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Chapter 1: Circuit Variables 1.1 Electrical Engineering: An Overview Electrical Engineers are concerned with the design, analysis, and operation of systems involving electrical signals. Examples: Communications/signal engineering Computer systems Control systems/robotics Power systems Microelectronics Theoretical Basis for Electrical Engineering: Electromagnetics and Maxwell‟s Equations. We will not deal with this topic here. Instead we will talk about a specialization of electromagnetics… Circuit Theory is an important special case of electromagnetics. It is appropriate where the spatial dimensions of the electrical system are small compared with the wavelength of an electromagnetic signal, i.e., where the shape of the circuit does not matter. 1 1.2 Units We will use the International System of units: Length: meters (m) Mass: kilogram (kg) Time: second (s) Current: Ampere (A) Temperature: Kelvin (K) Also: 1 Ampere is the current that, maintained in two straight, parallel, infinite conductors of negligible circular cross section placed 1 m apart in vacuum, would produce a force between the conductors of 2 x 10 -7 N/m. The Coulomb is a unit of charge derived from the Ampere; more below. 1.3 Circuit Analysis: An Overview Idea: We want to be able to make quantitative predictions of electrical circuit behavior. We do this using idealized interconnections of circuit elements called circuit models. Modeling is an important concept: we will use simple components (resistors, Rcase batteries, capacitors…) to construct models of electrical systems. These models help us think about and design real electrical systems. They tell us how the system will behave if we make changes to it, for example. The vbattery Rbulb diagram to the right is a circuit model for a flashlight; we will be using it in the next chapter to discuss basic circuit analysis concepts. 2 iC RC RB1 is biB vCE vo iB RB2 RE iE This is a circuit model taken from the text book, Chapter 2. We will have a few things to say about it in class… 3 1.4 Voltage and Current Voltage and current are the important variables for electrical circuits. Before discussing them, we look at a more fundamental concept: charge. Charge is a basic property of matter. The smallest “piece” of charge is the charge on an electron, with magnitude q = 1.6 x 10-19 Coulombs (Coul). We find that... There are two types of charge: positive and negative; Like charges (i.e., both positive or both negative) repel one another, while opposites attract. Voltage and current arise from the phenomenon of charge. To separate charges from one another requires that we exert energy, and results in the presence of a potential, or voltage. Separation of charge voltage If charge is moving, we have a current. Movement of charge current Formal Definitions: Potential and Voltage: To define electric potential, we imagine moving a hypothetical test charge from point A to point B in a region where electric forces are present. This requires work (energy). The work required is the difference in electric potential energy U between point A and point B. Electric potential is the work per unit 4 charge required to move the test charge from A to B. It is defined in the limit of a vanishingly small test charge (i.e., as q becomes 0): U U A vB v A B q0 q In the equation above, vB - vA is the difference in potential, or voltage v, between A and B; UB – UA is the work w done in moving the charge from A to B; q is the charge in Coulombs. So in differential form we have dw v . dq Units: 1 Volt = 1 Joule/Coulomb 1 Joule = 1 kg m2/s2 1 Coulomb = 1 Ampere . 1 s. It is important to note that voltage, like potential energy, can only be defined as a difference. Even though we write „v‟, we mean a change in electrical potential from point A to point B. Current: When charge flows in a conducting material, a current exists. dq i dt In this equation, i is the current; q is the charge; t is time. 5 Units: 1 Ampere = 1 Coulomb/s We will not be concerned with the details of how current flows. We usually think of it as a flow of electrons in a wire, for example, and that will be good enough for us here. The Water Analogy To some extent, electricity flowing in wires has an analogue in water flowing through pipes. Voltage (electric potential) height of water (gravitational potential) Current water flowing in pipes So what is voltage? It is an electric potential difference. It describes the ability of the system to make something happen, like move charge through a wire, just as gravity has the potential to move water through the pipes. What about current? Current is the flow of charge. If charge is moving, we have a current. For circuit analysis, we don‟t need to worry about what exactly is moving, or how it is moving. 6 A simple circuit + - The figure to the right shows a battery. The battery provides a voltage of 9 V, with “positive” and Ever- “negative” terminals as indicated. Chemical forces in the battery maintain an electric potential Hardy 9 difference between the terminals. For now, we can think of the terminals as being analogous to “up” Volt and “down” in the water tower. More on that later… The battery doesn‟t “do” anything until we include it in a complete circuit. Let‟s look at a simple example. + - What‟s going on here? The battery is providing a voltage of 9 Volts; when wires and a light bulb are connected across it, the voltage causes a current to flow Ever- Hardy 9 through the wires. The current is causing the light to glow. Note that we need a Volt connection to and from the bulb for the circuit to work; that is, we need a complete path for current to flow. How does this work? We can go back to our Physics text to find out that the battery exerts a force on the charges in the wire and causes them to move, provided the circuit is complete. We do not need to know about these forces to do circuit analysis. Instead of forces, we talk about voltage. Before going any further, let‟s talk more about voltage and current, and about voltage polarity and current direction. These are important concepts. We begin by defining a basic circuit element. 7 1.5 The Ideal Basic Circuit Element We will use the box to represent several kinds of things. It may represent just one or perhaps many electrical components, or even an entire circuit or electrical system. For now, we assume it represents an ideal basic circuit element. Circuit elements are the building blocks of circuit models (resistors, batteries, …). We assume that the box is connected to something (at terminals 1 and 2) that is not shown. Otherwise it‟s hard to imagine how a current can be flowing through it. 1 i Properties of the ideal basic circuit element (pictured on the right): + It has two terminals (labeled “1” and “2”). It can be described mathematically in terms of a voltage v and/or a v current i. - 2 It cannot be subdivided into other circuit elements, hence it is basic. It is ideal in the sense that it has idealized properties that do not necessarily hold for the device it models. We will define these properties as we go on. Our circuit element has a voltage „v‟ and current „i‟ associated with it. The voltage is indicated as a difference in potential between terminals 1 and 2, with terminal 1 indicated as being “positive” with respect to terminal 2. Recall that water in the tower was at a different height than the house. It is the difference in height that is important, and which causes the water to flow, not the absolute position of the water tower. Similarly, it is the difference in potential between the terminals that‟s important. It is also important to note that the voltage v can exist for several reasons, as we will discuss; it does not necessarily mean that the circuit element is a battery, or even that the box contains a battery, although it could. The circuit element also has a current i flowing through it, as indicated by the arrow. The current is flowing into the circuit element by way of terminal 1. Since there is no other exit, it must be coming out by way of terminal 2. We can describe this by saying that current is flowing from terminal 1 to terminal 2. 8 Interpretation of the symbols The „+‟ and „-„ associated with v in the figure indicate the reference polarity for the voltage. They indicate a reference, just as an axis on an x-y graph indicates “positive x”, and not necessarily the actual polarity. It is important to note that v has both a value (a magnitude) and a sign (positive or negative). Both of these are required if we want to know the potential difference between terminals 1 and 2. Some examples will clarify the idea. Consider the circuit element to the right. 1 + a) Given: v = 5 V. Then, terminal 1 is 5 V higher in potential than terminal 2. v b) Given: v = -3 V. Then, terminal 1 is 3 V lower in potential than terminal 2. - 2 So the „+‟ and „ – „ give me a reference that allows me to determine what the value and sign of v mean. [I could say for case a) that terminal 2 is 5 V lower in potential than terminal 1; this would mean the same thing as I‟ve written above.] Let‟s look at another example. 1 - a) Given: v = 4 V. Then, terminal 1 is 4 V lower in potential than terminal 2. v b) Given: v = -12 V. Then, terminal 1 is 12 V higher in potential than terminal 2. + 2 Bottom line: The „+‟ and „-„ signs tell us the reference polarity. We don‟t know the actual polarity until we are given (or calculate) the sign of v. This is an important concept. Sometimes we will have to label a voltage even before we know what the actual polarity is. That label will be have to include a reference polarity. 9 Note that in the first figure, case a), I would get v = 5 V if terminal 1 were at 10 V and terminal 2 at 5 V. I would get the same result if terminal 1 were at -3 V and terminal 2 at – 8 V, or if terminal 1 were at 5 V and terminal 2 at 0 V (I could go on…). Notation: I have been using a lower case v for my voltage label and an upper case V for the abbreviation for Volts. I would like you to do the same thing. 1 i We can do the same thing for the current. The arrow here indicates the reference current direction. So… a) Given i = 30 mA. Then, 30 mA of current is entering terminal 1. 2 b) Given: i = -25 mA. Then, 25 mA of current is leaving terminal 1. Another example: 1 i a) Given: i = 5 mA. Then, 5 mA of current is leaving terminal 1. b) Given: i = - 100 mA. Then, 100 mA of current is entering terminal 1. 2 Bottom Line: The arrow indicates a reference current direction. I don‟t know the actual current direction until I know the sign of i. Again, this is important. We will often have to label currents even though we don‟t know which way they are going. These labels will need to include reference directions. Notation and units: I am using a lower case i for current, and an upper case A as the abbreviation for Amperes. The current values are milli-Amperes (mA). We have that 1 mA = 0.001 A. 10 Another consideration It turns out that a positive current is defined as the direction in which positive charge carriers are moving. In other words, if the current in the last figure above (second current example) is positive, it means either that positive charges are leaving the box through terminal 1, or else that negative charges (say, electrons) are entering the box through terminal 1. This is a nuisance, since we usually think of current in metal wires as being due to electrons, and so current is opposite to the direction of the charge carriers in most situations. Fortunately, for circuit analysis we don‟t need to worry about this; we don‟t care what is carrying the current or which direction it is moving. All we need to know is which way the arrow is pointing and what the sign of the current is. Voltage Drop and Voltage Rise We give here a couple of definitions that will be useful later. Referring to our basic circuit elements: when the voltage at terminal 1 is higher than the voltage at terminal 2, we say there is a voltage drop from terminal 1 to terminal 2. If the terminal 1 voltage is lower than the terminal 2 voltage, there is a voltage rise from terminal 1 to terminal 2. Some examples: 1 If v = 5 V, there is a voltage drop (of 5 V) from terminal 1 to terminal 2. I could + also say there is a voltage rise from terminal 2 to terminal 1. v - 2 If v = -3 V, there is voltage rise from terminal 1 to terminal 2. I could also say there is a voltage drop from terminal 2 to terminal 1. 11 1.6 Power and Energy Not surprisingly, power and energy are important concepts for electrical engineers. Most of the time we will be talking about electrical energy, as opposed to sound or light energy. Some important ideas: Electrical energy can be either delivered or absorbed. For example: o A glowing flashlight bulb is absorbing electrical energy. [It is giving off (delivering) light and heat energy.] o A flashlight battery is delivering energy to the bulb (assuming the flashlight is on). o Your camera battery can (probably) be put into a battery charger, during which time the camera battery is absorbing electrical energy. Power is the rate at which energy is delivered or absorbed. That is, dw p dt Units: w is the energy in Joules, t is the time in seconds, and p is the power in Watts. Since electrical energy can be delivered or absorbed, electrical power can be delivered or absorbed. It is very important that we keep track of whether a circuit element is delivering or absorbing power. We can relate electrical power to voltage and current. dw dw dq p v.i dt dq dt 12 So the electrical power associated with a thing, say, a battery or a light bulb or a resistor, is obtained by multiplying the voltage and current for that thing. But how can we tell whether power is being delivered or absorbed? Sometimes it will be obvious but sometimes it will not. To get that right, we need a rule for signs. Let‟s look at our circuit element again and calculate the power absorbed by this element (an arbitrary choice). Here is the rule for that: In the diagram on the left below, the current is entering the positive terminal and leaving the negative terminal; we say that the current is in the direction of the voltage drop. In that case, we write pabs = v.i. Here “pabs” means “the power being absorbed”, which is what we said we would calculate. i i + - v v - + pabs vi pabs vi For the diagram on the right, where the current and the voltage drop are in opposite directions, we write p abs = - v.i. The idea is that we need to use the appropriate sign, which we get by looking at the diagram, and in particular at which way the current is going relative to the voltage drop. Some examples: For the case on the left: If v = 3 V and i = 25 mA, then pabs = v.i = (3) (0.025) = 0.075 W = 75 mW. If v = -2 V and i = 10 mA, then pabs = v.i = (-2) (0.010) = - 20 mW. For the case on the right: If v = 2 V and i = - 250 mA, pabs = - v.i = - (2) (-0.250) = 0.5 W. If v = - 10 V and i = - 35 mA, pabs = - v.i = - (-10) (-0.035) = - 350 mW. 13 But wait: what does it mean that the absorbed power is negative? We interpret a negative absorbed power to mean that power is in fact being delivered, not absorbed. We could also have calculated delivered power as pdel = - pabs. Important notes: Our choice of formula (pabs = v.i or pabs = - v.i) depends only on the diagram (which shows the current reference direction and voltage reference polarity). Regardless of which formula I use, I must keep track of the signs of v and i, and include them in the calculation. I may not know before hand whether power is being absorbed or delivered, but it doesn‟t matter. I can calculate the power absorbed in any case, and if the answer is negative, I conclude that in fact power is being delivered. Alternatively, I can calculate power delivered (by changing the signs on the formulas above). As long as I am clear about what I am doing, it doesn‟t matter. It is very, very important to keep track of whether power is being delivered or absorbed. Therefore every time you calculate power, I want you to indicate whether you are calculating absorbed power or delivered power. Do this by writing either pabs or pdel, or some other convenient and clear notation. I am using a lower case p for power. I want you to do the same thing. I want you to be very careful about notation, including any subscripts on v and i. See the example given after the summary below. Summary To calculate power… Choose whether to calculate absorbed power or delivered power; this is your choice, even if you have no idea which it “actually” is. Then, use the appropriate formula, which is based on a diagram showing reference current direction and reference voltage polarity (as shown above). Plug voltage and current into the appropriate formula, including the signs of v and i. Decide whether power is actually being absorbed or delivered based on the sign of the result. 14 Example ix Given: vx = - 12 V; ix = - 0.2 A. - vx Problem: Calculate the power absorbed by the circuit element. + Solution: We note that the current is in the direction of the voltage rise, so we write pabs = - vx.ix = - (-12) (- 0.2) = - 2.4 W. So power is in fact being delivered. Thus, the delivered power is 2.4 W; the absorbed power is -2.4 W. Important Note: The voltage and current labels given in the problem above include subscripts „x‟. I must keep these subscripts in my calculation if I am to be clear about what I am calculating. The people who grade my homework, quizzes, and exams are going to be very concerned about the clarity of my work. Therefore they will take off points if I fail to use the proper notation. In particular, if I were to write “p = v.i” for the problem above, I would have failed to indicate whether I‟m calculating absorbed or delivered power, and I would have failed to use correct notation for v and i. In that case, I would have lost significant credit for the problem, perhaps half the credit. The meaning of “absorbed” power and energy i Let‟s look more closely at what it means for an object to absorb electrical + power or energy. Assume in the figure to the right that we know both i and v are positive. Assume also that the current is carried by positive charge, i.e., v that positive charge is entering the upper terminal and leaving via the lower. - In going from the upper to the lower terminal, the electric potential of the 15 charges is being lowered, which means they are losing energy. They are losing energy to the circuit element, which means the circuit element is absorbing energy (and therefore absorbing power). Sign Relationships We finish with one last definition. The issue of whether or not the reference direction for the current is in the direction of the voltage drop will come up in future discussions, so we need a way to refer to it. i + Passive Sign Convention: When the reference current direction is in the v direction of the voltage drop, as it is in the figure to the right, we say that we are using the passive sign convention. - i - Active Sign Convention: The alternative is a situation in which the reference v current direction is in the direction of the voltage rise, as it is n the figure to the + left. This is the active sign convention. The passive and active sign conventions provide names for dealing with these issues. 16