Wolfgang Bauer 0. Milo D. Koretsky 0. Randall D. Knight 0. George Odian 0. John Kenkel 0. Trott 0. Carl S. Warren 2. Warren 0. Abraham Silberschatz 1. Frederick S. Hillier 1. William Stallings 1. Morris Mano 1. David Irwin 0. Morris Mano 0. Michael F. Ashby 0. William Thomson 0. Gene Mathers 0. Jack C. McCormac 1. William T. Segui 0. Richard T. Evans 0. Bill W. Tillery 0. Giorgio Rizzoni 0. Khurmi 1. Table of Contents Chapter 1. Introduction and Basic Concepts 1. Role and importance of circuits in engineering 2.
Fields, load and current Fields Load Current 3. Voltage 4. Conversion of energy in an electric circuit 5. Ideal sources of voltage and current 7. Resistance, Law of Ohm and Power a Repetition 8.
Introduction 2. Terminology: Parallel, Series, Node, Branch, and so on 3. The Current Law of Kirchhoff 4. Kirchhoff's Law of Voltage 5. Parallel Resistors and Current Division 7. Series-Parallel Interconnections 8. Revised dependent sources 9. Model for a noisy battery and battery capacity Nodal and Loop Analysis 1. Introduction, Review and Terminology 2. Concepts of nodal and loop analysis 3. Nodal analysis I: Ground voltage sources 4. Loop Analysis 6. Modified Nodal Analysis 7.
The Operational Amplifier 1. The Ideal Operational Amplifier 3. Saturation and the Active Region of Op Amp 5. Linearity, superposition, and source transformations 1.
Linearity 3. Overlap and proportionality 4. Role and Importance o f Circuits in Engineering 2. Charge and Current 3. Voltage 4. Circuit Elements 5. Voltage, Current, Power, Energy, Relationships 6. Ideal Voltage and Current Sources 7. Introduce and investigate three basic electrical quantities: charge, current, and voltage, and the conventions for their reference directions. Define a two-terminal circuit element. Define and investigate power and energy conversion in electric circuits, and demonstrate that these quantities are conserved.
Investigate power dissipation in a resistor. Classify memoryless circuit elements by dieir terminal voltage-current relationships. Explain the difference between a device and its circuit model.
Are you curious about how fuses blow? About the meaning o f different wattages on Hght bulbs? About the heating elements in an oven? And how is the presence o f your car sensed at a stoplight? Circuit theory, the focus o f this text, provides answers to all these questions. In this text, we define and analyze common circuit elements and describe their interaction. But they also have broad applicability in other fields. For instance, disciplines such as bioengineering and mechanical engineering have similar patterns o f analysis and often utilize circuit analogies.
Similarly, a T V reception antenna or a satellite dish can pick up and direct this energy to a T V set. T h e T V contains circuits Figure 1. Charge is an electrical property o f matter. Matter consists o f atoms. Roughly speaking, an atom contains a nucleus that is made up o f positively charged protons and neutrons which have no charge.
T he nucleus is surrounded by a cloud o f negatively charged electrons. Th e accumulated charge on 6. Thus, the charge on an electron is Particles with opposite charges attract each other, whereas those with similar charges repel. The force o f attraction or repulsion between two charged bodies is inversely proportional to the square o f the distance between them, assuming the dimensions o f the bodies are very small compared with the distance o f separation. The force is attractive if the particles have opposite charges.
How many electrons have a combined charge o f -5 3. A conductor refers to a material in which electrons can move to neighboring atoms with relative ease. Metals, carbon, and acids are common conductors. An ideal conductor offers zero resistance to electron movement. Wires are assumed to be ideal conductors, unless otherwise indicated. Insulators oppose electron movement.
Common insulators include dry air, dry wood, ceramic, glass, and plastic. An ideal insulator offers infinite opposition to electron movement. Current refers to the net flow o f charge across any cross section o f a conductor. The direction o f current flow is taken by convention as opposite to the direction o f electron flow, as illustrated in Figure 1. This is because early in the history o f electricity, scientists erroneously believed that current was the movement o f only positive charges, as illustrated in Figure 1.
In metallic conductors, current consists solely o f the movement o f electrons. By convention, the positive current direction is taken as left to right. One Ampere of Current. One Coulom b One of positive 'second charge later. Both Figures 1. In circuit analysis, we do not distinguish between these two cases: each is represented symbolically, as in Figure 1. The arrowhead serves as a reference for determining the true direction o f the current. A positive value o f current means the current flows in the same direction as the arrow.
A current o f negative value implies flow is in the opposite direction o f the arrow. For example, in both Figures 1. In Figure 1. The wall socket in a typical home is a source o f alternating current, which changes its sign periodically, as we will describe shortly. These situations require the notion o f a negative current.
Figure 1. Positive charge carriers move from left to right at the rate o f 0. Negative charge carriers move from right to left at the rate o f 0. S o lu tio n a The current from left to right, due to the movement o f the positive charges, is 0. The current from left to right, due to the movement o f the negative charges, is 0. Since ly is the current from right to left, its value is then Therefore, at boundaries A and B, negative charges carried by electrons move from right to left at the rate o f 0.
In Example 1. The equivalent integral counterpart o f Equation 1. S o lutio n As per Equation 1. Compute the current flow. Suppose the current through a cross section o f conductor is given in Figure 1.
There are two very important current types: direct current do and alternating current ac. Constant current i. Household current is ac. Lastly, Figure 1. There are many other types o f waveforms. A; c neither ac nor dc. Because the value o f an ac waveform changes with time, ac is measured in different ways. Suppose the instantaneous value o f the current at time t is A!
The term peak value refers to K in K sin co? The peak-to-peak value is 2K. Another measure o f the alternating current, indicative o f its heating effect, is the root mean square rms , or effective value. A special instrument called an ammeter measures current.
Some ammeters read the peak value, whereas some others read the rms value. One type o f ammeter, based on the interaction between the current and a permanent magnet, reads the average value o f a current. From calculus, Fave! For a general ac waveform, the average value is zero. However, ac signals are often rectified, i.
For such circuits, the average value o f the rectified signal is important. From Equation 1. W hat causes current to flow? An analogous question might be. W hat causes water to flow in a pipe or a hose? W ithout pressure from either a pump or gravity, water in a pipe is still. Strictly speaking, water flows from a point o f higher pressure— say, p o in ts — to a point o f lower pressure— say, point 5 — along a pipe.
Between the two points and B, there is said to be a pressure drop. Gravity forces the water to flow from a higher elevation to a lower elevation. O n the top plate is a fixed amount o f positive static charge. On the bottom plate is an equal amount o f negative static charge. Suppose a small positive charge were placed between the plates. This small charge would experience a force directed toward the negatively charged bottom plate.
Part o f the force is due to repulsion by the positive charges on the top plate, and part is due to the attraction by the negative charges on the bottom plate. This repulsion and attraction marks the presence o f an electric field produced by the opposite sets o f static charges on the plates.
The electric field indicated in Figure 1. The point is that current flow is induced by an electric pressure called a voltage drop.
As mentioned, in Figure 1. Equivalently, point 5 is at a lower potential than point A. Note, however, that if we turn the whole setup o f Figure 1.
Similarly, if a negative charge - q is placed at B, as in Figure 1. Just before q hits the bottom plate, the kinetic energy gained equals the constant force acting on q multiplied by the distance traveled in the direction o f the force.
The standard unit for measuring potential difference or voltage is the volt V. According to Equation 1. The following four statements illustrate this point in the context of Figure 1. The voltage between or across A a n d 5 is 2 V.
Voltage can be generated by chemical action, as in batteries. In a battery, chemical action causes an excess of positive charge to reside at a terminal marked with a plus sign and an equal amount of negative charge to reside at a terminal marked with a negative sign.
On the other hand, if is negative, then point 5 is at a higher potential than point A. Since stands for the voltage drop from point B to point A, o. The double-subscript convention is one o f three methods commonly used to unambiguously specify a voltage drop. O n the other hand, if Vq is negative, then 5 is at a higher potential than A.
A digital voltmeter DVM is connected across the terminals, as shown. The display reads -1 2 V. One final note: As with current, there are different types o f voltages— dc voltage, ac voltage, and general voltage waveforms. Circuits consist o f interconnections o f circuit elements. The most basic circuit element has two terminals, and is called a two-terminal circuit element, as illustrated in Figure 1.
The battery is a very common source, providing nearly constant voltage and the usually small current needed to operate small electronic devices. Car batteries, for example, are typically 12 volts and can produce large currents during starting. The wall outlet in a home can be thought o f as a -volt ac source. A current z r flows through the element.
Total water into a pipe equals total water out o f the pipe. Analogously, the current entering a two-terminal device must, by definition, equal the current leaving the two-terminal device.
The circuit element o f Figure 1. Such a labeling o f the voltage-current reference directions is called the passive sign convention. In contrast, the current iij flows from the minus terminal to the plus terminal through the battery; this labeling is conventional for sources but not for non-source circuit elements.
For a resistor, the amount o f current flow depends on a property called resistance; the smaller the resistance, the larger the current flow for a fixed voltage across the resistor.
A small-diameter pipe offers more resistance to water flow than a large-diameter pipe. A conductor that is designed to have a specific resistance is called a resistor. More on this shortly. The circuit elements called the capacitor and the inductor will be described later in the text. Also, future chapters will describe the operational amplifier and the transformer that are circuit elements having more than two terminals. The relationship between voltage across and current through a two-terminal element determines whether power and, thus, energy is delivered or absorbed.
The heating element in an electric oven can be thought o f as a resistor. The heating element absorbs electric energy and converts it into heat energy that cooks, among other things, turkey dinners.
W ith reference to Figures 1. Similarly, at each instant o f time, the battery can be viewed as delivering 60 watts o f power to the headlight. The power absorbedhj element 1 is, by definition, the rate at which it converts or absorbs energy. W hat is the energy absorbed by circuit element 1 in one minute?
W hat is the power absorbed by circuit element 1? W ith respect to Figure 1. Consequently, the energy, W , in J , absorbed during the time interval Tis. Now, let us reconsider Figure 1. As such, the remainder o f the circuit is said to generate electric energy.
Observe that the rate at which the remainder o f the circuit generates power precisely equals the rate at which circuit element 1 absorbs power.
How much power does circuit element 1 absorb? In general, whenever a two-terminal general circuit element is labeled according to the passive sign convention, as in Figure 1. As a general convention, non-source circuit elements are labeled according to the passive sign convention.
As mentioned, power is the rate o f change o f work per unit o f time. T he ability to determine the power absorbed by each circuit element is highly important because using a circuit element or some device beyond its power-handling capability could damage the device, cause a fire, or result in a serious disaster.
This is why households use circuit breakers to make sure electrical wiring is not overloaded. How much power can the car heater absorb before the 20 -amp fuse blows.
As mentioned earlier, the calculated value o f absorbed power P may be negative. If the absorbed power P is negative, then the circuit element actually generates power or, equivalently, delivers power to the remainder o f the circuit. In any circuit, some elements will have positive absorbed powers, whereas some others will have negative absorbed powers.
If one adds up the absorbed powers o f ALL elements, the sum is zero! This is a universal property called conservation o f power. Equivalently, the sum o f the absorbed powers equals the sum o f the generated powers at each instant o f time. For the present, we will simply use it to solve various problems. W a t t l e measures the power consumed by a bulb.
Typical wattages include 15, 25, 40, 60, 75, and W. W hat is the wattage o f the unlabeled bulb? S o lutio n From conservation o f power, the total power delivered by the battery equals the total power absorbed by all the bulbs. Therefore, the power absorbed by the unknown bulb is. Electroplating Apparatus. Suppose the electroplating apparatus o f Example 1.
W hat is the cost o f operation for a single h day? W hat is the cost o f operating for a 20 workday month? Compute the power absorbed by each circuit element. W hich elements are delivering power? S o lutio n Step 1. Compute power absorbed by each element. Using either Equation 1. Step 2. Verify conservation o f power. Since P 4 and Pg are negative, element 4 delivers 14 W, and element 6 delivers 20 W o f power.
T he remaining four elements absorb power. W hat is the new power absorbed by element 3? If the power absorbed by a circuit element is positive, the exact nature o f the element determines the type o f energy conversion that takes place.
If the circuit element is a battery that is being charged, then electric energy is converted into chemical energy within the battery. Consider Figure 1. Circuit Elem ent Absorbing Power p t.
This product also makes sense from a dimensional point o f view;. If, in Equation 1. Find W 0, oo. Since energy is the integral o f power, power is the rate o f change derivative o f energy. Differentiating both sides o f Equation 1. S o lutio n A simple graphical multiplication o f Figures 1. The wall socket o f a typical home represents a practical voltage source. After flipping the switch on an appliance plugged into a wall socket, a current flows through the internal circuitry o f the appliance, which, for a vacuum cleaner or dishwasher, converts electrical energy into mechanical energy.
The symbol is more commonly referred to as independent voltage source. T he reference polarity does not mean that v t is positive. An ideal battery produces a constant voltage under all operating conditions, i. Real batteries are not ideal but approximate the ideal case over a manufacturer-specified range o f current requirements. Practical sources i. One important respect is that the terminal voltage depends on the current delivered by the source.
There are two general battery categories: nonrechargeable and rechargeable. A discussion o f the dramatically advancing battery technology is beyond the scope o f this text. Besides batteries and ideal voltage sources, devices called ideal or independent current sources maintain fixed current waveforms into a circuit, as illustrated in Figure 1.
T he symbol o f an ideal current source is a circle with an arrow inside, indicating a reference current direction. In nature, lightning is an example o f an approximately ideal current source. W hen lightning strikes a lightning rod, the path to the ground is almost a short circuit, and very little voltage is developed between the top o f the rod and the ground.
However, if lightning strikes a tree, the path o f the current to the ground is impeded by the trunk o f the tree. A large voltage then develops from the top o f the tree to the ground. Independent sources have conventional labeling, as shown in Figure 1. Another type o f ideal source is a dependent source.
Such sources model real-world devices that are used in real circuits. In the text, the symbol for a dependent source is a diamond. If an arrow appears inside the diamond, it is a dependent current source, as illustrated in Figure 1.
There is dual terminology for dependent current sources. The configuration o f Figure 1. Source voltages or currents are called excitations, inputs, or input signals.
The units are volts, amperes, and so on. Smaller and larger quantities are expressed by the use o f prefixes, as defined in Standard Engineering Notation Table 1. TABLE 1. Engineering Notation for Large and Small Quantities. Different materials allow electrons to move from atom to atom with different levels o f ease. Suppose the same dc voltage is applied to two conductors, one carbon and one copper, o f the same size and shape. Two different currents will flow. A conductor designed to have a specific resistance is called a resistor.
Hence, a resistor is a device that impedes current flow. Just as dams impede water flow and provide flood control for rivers, resistors provide a means to control current flow in a circuit.
In , Ohm observed that for a connection like that o f Figure 1. Inserting a proportionality constant, one can write. The proportionality constant R is the resistance o f the conductor in ohms. A two-terminal device has a 1-Q resistance i f a 1-V excitation causes 1-A o f current to flow. In Equation 1. In this text, we try to adhere to the SI system. On the other hand, if a device or wire has infinite resistance zero conductance , it is called an open circuit. In most o f the literature on electronic circuits, resistor and resistance are used synonymously, and we will continue this practice.
However, it is true for all time-dependent waveforms exciting a linear resistor. Thus, we can generalize Equation 1. As an aid in writing the correct v-i relationship for a resistor.
For a resistor connected between terminals A and B, the voltage drop from A to B is equal to the resistance multiplied by the current flowing from A to B through the resistor. Find the resistance R for each o f the resistor configurations in Figure 1. Once the voltage and the current associated with a resistor are known, the power absorbed by the resistor is easily calculated. Assuming the passive sign convention, then combining Equation 1.
Equations 1. Intuitively speaking, electrons that flow through the resistor collide with other particles along the way. The process resembles the action in a pinball game: the pinball sue- cessively collides with various pegs as it rolls from a higher to a lower elevation. Electrical energy that is converted to heat or used to overcome friction is usually called a loss. A fuse is a short piece o f inexpensive conductor with a very low resistance and a predetermined current-carrying capacity.
When inserted in a circuit, it carries the current o f the equipment or appliances it must protect. Resistance o f a conductor depends on the material and its geometrical structure. For a specific temperature, R is proportional to the length I o f a conductor and inversely proportional to its cross-sectional area A,. Table 1.
Silver 0. Copper 1. Gold 1. Then find the voltage across each wire and the power absorbed given off as heat by each wire if a A direct current flows through feet o f each wire. S o lutio n The resistivities o f aluminum and nickel wire relative to copper are 1. Notice that every feet o f 16 AWG aluminum wire would absorb This absorbed power, given off as heat, is why nickel wire is used for heating elements in toasters and ovens.
Exercise, a If a constant current o f 10 A flows through 1, feet o f 16 AWG copper wire, how many watts o f heat are generated by the wire? Temperature also affects resistance. For most metallic conductors, resistance increases with increasing temperature— except carbon, which has a decrease in resistance as temperature rises.
Find the current through and the power absorbed by the bulb if it is connected across a constant V source, as illustrated in Figure 1. Equation 1. By Equation 1. Step 3. T he power delivered by the source is 90 x 0. Therefore, the power delivered by the source equals the power absorbed by the resistor. W hat is the power absorbed by the lamp? W hat is the power delivered by the battery?
Discover which light bulb has burned out. S o lutio n a From Equation 1. Hence, the watt bulb has gone dark. Repeat Example 1. W hen similarly connected, halogen light bulb 2 uses 60 watts o f power. Find the hot resistances. The hot resistances o f each bulb are given by. Find the current through each bulb, the power absorbed by each bulb, and the power delivered by the source. The circuit o f Figure 1.
By definition, in a nvo-terminal circuit element, the current entering each resistor equals the current leaving. To calculate these values, we need to know I. By conservation o f power, the power delivered by the source is the sum o f the absorbed powers, i. A potential problem with series connections o f light bulbs is circuit failure.
If one bulb burns out, i. Parallel circuits continue to operate in the presence o f open-circuit failures and are easier to fix: only the unlit bulb must be replaced. The slope o f the line in the v-i plane is the value o f the resistance. Recall that an ideal voltage source maintains a given voltage, irrespective o f the current demands o f the attached circuit. For now, we must be content with this brief discussion. Chapter 2 will reiterate and expand on these ideas.
Analogously, an ideal current source maintains the given current, irrespective o f the voltage requirements o f the attached circuit. For constant-current sources, as in Figure 1.
A rigorous treatment would require field theory and quantum electronics. A dependent source produces a voltage or current proportional to a voltage across or a current through some other element o f the circuit. The various types o f dependent sources are summarized in Table 1. This absorbed power is dissipated as heat.
Hence, the passive resistor models the heating elements in a stove or toaster oven quite well. In addition, the resistor models the hot resistance o f a light bulb. Throughout the text, the resistor will often represent a fixed electrical load.
However, such a device is rather complex to build and requires such things as the operational amplifier covered in Chapter 4. The various quantities defined and used throughout the chapter have various units. Some simple resisrive circuits were analyzed. These basic laws o f circuit theory are set forth in the next chapter. Alternating current: a sinusoidally time-varying current signal having the form A'sin co? Charge: an electric property o f matter, measured in Coulombs.
Like charges repel, and unlike charges attract each other. Each electron carries the smallest known indivisible amount o f charge equal to - 1. Conductance: reciprocal o f resistance, with siemens S or formerly, mhos as its unit. Conductor: a material, usually a metal, in which electrons can move to neighboring atoms with relative ease.
Conservation o f power energy : the sum o f powers generated by a group o f circuit elements is equal to the sum o f powers absorbed by the remaining circuit elements. Current: the movement o f charges constitutes an electric current. Current is measured in Amperes. One Ampere means movement o f charges through a surface at the rate o f 1 Coulomb per second. Current source: a device that generates electrical current. Dependent controlled current source: a current source whose output current depends on the voltage or current o f some other element in the circuit.
Direct current: a current constant with time. Ideal conductor: offers zero resistance to electron movement. Ideal insulator: offers infinite resistance to electron movement. Independent ideal current source: an ideal device that delivers current as a prescribed function of time, e.
Insulator: a material that opposes easy electron movement. Mho: historical unit of conductance equal to the reciprocal of an ohm. Ohm: unit of resistance. One ohm equals the ratio of IV to lA. Open circuit: connection of infinite resistance or zero conductance. Power: rate of change of work per unit of time.
The proportionality constant R is called the resistance, i. Resistance is measured in ohms: 1 ohm means the voltage is 1 V when the current is 1 A.
Resistivity: the resistance of a conductor is proportional to its length and inversely proportional to its cross-sectional area. The proportionality constant p is called the resistivity of the material. Resistor: physical device that obeys Ohms law. Resistors convert electric energy into heat. Short circuit: connection of zero resistance or infinite conductance.
Siemens: unit of conductance formerly, mho or inverse ohms. Voltage potential difference : positive charge, without obstruction, will move from a higher potential point to a lower potential point, accompanied by a conversion of energy. Voltage is measured in volts; 1 volt between two points A and B means that the energy converted when moving 1 Coulomb of charge between A and B is 1 joule.
Wattage: measure of power consumption. Consider the diagram o f Figure P I. In what direction would the current flow at? Reconsider Figure 1. Plot the current i t through the wire. See Example 1. Figure P 1. A plot o f the current flowing past point A is 8.
Find i t when the charge transported across a surface cutting a conductor is shown in Figure shown on the graph o f Figure P I.
Find the net positive charge transferred in the direction P I. Figure P I. In Figure P I. Find the values of il' 0 ' Figure P I. Consider the circuit o f Figure P I.
Be careftil o f sign. In the circuit shown in Figure P I. In the circuit o f Figure P I. Suppose energy cost in Indiana is 10 cents tors. Using Equation 1. For the circuit o f Figure P 1. The switch S in Figure P I. Use intuitive reasoning. Repeat problem 23 when the switch is In Figure P1. Find the average value o f i tj. T he power delivered by the source in the b W hat is the safe maximum current o f circuit o f Figure P I.
Find the value o f the hot Figure P I. The Find the hot resistance o f each bulb. The power absorbed by the resistor R in the circuit o f Figure P I. Figure PI. Figure P1. A car with such a battery is parked at For music, the car stereo is playing, which shows three lamps, AA, BB, using watts, and some o f the lights are on and C C in a parallel circuit. This is a using watts. W ith this load, the battery will simplified example o f a light circuit on supply approximately 3 M J o f energy before it a car, in your house, or possibly on a will have insufficient stored energy to start the Christmas tree.
Halogen bulb AA uses car. Consider the circuit in Figure P I. W hat is R, the power delivered by each source? For the circuit in Figure P I. For the circuit of Figure P I.
For the circuit of Figure P 1. Find the resistor values. One use o f resistors in electronic circuits is to control current flow, just as dams control water flow along rivers. In this diagram, three resistors are connected in series, and their connecting points are attached to a switch.
As we will learn in this chapter, the resistance o f a series connection is the sum o f the resistances. So with the switch in the low position, the V car battery sees three resistors in series with the motor. W ith less current, there is less power, and the fan motor speed is slow. More current flow increases the fan motor speed. Each successive switch position removes resistance from the circuit, and the fan motor speed increases accordingly.
KirchhofFs Current Law 3. Series Resistances and Voltage Division 5. Parallel Resistances and Current Division 6. Series-Parallel Interconnections 7. Dependent Sources Revisited 8. Model for a Non-ideal Battery 9. Introduce series and parallel resistive circuits. Show that a parallel connection o f resistors has an equivalent conductance equal to the sum o f the conductances in the parallel connection. Explore the calculation o f voltages, currents, and power in a series-parallel connection o f resistances.
Revisit the notion o f a dependent source and use a V C C S to model an amplifier circuit. Describe a practical battery source and look at a general practical source model. The circuits studied in Chapter 1 were interconnections o f resistors and sources that were two-terminal circuit elements. These laws govern the voltage relationships and the current relationships, respectively, o f interconnections o f two- terminal circuit elements. Figure 2. In general, a node is the connection point o f one or more circuit elements.
An important property o f the series connection o f Figure 2. Similarly, in a parallel connection, such as Figure 2. Sources interconnected with circuit elements produce currents through the elements and voltages across the elements. For example, a voltage source connected across Figure 2. Similarly, a current source connected across the circuit o f Figure 2. T he endpoints o f a branch the terminals o f the circuit element are called nodes, as in Figure 2.
The voltage polarity and current direction for the branches in Figures 2. In general, reference directions can be assigned arbitrarily. The conventional assignment o f voltage polarity and current direction to voltage and current soiurces is given in Figure 2. Imagine a number o f branches connected at a common point, as at node A o f Figure 2. The current through each branch has a reference direction indicated by an arrow. If the arrow points toward the node, the reference direction o f the current is entering the node; if the arrow points away from the node, the reference direction o f the current is leaving.
If a current is referenced as leaving a node, then the negative o f the current enters the node, and conversely. S tatem ent 2 : Equivalently, the algebraic sum o f the currents leaving a node is zero for every instant o f time. Further, from physics we know that charge is neither created nor destroyed.
Thus, the charge transported into the node must equal the charge leaving the node because charge cannot accumulate at a node. Moreover, KCL specifies how branch currents interact at a node, regardless o f the type o f element connected to the node.
Referring to Figure 2. S o lutio n By KCL, the sum o f the currents entering the node must be zero. Suppose the current through the voltage source in Figure 2. Three branches connect at a node. All branch currents have reference directions leaving the node.
Two implications o f KCL are o f immediate interest. First, as a general rule, KCL forbids the series connection o f current sources. An open circuit has infinite resistance, or zero conductance. This means that a current source has infinite internal resistance. Find the current through each resistor and the current, supplied by the voltage source.
In Figure 2. Suppose the source voltage in the circuit o f Figure 2. A Gaussian curve or surface is a closed curve such as a circle in a plane or a closed surface such as a sphere or ellipsoid in three dimensions. A Gaussian curve or surface has a well-defined inside and outside. For the two-terminal circuit element o f Figure 2. For the three-terminal device o f Figure 2. Finally, for Figure 2. From these illustrations, one might imagine that the use o f Gaussian surfaces might simplify or provide a short cut to certain branch current computations.
Our objective is to find the current without having to solve a set o f complex circuit equations. In the next chapter, circuits such as the one in Figure 2. Draw a Gaussian surface on the circuit in Figure 2. Before conveying four equivalent versions o f KVL, we first set forth several necessary background concepts.
The first is the notion o f a closed path. In a circuit, a closed path is a connection o f two-terminal elements that ends and begins at the same node and which traverses each node in the connection only once.
A node voltage o f a circuit is the voltage drop from a given node to a reference node. The reference node is usually indicated on the circuit or is taken as ground. The circuit o f Figure 2. The voltage denotes the voltage drop from node A to node E-, denotes the voltage drop from node D to node E, and similarly for the remaining node voltages. Node E, being the reference node, has zero as its node voltage. Statem ent 1: The algebraic sum o f the voltage drops around any closed path is zero at every instant o f time.
Herey and k stand for arbitrary node indices. For example, in Figure 2. Referring back to Figure 2. Thus, by knowing the node voltages o f a circuit, one can easily compute the branch voltages. Again, with reference to Figure 2. A third concept needed for two further equivalent statements o f KVL is that o f a closed node sequence. A closed node sequence is a finite sequence o f nodes that begins and ends at the same node.
A closed node sequence generalizes the notion o f a closed path. Finally, we define the notion o f a connected circuit. In a connected circuit, each node can be reached from any other node by some path through the circuit elements.
Figures 2. However, in Figure 2. Statem ent 3 ; For connected circuits and any node sequence, say A -D -B S tatem ent 4 : For connected circuits, the algebraic sum o f the node-to-node voltages for any closed node sequence is zero for every instant o f time.
Now, consider the closed node sequence E-C-D-E. Find at? Node E is taken as the reference node. Two further implications o f the KVL are o f immediate interest.
O n the other hand, two voltage sources in series can be combined to form a single source, as illustrated in Figure 2. Second, a voltage source supplying 0 V is equivalent to a short circuit, as illustrated in Figure 2.
Also, the internal resistance o f a voltage source is zero. These ideas are dual to those expressed for current sources earlier. The justification is more readily comprehended via the analogy o f the gravitational field, also developed in that section. The distribution o f voltages around closed paths can be viewed as a special case o f this general statement.
During holidays, one often sees strings o f lights hanging between poles or trees. Sometimes these strings consist o f a series connection o f light bulbs. The series connection o f bulbs can be modeled by a series connection o f resistors, with each resistor paired with a specific bulb.
Computing the voltage across each light a very important type o f calculation would then be equivalent to finding the voltage across each o f the resistors in the equivalent circuit model. It is quite common to model electrical loads, such as a light, by resistors. Express the voltage across each resistor in terms o f the input current. For the circuit o f Figure 2. By KVL, the source voltage equals the sum o f the resistor voltages, i. Equation 2. This formula imphes that if a resistance R.
O n the other hand, if a resistance R. One concludes that the voltage distributes around a loop o f resistors in proportion to the value o f each resistance. The equivalent resistance seen by the voltage source for a resistive circuit is implicidy defined by Ohm s law, i. By Equation 2.
A formal discussion o f equivalent resistance and its generalization the Thevenin resistance is taken up in Chapter 6. Example 2. Consider Figure 2.
This means that resistances in series add, i. Find the equivalent resistance seen by the source and the voltage across each resistor in terms o f the source voltages. Express the branch voltages in terms o f and substitute into Equation 2. Subsrimting into Equation 2. Notice that the dependent source increases the resistance o f the two series resistors by 4 Q. Dependent sources can increase or decrease the resistance o f the circuit.
W ith dependent sources, it is even possible to make the equivalent resistance negative. Find the power absorbed by the Q. Suppose the dependent source in the circuit o f Figure 2. Many o f the electrical outlets in the average home are connected in parallel. W hen too many appliances are connected to the same outlet or set o f oudets on the same fused circuit, a fuse will blow or a circuit breaker will open. Although each appliance uses only a portion o f the maximum allowable current for the fused circuit, together, the total current exceeds the allowable limit.
To keep the analysis simple, consider a set o f three parallel resistors driven by a current source. It says that currents distribute through the branches o f a parallel resistive circuit in proportion to the conductance o f the particular branch G. Step 4. Compute the equivalent resistance seen by the source.
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