shall meet this and the geometrical significance of ω later. Resistors and Ohm's law in AC circuits The voltage v across a resistor is proportional to the current i travelling through it. (See the page on drift velocity and Ohm's law.) Further, this is true at all times: v = Ri. So, if the current in a resistor is i = Im . sin (ωt) , we write: v = R.i = R.Im sin (ωt) v = Vm. sin (ωt) where Vm = R.Im So for a resistor, the peak value of voltage is R times the peak value of current. Further, they are in phase: when the current is a maximum, the voltage is also a maximum. (Mathematically, φ = 0.) The first animation shows the voltage and current in a resistor as a function of time. The rotating lines in the right hand part of the animation are a very simple case of a phasor diagram (named, I suppose, because it is a vector representation of phase). With respect to the x and y axes, radial vectors or phasors representing the current and the voltage across the resistance rotate with angular velocity ω. The lengths of these phasors represent the peak current Im and voltage Vm. The y components are Im sin (ωt) = i(t) and voltage Vm sin (ωt)= v(t). You can compare i(t) and v(t) in the animation with the vertical components of the phasors. The animation and phasor diagram here are simple, but they will become more useful when we consider components with different phases and with frequency dependent behaviour. (See Physclips for a comparison of simple harmonic motion and circular motion) What are impedance and reactance? Circuits in which current is proportional to voltage are called linear circuits. (As soon as one inserts diodes and transistors, circuits cease to be linear, but that's another story.) The ratio of voltage to current in a resistor is its resistance. Resistance does not depend on frequency, and in resistors the two are in phase, as we have seen in the animation. However, circuits with only resistors are not very interesting. In general, the ratio of voltage to current does depend on frequency and in general there is a phase difference. So impedance is the general name we give to the ratio of voltage to current. It has the symbol Z. Resistance is a special case of impedance. Another special case is that in which the voltage and current are out of phase by 90°: this is an important case because when this happens, no power is lost in the circuit. In this case where the voltage and current are out of phase by 90°, the ratio of voltage to current is called the reactance, and it has the symbol X. We return to summarise these terms and give expressions for them below in the section Impedance of components, but first let us see why there are frequency dependence and phase shifts for capacitors and for inductors. Capacitors and charging The voltage on a capacitor depends on the amount of charge you store on its plates. The current flowing onto the positive capacitor plate (equal to that flowing off the negative plate) is by definition the rate at which charge is being stored. So the charge Q on the capacitor equals the integral of the current with respect to time. From the definition of the capacitance, vC = q/C, so Now remembering that the integral is the area under the curve (shaded blue), we can see in the next animation why the current and voltage are out of phase. Once again we have a sinusoidal current i = Im . sin (ωt), so integration gives (The constant of integration has been set to zero so that the average charge on the capacitor is 0). Now we define the capacitive reactance XC as the ratio of the magnitude of the voltage to magnitude of the current in a capacitor. From the equation above, we see that XC = 1/ωC. Now we can rewrite the equation above to make it look like Ohm's law. The voltage is proportional to the current, and the peak voltage and current are related by Vm = XC.Im. Note the two important differences. First, there is a difference in phase: the integral of the sinusoidal current is a negative cos function: it reaches its maximum (the capacitor has maximum charge) when the current has just finished flowing forwards and is about to start flowing backwards. Run the animation again to make this clear. Looking at the relative phase, the voltage across the capacitor is 90°, or one quarter cycle, behind the current. We can see also see how the φ = 90° phase difference affects the phasor diagrams at right. Again, the vertical component of a phasor arrow represents the instantaneous value of its quanitity. The phasors are rotating counter clockwise (the positive direction) so the phasor representing VC is 90° behind the current (90° clockwise from it). Recall that reactance is the name for the ratio of voltage to current when they differ in phase by 90°. (If they are in phase, the ratio is called resistance.) Another difference between reactance and resistance is that the reactance is frequency dependent. From the algebra above, we see that the capacitive reactance XC decreases with frequency . This is shown in the next animation: when the frequency is halved but the current amplitude kept constant, the capacitor has twice as long to charge up, so it generates twice the potential difference. The blue shading shows q, the integral under the current curve (light for positive, dark for negative). The second and fourth curves show VC = q/C . See how the lower frequency leads to a larger charge (bigger shaded area before changing sign) and therefore a larger VC. Thus for a capacitor, the ratio of voltage to current decreases with frequency. We shall see later how this can be used for filtering different frequencies. Inductors and the Farady emf An inductor is usually a coil of wire. In an ideal inductor, the resistance of this wire is negligibile, as is its capacitance. The voltage that appears across an inductor is due to its own magnetic field and Faraday's law of electromagnetic induction. The current i(t) in the coil sets up a magnetic field, whose magnetic flux φB is proportional to the field strength, which is proportional to the current flowing. (Do not confuse the phase φ with the flux φB.) So we define the (self) inductance of the coil thus: φB(t) = L.i(t) Faraday's law gives the emf EL = - dφB/dt. Now this emf is a voltage rise, so for the voltage drop vL across the inductor, we have: Again we define the inductive reactance XL as the ratio of the magnitudes of the voltage and current, and from the equation above we see that XL = ωL. Again we note the analogy to Ohm's law: the voltage is proportional to the current, and the peak voltage and currents are related by Vm = XL.Im. Remembering that the derivative is the local slope of the curve (the purple line), we can see in the next animation why voltage and current are out of phase in an inductor. Again, there is a difference in phase: the derivative of the sinusoidal current is a cos function: it has its maximum (largest voltage across the inductor) when the current is changing most rapidly, which is when the current is intantaneously zero. The animation should make this clear. The voltage across the ideal inductor is 90° ahead of the current, (ie it reaches its peak one quarter cycle before the current does). Note how this is represented on the phasor diagram. Again we note that the reactance is frequency dependent XL = ωL. This is shown in the next animation: when the frequency is halved but the current amplitude kept constant, the current is varying only half as quickly, so its derivative is half as great, as is the Faraday emf. For an inductor, the ratio of voltage to current increases with frequency, as the next animation shows. Impedance of components Let's recap what we now know about voltage and curent in linear components. The impedance is the general term for the ratio of voltage to current. Resistance is the special case of impedance when φ = 0, reactance the special case when φ = ± 90°. The table below summarises the impedance of the different components. It is easy to remember that the voltage on the capacitor is behind the current, because the charge doesn't build up until after the current has been flowing for a while. The same information is given graphically below. It is easy to remember the frequency dependence by thinking of the DC (zero frequency) behaviour: at DC, an inductance is a short circuit (a piece of wire) so its impedance is zero. At DC, a capacitor is an open circuit, as its circuit diagram shows, so its impedance goes to infinity. RC Series combinations When we connect components together, Kirchoff's laws apply at any instant. So the voltage v(t) across a resistor and capacitor in series is just vseries(t) = vR(t) + vC(t) however the addition is complicated because the two are not in phase. The next animation makes this clear: they add to give a new sinusoidal voltage, but the amplitude is less than VmR(t) + VmC(t). Similarly, the AC voltages (amplitude times √2) do not add up. This may seem confusing, so it's worth repeating: vseries = vR + vC but Vseries < VR + VC. This should be clear on the animation and the still graphic below: check that the voltages v(t) do add up, and then look at the magnitudes. The amplitudes and the RMS voltages V do not add up in a simple arithmetical way. Here's where phasor diagrams are going to save us a lot of work. Play the animation again (click play), and look at the projections on the vertical axis. Because we have sinusoidal variation in time, the vertical component (magnitude times the sine of the angle it makes with the x axis) gives us v(t). But the y components of different vectors, and therefore phasors, add up simply: if rtotal = r1 + r2, then ry total = ry1 + ry2. So v(t), the sum of the y projections of the component phasors, is just the y projection of the sum of the component phasors. So we can represent the three sinusoidal voltages by their phasors. (While you're looking at it, check the phases. You'll see that the series voltage is behind the current in phase, but the relative phase is somewhere between 0 and 90°, the exact value depending on the size of VR and VC. We'll discuss phase below.) Now let's stop that animation and label the values, which we do in the still figure below. All of the variables (i, vR, vC, vseries) have the same frequency f and the same angular frequency ω, so their phasors rotate together, with the same relative phases. So we can 'freeze' it in time at any instant to do the analysis. The convention I use is that the x axis is the reference direction, and the reference is whatever is common in the circuit. In this series circuit, the current is common. (In a parallel circuit, the voltage is common, so I would make the voltage the horizontal axis.) Be careful to distinguish v and V in this figure! (Careful readers will note that I'm taking a shortcut in these diagrams: the size of the arrows on the phasor diagrams are drawn the same as the amplitudes on the v(t) graphs. However I am just calling them VR, VC etc, rather than VmR, VmR etc. The reason is that the peak values (VmR etc) are rarely used in talking about AC: we use the RMS values, which are peak values times 0.71. Phasor diagrams in RMS have the same shape as those drawn using amplitudes, but everything is scaled by a factor of 0.71 = 1/√2.) The phasor diagram at right shows us a simple way to calculate the series voltage. The components are in series, so the current is the same in both. The voltage phasors (brown for resistor, blue for capacitor in the convention we've been using) add according to vector or phasor addition, to give the series voltage (the red arrow). By now you don't need to look at v(t), you can go straight from the circuit diagram to the phasor diagram, like this: From Pythagoras' theorem: V2mRC = V2mR + V2mC If we divide this equation by two, and remembering that the RMS value V = Vm/√2, we also get: Now this looks like Ohm's law again: V is proportional to I. Their ratio is the series impedance, Zseries and so for this series circuit, Note the frequency dependence of the series impedance ZRC: at low frequencies, the impedance is very large, because the capacitive reactance 1/ωC is large (the capacitor is open circuit for DC). At high frequencies, the capacitive reactance goes to zero (the capacitor doesn't have time to charge up) so the series impedance goes to R. At the angular frequency ω = ωo = 1/RC, the capacitive reactance 1/ωC equals the resistance R. We shall show this characteristic frequency on all graphs on this page. Remember how, for two resistors in series, you could just add the resistances: Rseries = R1 + R2 to get the resistance of the series combination. That simple result comes about because the two voltages are both in phase with the current, so their phasors are parallel. Because the phasors for reactances are 90° out of phase with the current, the series impedance of a resistor R and a reactance X are given by Pythagoras' law: Zseries2 = R2 + X2 . Ohm's law in AC. We can rearrange the equations above to obtain the current flowing in this circuit. Alternatively we can simply use the Ohm's Law analogy and say that I = Vsource/ZRC. Either way we get where the current goes to zero at DC (capacitor is open circuit) and to V/R at high frequencies (no time to charge the capacitor). So far we have concentrated on the magnitude of the voltage and current. We now derive expressions for their relative phase, so let's look at the phasor diagram again. From simple trigonometry, the angle by which the current leads the voltage is tan-1 (VC/VR) = tan-1 (IXC/IR) = tan-1 (1/ωRC) = tan-1 (1/2πfRC). However, we shall refer to the angle φ by which the voltage leads the current. The voltage is behind the current because the capacitor takes time to charge up, so φ is negative, ie φ = −tan-1 (1/ωRC) = tan-1 (1/2πfRC). (You may want to go back to the RC animation to check out the phases in time.) At low frequencies, the impedance of the series RC circuit is dominated by the capacitor, so the voltage is 90° behind the current. At high frequencies, the impedance approaches R and the phase difference approaches zero. The frequency dependence of Z and φ are important in the applications of RC circuits. The voltage is mainly across the capacitor at low frequencies, and mainly across the resistor at high frequencies. Of course the two voltages must add up to give the voltage of the source, but they add up as vectors. V2RC = V2R + V2C. At the frequency ω = ωo = 1/RC, the phase φ = 45° and the voltage fractions are VR/VRC = VC/VRC = 1/2V1/2 = 0.71. So, by chosing to look at the voltage across the resistor, you select mainly the high frequencies, across the capacitor, you select low frequencies. This brings us to one of the very important applications of RC circuits, and one which merits its own page: filters, integrators and differentiators where we use sound files as examples of RC filtering. RL Series combinations In an RL series circuit, the voltage across the inductor is aheadof the current by 90°, and the inductive reactance, as we saw before, is XL = ωL. The resulting v(t) plots and phasor diagram look like this. It is straightforward to use Pythagoras' law to obtain the series impedance and trigonometry to obtain the phase. We shall not, however, spend much time on RL circuits, for three reasons. First, it makes a good exercise for you to do it yourself. Second, RL circuits are used much less than RC circuits. This is because inductors are always* too big, too expensive and the wrong value, a proposition you can check by looking at an electronics catalogue. If you can use a circuit involving any number of Rs, Cs, transistors, integrated circuits etc to replace an inductor, one usually does. The third reason why we don't look closely at RL circuits on this site is that you can simply look at RLC circuits (below) and omit the phasors and terms for the capacitance. * Exceptions occur at high frequencies (~GHz) where only small value Ls are required to get substantial ωL. In such circuits, one makes an inductor by twisting copper wire around a pencil and adjusts its value by squeezing it with the fingers. RLC Series combinations Now let's put a resistor, capacitor and inductor in series. At any given time, the voltage across the three components in series, vseries(t), is the sum of these: vseries(t) = vR(t) + vL(t) + vC(t), The current i(t) we shall keep sinusoidal, as before. The voltage across the resistor, vR(t), is in phase with the current. That across the inductor, vL(t), is 90° ahead and that across the capacitor, vC(t), is 90° behind. Once again, the time-dependent voltages v(t) add up at any time, but the RMS voltages V do not simply add up. Once again they can be added by phasors representing the three sinusoidal voltages. Again, let's 'freeze' it in time for the purposes of the addition, which we do in the graphic below. Once more, be careful to distinguish v and V. Look at the phasor diagram: The voltage across the ideal inductor is antiparallel to that of the capacitor, so the total reactive voltage (the voltage which is 90° ahead of the current) is VL - VC, so Pythagoras now gives us: V2series = V2R + (VL - VC)2 Now VR = IR, VL = IXL = ωL and VC = IXC= 1/ωC. Substituting and taking the common factor I gives: where Zseries is the series impedance: the ratio of the voltage to current in an RLC series ciruit. Note that, once again, reactances and resistances add according to Pythagoras' law: Zseries2 = R2 + Xtotal2 = R2 + (XL − XC)2. Remember that the inductive and capacitive phasors are 180° out of phase, so their reactances tend to cancel. Now let's look at the relative phase. The angle by which the voltage leads the current is φ = tan-1 ((VL - VC)/VR). Substiting VR = IR, VL = IXL = ωL and VC = IXC= 1/ωC gives: The dependence of Zseries and φ on the angular frequency ω is shown in the next figure. The angular frequency ω is given in terms of a particular value ωo, the resonant frequency (ωo2 = 1/LC), which we meet below. (Setting the inductance term to zero gives back the equations we had above for RC circuits, though note that phase is negative, meaning (as we saw above) that voltage lags the current. Similarly, removing the capacitance terms gives the expressions that apply to RL circuits.) The next graph shows us the special case where the frequency is such that VL = VC. Because vL(t) and vC are 180° out of phase, this means that vL(t) = − vC(t), so the two reactive voltages cancel out, and the series voltage is just equal to that across the resistor. This case is called series resonance, which is our next topic. Resonance Note that the expression for the series impedance goes to infinity at high frequency because of the presence of the inductor, which produces a large emf if the current varies rapidly. Similarly it is large at very low frequencies because of the capacitor, which has a long time in each half cycle in which to charge up. As we saw in the plot of Zseriesω above, there is a minimum value of the series impedance, when the voltages across capacitor and inductor are equal and opposite, ie vL(t) = − vC(t) so VL(t) = VC, so ωL = 1/ωC so the frequency at which this occurs is where ωo and fo are the angular and cyclic frequencies of resonance, respectively. At resonance, series impedance is a minimum, so the voltage for a given current is a minimum (or the current for a given voltage is a maximum). This phenomenon gives the answer to our teaser question at the beginning. In an RLC series circuit in which the inductor has relatively low internal resistance r, it is possible to have a large voltage across the the inductor, an almost equally large voltage across capacitor but, as the two are nearly 180° degrees out of phase, their voltages almost cancel, giving a total series voltage that is quite small. This is one way to produce a large voltage oscillation with only a small voltage source. In the circuit diagram at right, the coil corresponds to both the inducance L and the resistance r, which is why they are drawn inside a box representing the physical component, the coil. Why are they in series? Because the current flows through the coil and thus passes through both the inductance of the coil and its resistance. You get a big voltage in the circuit for only a small voltage input from the power source. You are not, of course, getting something for nothing. The energy stored in the large oscillations is gradually supplied by the AC source when you turn on, and it is then exchanged between capacitor and inductor in each cycle. For more details about this phenomenon, and a discussion of the energies involved, go to LC oscillations. Bandwidth and Q factor At resonance, the voltages across the capacitor and the pure inductance cancel out, so the series impedance takes its minimum value: Zo = R. Thus, if we keep the voltage constant, the current is a maximum at resonance. The current goes to zero at low frequency, because XC becomes infinite (the capacitor is open circuit for DC). The current also goes to zero at high frequency because XL increases with ω (the inductor opposes rapid changes in the current). The graph shows I(ω) for circuit with a large resistor (lower curve) and for one with a small resistor (upper curve). A circuit with low R, for a given L and C, has a sharp resonance. Increasing the resistance makes the resonance less sharp. The former circuit is more selective: it produces high currents only for a narrow bandwidth, ie a small range of ω or f. The circuit with higher R responds to a wider range of frequencies and so has a larger bandwidth. The bandwidth Δω (indicated by the horiztontal bars on the curves) is defined as the difference between the two frequencies ω+ and ω- at which the circuit converts power at half the maximum rate. Now the electrical power converted to heat in this circuit is I2R, so the maximum power is converted at resonance, ω = ωo. The circuit converts power at half this rate when the current is Io/√2. The Q value is defined as the ratio Q = ωo/Δω.

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