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| - | --> Real Analog: Chapter 1# | + | {{: |
| - | * {{ : | + | ====== Real Analog: Circuits |
| - | * {{ :learn: | + | "Real Analog: Circuits |
| - | * {{ : | + | * A 12 chapter |
| - | * {{ : | + | * Exercises designed to reinforce textbook |
| - | * {{ :learn: | + | * Homework assignments for every chapter |
| - | * {{ : | + | * Multiple design projects that reinforce |
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| - | | + | **Note:** The textbook can be downloaded as a PDF from the link at the bottom of the page, accessed online via the individual chapters below, or a [[https:// |
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| - | ====== 1. Introduction and Chapter Objectives ====== | ||
| - | In this chapter, we introduce all fundamental concepts associated with circuit analysis. Electrical circuits are constructed in order to direct the flow of electrons to perform a specific task. In other words, in circuit analysis and design, we are concerned with transferring electrical energy in order to accomplish a desired objective. For example, we may wish to use electrical energy to pump water into a reservoir; we can adjust the amount of electrical energy applied to the pump to vary the rate at which water is added to the reservoir. The electrical circuit, then, might be designed to provide the necessary electrical energy to the pump to create the desired water flow rate. | ||
| - | This chapter begins with introduction to the basic parameters which describe the energy in an electrical circuit: //charge//, // | ||
| - | Electrical circuits are composed of interconnected // | + | Design projects use Digilent' |
| - | Finally, we introduce | + | Real Analog: Circuits 1, the Analog Discovery 2, and Analog Parts Kit form the core of a world-class engineering educational program that can be used by themselves or in support of existing curricular materials. Students with their own design kits learn more, learn faster, retain information longer, and have a more enjoyable experience - now every student can take charge of their education for less than the cost of a traditional engineering textbook! |
| - | Please pay special attention to the //passive sign convention// | + | --> |
| + | In this chapter, we introduce all fundamental concepts associated with circuit analysis. Electrical circuits | ||
| + | * [[/ | ||
| - | In summary, this chapter introduces virtually all the basic concepts which will be used throughout this textbook. After this chapter, little information specific to electrical circuit analysis remains to be learned – the remainder of the textbook is devoted to developing analysis methods used to increase the efficiency of our circuit analysis and | + | <-- |
| - | introducing additional circuit components such as capacitors, inductors, and operational amplifiers. The student should be aware, however, that all of our circuit analysis is based on energy transfer among circuit components; this energy transfer is governed by Kirchhoff’s Current Law and Kirchhoff’s Voltage Law and the circuit components are modeled by their voltage-current relationships. | + | |
| - | ---- | + | -->Chapter 2: Circuit Reduction# |
| + | In this chapter, we introduce analysis methods based on circuit reduction. Circuit reduction consists of combining resistances in a circuit to a smaller number of resistors, which are (in some sense) equivalent to the original resistive network. Reducing the number of resistors, of course, reduces the number of unknowns in a circuit. | ||
| + | * [[/ | ||
| - | ==== After Completing this Chapter, You Should be Able to: ==== | + | <-- |
| - | * Define voltage and current in terms of electrical charge | + | |
| - | * State common prefixes and the symbols used in scientific notation | + | |
| - | * State the passive sign convention from memory | + | |
| - | * Determine the power absorbed or generated by an circuit element, based on the current and voltage provided to it | + | |
| - | * Write symbols for independent voltage and current sources | + | |
| - | * State from memory the function of independent voltage and current sources | + | |
| - | * Write symbols for dependent voltage and current sources | + | |
| - | * State governing equations for the four types of dependent sources | + | |
| - | * State Ohm’s Law from memory | + | |
| - | * Use Ohm’s Law to perform voltage and current calculations for resistive circuit elements | + | |
| - | * Identify nodes in an electrical circuit | + | |
| - | * Identify loops in an electrical circuit | + | |
| - | * State Kirchhoff’s current law from memory, both in words and as a mathematical expression | + | |
| - | * State Kirchhoff’s voltage law from memory, both in words and as a mathematical expression | + | |
| - | * Apply Kirchhoff’s voltage and current laws to electrical circuits | + | |
| - | ---- | + | -->Chapter 3: Nodal and Mesh Analysis# |
| + | In cases where circuit reduction is not feasible, approaches are still available to reduce the total number of unknowns in the system. Nodal analysis and mesh analysis are two of these. | ||
| + | * [[/ | ||
| - | ===== 1.1 Basic Circuit Parameters and Sign Conventions ===== | + | <-- |
| - | This section introduces the basic engineering parameters for electric circuits: // | + | |
| - | This section also introduces the //passive sign convention// | ||
| - | ==== Electrical Charge ==== | + | --> |
| - | Electron flow is fundamental to operation of electric circuits; | + | In this chapter, we introduce |
| + | * [[/learn/courses/real-analog-chapter-4/start|Real Analog - Chapter Four]] | ||
| - | ==== Voltage ==== | + | <-- |
| - | //Voltage// is energy per unit charge. Energy is specified relative to some reference level; thus, voltages are more accurately specified as voltage // | + | |
| - | voltage are volts, abbreviated V. The voltage difference between two points indicates the energy necessary to move a unit charge from one of the points to the other. Voltage differences can be either positive or negative. | + | |
| - | Mathematically, | ||
| - | $$ v=\frac{dw}{dq}\ | ||
| - | where //v// is the voltage difference | + | --> |
| + | Operational amplifiers | ||
| + | * [[/learn/courses/real-analog-chapter-5/start|Real Analog - Chapter Five]] | ||
| - | ==== Current ==== | + | <-- |
| - | Current is the rate at which charge is passing a given point. Current is specified at a particular point in the circuit, and is not relative to a reference. Since current is caused by charge in motion, it can be thought of as indicating //kinetic// energy. | + | |
| - | Mathematically, | ||
| - | $$ i=\frac{dq}{dw}\ | + | --> |
| + | This chapter begins with an overview of the basic concepts associated with energy storage. This discussion focuses not on electrical systems, but instead introduces the topic qualitatively in the context of systems with which the reader is already familiar. The goal is to provide a basis for the mathematics, | ||
| + | * [[/ | ||
| - | Where //i// is the current in amperes, //q// is the charge in coulombs, and //t// is the time in seconds. Thus, current is the time rate of change of charge and units of charge are coulombs per second, or //amperes// (abbreviated as // | + | <-- |
| - | ==== Power ==== | ||
| - | An electrical system is often used to drive a non-electrical system (in an electric stove burner, for example, electric energy is converted to heat). Interactions between electrical and non-electrical systems are generally described in terms of //power//. Electrical power associated with a particular circuit element is the product of the current passing through the element and the voltage difference across the element. This is often written as: | ||
| - | $$ p(t)=v(t) \cdot i(t) (Eq. 1.3) $$ | + | --> |
| + | First order systems are, by definition, systems whose input-output relationship is a first order differential equation. A first order differential equation contains a first order derivative but no derivative higher than first order - the order of a differential equation is the order of the highest order derivative present in the equation. | ||
| + | * [[/ | ||
| - | Where //p(t)// is the // | + | <-- |
| - | ==== International System of Units and Prefixes ==== | ||
| - | We will use the international system of units (SI). The scales of parameters that are of interest to engineers can vary over many orders of magnitudes. For example, voltages experienced during lightning strikes can be on the order of 10< | ||
| - | ^ Multiple | + | -->Chapter 8: Second Order Circuits# |
| - | | 10<sup>9</ | + | Second order systems are, by definition, systems whose input-output relationship is a second order differential equation. A second order differential equation contains a second order derivative but no derivative higher than second order. |
| - | | 10< | + | * [[/learn/courses/real-analog-chapter-8/start|Real Analog |
| - | | 10< | + | |
| - | | 10< | + | |
| - | | 10< | + | |
| - | | 10< | + | |
| - | | 10< | + | |
| - | //Table 1.1. SI prefixes.// | + | <-- |
| - | ==== Passive Sign Convention ==== | ||
| - | A general two-terminal electrical circuit element is shown in Fig. 1.1. In general, there will be some current, //i//, flowing through the element and some voltage difference, //v//, across its terminals. Note that we are currently representing both voltage and current as constants, but none of the assertions made in this chapter change if they are functions of time. | ||
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| - | {{ : | ||
| - | The __assumed__ direction of the current, //i//, passing through | + | --> |
| + | In this chapter, we will provide a very brief introduction to the topic of state variable modeling. The brief presentation provided here is intended simply | ||
| + | * [[/learn/courses/real-analog-chapter-9/start|Real Analog - Chapter Nine]] | ||
| + | <-- | ||
| - | ---- | ||
| - | === Example 1.1: === | + | --> |
| - | Three amperes (3 A) of current is passing through a circuit element connecting nodes a and b. The current is flowing from node a to node b. The physical situation can be represented schematically by any of the figures shown below - all four figures represent the same current flow and direction. | + | In this chapter we will study dynamic systems which are subjected |
| + | * [[/ | ||
| - | {{ : | + | <-- |
| - | ---- | ||
| - | The __assumed__ polarity | + | --> |
| + | In this chapter we discuss representation | ||
| + | * [[/learn/courses/real-analog-chapter-11/start|Real Analog - Chapter Eleven]] | ||
| + | <-- | ||
| - | ---- | ||
| - | === Example 1.2: === | + | --> |
| - | A 5 volt (5 V) voltage potential difference is applied across a circuit element connecting nodes a and b. The voltage at node a is positive relative to the voltage at node b. The physical situation can be represented schematically by either | + | In this chapter we will address the issue of power transmission via sinusoidal |
| + | * [[/ | ||
| - | {{ : | + | <-- |
| - | ---- | ||
| - | The assumed voltage polarity and current direction are not individually significant | + | --> |
| + | Download | ||
| + | * {{ : | ||
| + | <-- | ||
| - | ---- | + | -->Complete Chapters Solutions# |
| - | + | Download | |
| - | === Example 1.3: === | + | * {{ : |
| - | The passive sign convention is satisfied for either of the two voltage-current definitions shown below - the current is assumed to enter the positive voltage node. | + | |
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| - | The passive sign convention is __not__ satisfied for either of the two voltage current definitions shown below - the current is assumed to enter the negative voltage node. | + | |
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| - | **Note**: | + | |
| - | Many students attempt to choose current directions and voltage polarities so that their calculations result in positive values for voltages and currents. In general, this is a wast of time - it is best to arbitrarily assume __either__ a voltage polarity of current direction for each circuit element. | + | |
| - | + | ||
| - | Choice of a positive direction for current dictates the choice of positive voltage polarity, per Fig. 1.1. Choice of a positive voltage polarity dictates the choice of positive current direction, per Fig. 1.1. | + | |
| - | + | ||
| - | Analysis of the circuit is performed using the above assumed signs for voltage and current. The sign of the results indicates whether the assumed choice of voltage polarity and current direction was correct. A positive magnitude of a calculated voltage indicates that the assumed sign convention is correct; a negative magnitude indicates that the actual voltage polarity is opposite to the assumed polarity. Likewise, a positive magnitude of a calculated current indicates that the assumed current direction is correct; a negative magnitude indicates that the current direction is opposite to that assumed. | + | |
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| - | + | ||
| - | ==== Voltage Subscript and Sign Conventions ==== | + | |
| - | The assumed sign convention for voltage potentials is sometimes expressed by using subscripts. The first subscript denotes the node at which the __positive__ voltage polarity is assumed and the second subscript is the __negative__ voltage polarity. For example, //v<sub>ab</ | + | |
| - | + | ||
| - | ==== Reference Voltages and Ground ==== | + | |
| - | For convenience, | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | + | ||
| - | ---- | + | |
| - | === Example 1.4: === | + | |
| - | The two figures below show identical ways of specifying the voltage across a circuit element. In the circuit to the left, the voltage //v// is the voltage potential between nodes a and b, with the voltage at node a being assumed positive relative to voltage at node b. This can be equivalently specified as // | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Power and Sign Conventions ==== | + | |
| - | The sign of the voltage across an element relative to the sign of the current through the element governs the sign of the power. Equation (1.3) above defines power as the product of the voltage times current: | + | |
| - | + | ||
| - | $$ P=vi $$ | + | |
| - | + | ||
| - | The power is // | + | |
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| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | === Example 1.5: === | + | |
| - | In Fig. (a) below, the element agrees with the passive sign convention since a positive current is entering the positive voltage node. Thus, the element of Fig. (a) is absorbing energy. In Fig. (b), the element is absorbing power - positive current is leaving the negative voltage node, which implies that positive current enters the positive voltage node. The element of Fig. (c) generates power; negative current enters the positive voltage node, which disagrees with the passive sign convention. Fig. (d) also illustrates an element which is generating power, since positive current is entering a negative voltage node. | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Section Summary ==== | + | |
| - | * In this text, we will be primarily concerned with the movement of electrical charge. Electrical charge motion is represented by voltage and current. Voltage indicates the energy change associated with the movement of a charge from one location to another, while current is indicative of the rate of current motion past a particular point. | + | |
| - | * Voltage is an energy difference between two physically separated points. The polarity of a voltage is used to indicate which point is to be assumed to be at the higher energy level. The positive terminal (+) is assumed to be at a higher voltage than the negative terminal (-). A negative voltage value simply indicates that the actual voltage polarity is opposite to the assumed polarity. | + | |
| - | * The sign of the current indicates the assumed direction of charge motion past a point. A change in the sign of the current value indicates that the current direction is opposite to the assumed direction. | + | |
| - | * The assumed polarity of the voltage across a passive circuit element must be consistent with the __assumed__ current direction through the element. The assumed positive direction for current must be such that positive current enters the positive voltage terminal of the element. Since this sign convention is applied only to passive elements, it is known as the //passive sign convention// | + | |
| - | * The assumed current direction __or__ the assumed voltage polarity can be chosen arbitrarily, | + | |
| - | * The power absorbed or generated by an electrical circuit component is the product of the voltage difference across the element and the current through the element: $ p=iv $. The relative sign of the voltage and current are set according to the passive sign convention. Positive power implies that the voltage and current are consistent with the passive sign convention (the element absorbs or dissipates energy) while negative power indicates that the relative signs between voltage and current are opposite to the passive sign convention (the element generates or supplies energy to the circuit). | + | |
| - | + | ||
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| - | ---- | + | |
| - | + | ||
| - | === Exercises === | + | |
| - | 1. Assign reference voltage and current directions to the circuit elements represented by the shaded boxes in the circuits below. | + | |
| - | {{ : | + | |
| - | 2. Either the reference voltage polarity or the reference current direction is provided for the circuit elements below. Provide the appropriate sign convention for the missing parameters. | + | |
| - | {{ : | + | |
| - | 3. Determine the magnitude and direction of the current in the circuit element below if the element absorbs 10W. | + | |
| - | {{ : | + | |
| - | 4. Determine the power absorbed or supplied by the circuit element below. State whether the power is absorbed or supplied. | + | |
| - | {{ : | + | |
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| - | ---- | + | |
| - | + | ||
| - | ===== 1.2 Power Sources ===== | + | |
| - | Circuit elements are commonly categorized as either //passive// or //active//. A circuit element is passive if the __total__ amount of energy it delivers to the rest of the circuit (over all time) is non-positive (passive elements can __temporarily__ deliver energy to a circuit, but only if the energy was previously stored in the passive element by the circuit). An active circuit element has the ability to create and provide power to a circuit from mechanisms __external__ to the circuit. Examples of active circuit elements are batteries (which create electrical energy from chemical processes) and generators (which create electrical energy from mechanical processes, such as spinning a turbine). | + | |
| - | + | ||
| - | In this section we consider some very important active circuit elements: voltage and current sources. We will discuss two basic types of sources: // | + | |
| - | + | ||
| - | ==== Independent Voltage Sources ==== | + | |
| - | An independent voltage source maintains a specified voltage across its terminals. The symbol used to indicate a voltage source delivering a voltage // | + | |
| - | + | ||
| - | Note that the sign of the voltage being applied by the source is provided on the source symbol - there is no need to assume a voltage polarity for voltage sources. The current direction, however, is unknown and must be determined (if necessary) from an analysis of the overall circuit. | + | |
| - | + | ||
| - | //Ideal// voltage sources provide a specified voltage regardless of the current flowing through the device. Ideal sources can, obviously, provide infinite power; all real sources will provide only limited power to the circuit. We will discuss approaches for modeling non-ideal sources in later chapters. | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | ==== Independent Current Sources ==== | + | |
| - | An independent current source maintains a specified current. This current is maintained regardless of the voltage differences across the terminals. The symbol used to indicate a current source delivering a current // | + | |
| - | + | ||
| - | Note that the sign of the current being applied by the source is provided on the source symbol - there is no need to assume a current direction. The voltage polarity, however, is unknown and must be determined (if necessary) from an analysis of the overall circuit. | + | |
| - | + | ||
| - | //Ideal// current sources provide a specified current regardless of the voltage difference across the device. Ideal current sources can, like ideal voltage sources, provide infinite power; all real sources will provide only limited power to the circuit. We will discuss approaches for modeling non-ideal current sources in later chapters. | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | ==== Dependent Sources ==== | + | |
| - | Dependent sources can be either voltage or current sources; Fig. 1.5(a) shows the symbol for a dependent voltage source and Fig. 1.5(b) shows the symbol for a dependent current source. Since each type of source can be controlled by either a voltage or current, there are four types of dependent current sources: | + | |
| - | * Voltage-controlled voltage source (VCVS) | + | |
| - | * Current-controlled voltage source (CCVS) | + | |
| - | * Voltage-controlled current source (VCCS) | + | |
| - | * Current-controlled current source (CCCS) | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | Figure 1.6 illustrates the voltage-controlled dependent sources, and Fig. 1.7 illustrates the current-controlled dependent sources. In all cases, some electrical circuit exists which has some voltage and current combination at its terminals. Either the voltage or current at these terminals is used to set the voltage or current of the dependent source. The parameters µ and β in Figs. 1.6 and 1.7 are dimensionless constants. //µ// is the //voltage gain// of a VCVS and //β// is the //current gain// of a CCCS. The parameter //r// is the voltage-to-current ratio of a CCVS and has units of volts/ | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | ==== Section Summary ==== | + | |
| - | * Circuit elements can be either active or passive. Active elements provide electrical energy from a circuit from sources outside the circuit; active elements can be considered to create energy (from the standpoint of the circuit, anyway). Passive elements will be discussed in section 1.3, when we introduce resistors. Active circuit elements introduced in this section are ideal independent and dependent voltage and current sources. | + | |
| - | * Ideal independent sources presented in this section are voltage and current sources. Independent voltage sources deliver the specified voltage, regardless of the current demanded of them. Independent current sources provide the specified current, regardless of the voltage levels required to provide this current. Devices such as batteries are often modeled as independent sources. | + | |
| - | * Dependent sources provide a voltage or current which is controlled by a voltage or current elsewhere in the circuit. Devices such as operational amplifiers and transistors are often modeled as dependent sources. We will revisit the subject of dependent sources in chapter 5 of this text, when we discuss operational amplifier circuits. | + | |
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| - | ==== Exercises ==== | + | |
| - | 1. The ideal voltage source shown in the circuit below delivers 12V to the circuit element shown. What is the current //I// through the circuit element? | + | |
| - | {{ : | + | |
| - | 2. The ideal current source shown in the circuit below delivers 2A to the circuit element shown. What is the voltage difference //V// across the circuit element? | + | |
| - | {{ : | + | |
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| - | ===== 1.3 Resistors and Ohm's Law ===== | + | |
| - | Resistance is a property of all materials - this property characterizes the loss of energy associated with passing an electrical current through some conductive element. Resistors are circuit elements whose characteristics are dominated by this energy loss. Since energy is always lost when current is passed through an electrical circuit element, all electrical elements exhibit resistive properties which are characteristic of resistors. Resistors are probably the simplest and most commonly used circuit elements. | + | |
| - | + | ||
| - | All materials impede the flow of current through them to come extent. Essentially, | + | |
| - | + | ||
| - | The relationship between voltage and current for a resistor is given by // | + | |
| - | + | ||
| - | $$ v(t)=Ri(t) | + | |
| - | + | ||
| - | Where voltage and current are explicitly denoted as functions of time. Note that in Fig. 1.8, the current is flowing from a higher voltage potential to a lower potential, as indicated by the polarity (+ and -) of the voltage and the arrow indicating direction of the current flow. The relative polarity between voltage and current for a resistor __must__ be as shown in Fig. 1.8; the current enters the node at which the voltage potential is highest. Values of resistance, //R// are __always__ positive, and resistors __always__ absorb power. | + | |
| - | + | ||
| - | **Note:** The voltage-current relationship for resistors always agrees with the passive sign convention. Resistors always absorb power. | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | Figure 1.9 shows a graph of //v// vs. //i// according to equation (1.5); the resulting plot is a straight line with slope //R//. Equation (1.5) thus describes the voltage-current relationship for a //linear// resistor. Linear resistors do not exist in reality - all resistors are // | + | |
| - | + | ||
| - | **Note:** | + | |
| - | For the most part, we will consider only linear resistors in this text. These resistors obey the linear voltage-current relationship shown in equation (1.5). All real resistors are nonlinear to some extent, but can often be assumed to operate as linear resistors over some reange of voltages and currents. | + | |
| - | + | ||
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | ==== Conductance ==== | + | |
| - | // | + | |
| - | + | ||
| - | $$ G=\frac{1}{R} | + | |
| - | + | ||
| - | The unit for conductance is siemens, abbreviated //S//. Ohm's law, written in terms of conductance, | + | |
| - | + | ||
| - | $$ i(t)=Gv(t) | + | |
| - | + | ||
| - | Some circuit analyses can be performed more easily and interpreted more readily if the elements' | + | |
| - | + | ||
| - | + | ||
| - | **Note:** | + | |
| - | In section 1.2, we characterized a current-controlled voltage source in terms of a parameter with units of ohms, since it had units of volts/amp. We characterized a voltage-controlled current source in terms of a parameter with units of siemens, since it had units of amps/volts. | + | |
| - | + | ||
| - | + | ||
| - | ==== Power Dissipation ==== | + | |
| - | Instantaneous power was defined by equation (1.3) in section 1.1 as: | + | |
| - | + | ||
| - | $$ P(t)=v(t) \cdot i(t)$$ | + | |
| - | + | ||
| - | For the special case of a resistor, we can re-write this (by substituting equation (1.5) into the above) as: | + | |
| - | + | ||
| - | $$ P(t)=Ri^2(t)= \frac{v^2(t)}{R} | + | |
| - | + | ||
| - | Likewise, we can write the power dissipation in terms of the conductance of a resistor as: | + | |
| - | + | ||
| - | $$ P(t)= \frac{i^2(t)}{G}=Gv^2(t) | + | |
| - | + | ||
| - | + | ||
| - | **Note:** | + | |
| - | We can write the power dissipation from a resistor in terms of the resistance or conductance of the resistor and __either__ the current through the resistor __or__ the voltage drop across the resistor. | + | |
| - | + | ||
| - | + | ||
| - | ==== Practical Resistors ==== | + | |
| - | All materials have some resistance, so all electrical components have non-zero resistance. However, circuit design often relies on implementing a specific, desired resistance at certain locations in a circuit; resistors are often placed in the circuit at these points to provide the necessary resistance. Resistors can be purchased in certain standard values. Resistors are manufactured in a variety of ways, though most commonly available commercial resistors are carbon composition or wire-wound. Resistors can have either a fixed or variable resistance. | + | |
| - | + | ||
| - | //Fixed// resistors provide a single specified resistance value and have two terminals, as shown in Fig. 1.5 above. // | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | Resistors, which are physically large enough, will generally have their resistance value printed directly on them. Smaller resistors generally will use a color code to identify their resistance value. The color coding scheme is provided in Fig. 1.12. The resistance values indicated on the resistor will provide a //nominal// resistance value for the component; the actual resistance value for the component will vary from this by some amount. The expected tolerance between the allowable actual resistance values and the nominal resistance is also provided on the resistor, either printed directly on the resistor or provided as an additional color band. The color-coding scheme for resistor tolerances is also provided in Fig. 1.12. | + | |
| - | + | ||
| - | {{ : | + | |
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| - | ---- | + | |
| - | + | ||
| - | ==== Example 1.6 ==== | + | |
| - | A resistor has the following color bands below. Determine the resistance value and tolerance: | + | |
| - | + | ||
| - | * First band (A): Red | + | |
| - | * Second band (B): Black | + | |
| - | * Exponent: Orange | + | |
| - | * Tolerance: Gold | + | |
| - | + | ||
| - | Resistance = (20+0)x10< | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Section Summary ==== | + | |
| - | * The relationship between voltage and current for a resistor is Ohm's Law: //v=iR//. Since a resistor only dissipates energy, the voltage and current for a resistor must always agree with the passive sign convention. | + | |
| - | * As noted in section 1.2, circuit elements can be either active or passive. Resistors are passive circuit elements. Passive elements can store or dissipate electrical energy provided to them by the circuit; they can subsequently return energy to the circuit which they have previously stored, but they cannot create energy. Resistors cannot store electrical energy, they can only dissipate energy by converting it to heat. | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Exercises ==== | + | |
| - | 1. The ideal voltage source shown in the circuit below delivers 18V to the resistor shown. What is the current //I// through the resistor? | + | |
| - | {{ : | + | |
| - | + | ||
| - | 2. The ideal current source shown in the circuit below delivers 4mA to the resistor shown. What is the voltage difference //V// across the resistor? | + | |
| - | {{ : | + | |
| - | + | ||
| - | 3. The ideal voltage source shown in the circuit below delivers 10V to the resistor shown. What is the current //I// in the direction shown? | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Kirchhoff' | + | |
| - | This section provides some basic definitions and background information for two important circuit analysis tools: Kirchhoff' | + | |
| - | + | ||
| - | We will use a // | + | |
| - | + | ||
| - | The lumped parameters approach toward modeling circuits is appropriate if the voltage and currents in the circui change slowly relative to the rate with which information can be transmitted through the circuit. Since information propagates in an electrical circuit at a rate comparable to the speed of light and circuit dimensions are relatively small, this modeling | + | |
| - | + | ||
| - | An alternate approach to circuit analysis is a // | + | |
| - | + | ||
| - | ==== Nodes ==== | + | |
| - | Identification of circuit nodes will be extremely important to the application of Kirchhoff' | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Example 1.7 ==== | + | |
| - | The circuit below has four nodes, as shown. A common error for beginning students is to identify points a, b, and c as being separate nodes, since they appear as separate points on the circuit diagram. However, these points are connected by perfect connectors (no circuit elements are between points a, b, and c) and thus the points are at the same voltage and are considered electrically to be at the same point. Likewise, points d, e, f, and g are at the same voltage potential and are considered to be the same node. Node 2 interconnects two circuit elements (a resistor and a source) and must be considered as a separate node. Likewise, node 4 interconnects two circuit elements and qualifies as a node. | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Loops ==== | + | |
| - | A //loop// is any closed path through the circuit which encounters no node more than once. | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Example 1.8 ==== | + | |
| - | There are six possible ways to loop through the circuit of the previous example. These loops are shown below. | + | |
| - | {{ : | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Kirchhoff' | + | |
| - | Kirchhoff' | + | |
| - | + | ||
| - | Kirchhoff' | + | |
| - | + | ||
| - | //**The algebraic sum of all currents entering (or leaving) a node is zero.**// | + | |
| - | + | ||
| - | A common alternate statement for KCL is: | + | |
| - | + | ||
| - | //**The sum of the currents entering any node equals the sum of the currents leaving the node.**// | + | |
| - | + | ||
| - | A general mathematical statement for Kirchhoff' | + | |
| - | + | ||
| - | $$ \sum_{k=l}^{N} i_k(t)=0 | + | |
| - | + | ||
| - | **Note**: Current directions (entering or leaving the node) are based on __assumed__ directions of currents in the circuit. As long as the assumed directions of the currents are consistent from node to node, the final result of the analysis will reflect the __actual__ current directions in the circuit. | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Example 1.9 ==== | + | |
| - | In the figure below, the assumed directions of // | + | |
| - | {{ : | + | |
| - | + | ||
| - | If we (arbitrarily) choose a sign convention such that the currents entering the node are positive, then currents leaving the node are negative and KCL applied at this node results in: | + | |
| - | + | ||
| - | $$ i_1(t) + i_2(t) - i_3(t)=0 $$ | + | |
| - | + | ||
| - | If, on the other hand, we choose a sign convention that currents entering the node are negative, then currents leaving the node are positive and KCL applied at this node results in: | + | |
| - | + | ||
| - | $$ -i_1(t) - i_2(t) + i_3(t) = 0 $$ | + | |
| - | + | ||
| - | These two equations are the same; the second equation is simply the negative of the first equation. Both of the above equations are equivalent to the statement: | + | |
| - | + | ||
| - | $$ i_1(t) + i_2(t) = i_3(t) $$ | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Example 1.10 ==== | + | |
| - | Use KCL to determine the value of the current //i// in the figure below: | + | |
| - | {{ : | + | |
| - | + | ||
| - | Summing the currents entering the node results in: | + | |
| - | + | ||
| - | $$ 4A - (-1A) - 2A - i = 0 \Rightarrow 4A + 1A - 2A = 3A $$ | + | |
| - | + | ||
| - | And //i=3A//, __leaving__ the node. | + | |
| - | + | ||
| - | In the figure below, we have reversed our assumed direction of //i// in the above circuit: | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | Now, if we sum currents entering the node: | + | |
| - | + | ||
| - | $$ 4A - (-1A) - 2A - i = 0 \Rightarrow i = -4A - 1A + 2A = -3A $$ | + | |
| - | + | ||
| - | So now //i=-3A//, __entering__ the node. The negative sign corresponds to a change in direction, so we can interpret this result to a +3A current __leaving__ the node, which is consistent with out previous result. Thus, the assumed current direction has not affected our results. | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | We can generalize Kirchhoff' | + | |
| - | + | ||
| - | //**The algebraic sum of all currents entering (or leaving) any enclosed surface is zero**//. | + | |
| - | + | ||
| - | Applying this statement to the circuit of Fig. 1 results in: | + | |
| - | + | ||
| - | $$ i_1 + i_2 + i_3 = 0 $$ | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | ==== Kirchhoff' | + | |
| - | Kirchhoff' | + | |
| - | + | ||
| - | Kirchhoff' | + | |
| - | + | ||
| - | //**The algebraic sum of all voltage differences around any closed loop is zero.**// | + | |
| - | + | ||
| - | An alternate statement of this law is: | + | |
| - | + | ||
| - | //**The sum of the voltage rises around a closed loop must equal the sum of the voltage drops around the loop.**// | + | |
| - | + | ||
| - | A general mathematical statement for Kirchhoff' | + | |
| - | + | ||
| - | $$ \sum_{k=l}^{N} V_k(t)=0 | + | |
| - | + | ||
| - | Where // | + | |
| - | + | ||
| - | **Note:** | + | |
| - | Voltage polarities are based on __assumed__ polarities of the voltage differences in the loop. As long as the assumed directions of the voltages are consistent from loop to loop, the final result of the analysis will reflect the __actual__ voltage polarities in the circuit. | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Example 1.11 ==== | + | |
| - | In the figure below, the assumed (or previously known) polarities of the voltages v< | + | |
| - | + | ||
| - | Our sign convention for applying signs to the voltage polarities in our KVL equations will be as follows: when traversing the loop, if the positive terminal of a voltage difference is encountered before the negative terminal, the voltage difference will be interpreted as __positive__ in the KVL equation. If the negative terminal is encountered first, the voltage difference will be interpreted as __positive__ in the KVL equation. We use this sign convention for convenience; | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | Applying KVL to the loop a-b-e-d-a, and using our sign convention as above results in: | + | |
| - | + | ||
| - | $$ v_1 - v_4 - v_6 - v_3 = 0 $$ | + | |
| - | + | ||
| - | The starting point of the loop and the direction that we loop in is abitrary; we could equivalently write the same loop equation as loop d-e-b-a-d, in which case out equation would become: | + | |
| - | + | ||
| - | $$ v_6 + v_4 - v_1 + v_3 = 0 $$ | + | |
| - | + | ||
| - | This equation is identical to the previous equation, the only difference is that the signs of all variables has changed and the variables appear in a different order in the equation. | + | |
| - | + | ||
| - | We now apply KVL to the loop b-c-e-b, which results in: | + | |
| - | + | ||
| - | $$ -v_2 +v_5 +v_4 = 0 $$ | + | |
| - | + | ||
| - | Finally, application of KVL to the loop a-b-c-e-d-a provides: | + | |
| - | + | ||
| - | $$ v_1 -v_2 + v_5 - v_6 - v_3 = 0 $$ | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Application Examples: Solving for Circuit Element Variables ==== | + | |
| - | Typically, when analyzing a circuit, we will need to determine voltages and/or currents in one or more elements in the circuit. In this chapter, we discuss use of the tools presented in previous chapters for circuit analysis. | + | |
| - | + | ||
| - | The complete solution of a circuit consists of determining the voltages and currents for __every__ elements in the circuit. A complete solution of a circuit can be obtained by: | + | |
| - | + | ||
| - | - Writing a voltage-current relationship for each element in the circuit (e.g. write Ohm's Law for the resistors). | + | |
| - | - Applying KCL at all but one of the nodes in the circuit. | + | |
| - | - Applying KVL for all but one of the loops in the circuit. | + | |
| - | + | ||
| - | This approach will typically result in a set of N equations in N unknowns, the unknowns consisting of the voltages and currents for each element in the circuit. Methods exist for defining a reduced set of equations or a complete analysis of a circuit; these approaches will be presented in later chapters. | + | |
| - | + | ||
| - | If KCL is written for __every__ node in the circuit and KVL written for __every__ loop in the circuit, the resulting set of equations will typically be over-determined and the resulting equations will, in general, not be independent. That is, there will be more than N equations in N unknowns and some of the equations will carry redundant information. | + | |
| - | + | ||
| - | Generally, we do not need to determine all the variables in a circuit. This often means that we can write fewer equations than those listed above. The equations to be written will, in these cases, be problem dependent and are often at the discretion of the person doing the analysis. | + | |
| - | + | ||
| - | Examples of using Ohm's Law, KVL, and KCL for circuit analysis are provided below. | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Example 1.12 ==== | + | |
| - | For the circuit below, determine // | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | We are free to arbitrarily choose wither the voltage polarity or the current direction in each element. Our choices are shown below: | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | Once the above voltage polarities and current directions are chosen, we must choose all other parameters in a way that satisfies the passive sign convention (current must enter the positive voltage polarity node). Our complete definition of all circuit parameters is shown below: | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | We now apply the steps outlined above for an exhaustive circuit analysis. | + | |
| - | + | ||
| - | - Ohm's aw, applied for each resistor, results in: $ v_1=(1 \Omega )i_1 $; $v_3=(3 \Omega )i_3 $; $v_6 = 6 \Omega )i_6$ | + | |
| - | - KCL, applied at node a: $ i_1 + i_3 - i_6 = 0 $ | + | |
| - | - KVL, applied over any two of the three loops in the circuit: $ -12V + v_1 - v_3 = 0 $; $v_3 + v_6 = 0 $ | + | |
| - | + | ||
| - | The above provide six equatoins in six unknowns. Solving these for // | + | |
| - | + | ||
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Example 1.13 ==== | + | |
| - | Determine // | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | We choose voltages and currents as shown below. Since // | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | KVL around the single loop in the ciruit does not help us - the voltage across the current source is unknown, so inclusion of this parameter in a KVL equation simply introduces an additional unknown to go with the equation we write. KVL would, however, be useful if we wished to determine the voltage across the current source. | + | |
| - | + | ||
| - | KCL at node a tells us that i< | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Section Summary ==== | + | |
| - | * Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL) govern the interactions between circuit elements. Governing equations for a circuit are created by applying KVL and KCL and applying the circuit element governing equations, such as Ohm’s Law. | + | |
| - | * Kirchhoff’s current law states that the sum of the currents entering or leaving a node must be zero. A node in a circuit is an point which has a unique voltage. | + | |
| - | * A node is a point of interconnection between two or more circuit elements. A circuit node has a particular voltage. Nodes can be “spread out” with perfect conductors. | + | |
| - | * Kirchhoff’s voltage law states that the sum of the voltage differences around any closed loop in a circuit must sum to zero. A loop in a circuit is any path which ends at the same point at which it starts. | + | |
| - | * A loop is a closed path through a circuit. Loops end at the same node at which they start, and typically are chosen so that no node is encountered more than once. | + | |
| - | + | ||
| - | ---- | + | |
| - | + | ||
| - | ==== Exercises ==== | + | |
| - | - For the circuit below, determine: | + | |
| - | - The current through the 2Ω resistor | + | |
| - | - The current through the 1Ω resistor | + | |
| - | - The power (absorbed or generated) by the 4V power source | + | |
| - | + | ||
| - | {{ : | + | |
| <-- | <-- | ||
| - | <-- Real Analog: Chapter 2# | ||
| - | * {{ : | ||
| - | * {{ : | ||
| - | * {{ : | ||
| - | * {{ : | ||
| - | * {{ : | ||
| - | * {{ : | ||
| - | * {{ : | ||
| - | * {{ : | ||
| - | * {{ : | ||
| - | * {{ : | ||
| - | * {{ : | ||
| - | |||
| - | * {{ : | ||
| - | * {{ : | ||
| - | |||
| - | ====== 2. Introduction and Chapter Objectives ====== | ||
| - | In [[https:// | ||
| - | |||
| - | In the next few chapters, we will still apply Kirchhoff' | ||
| - | |||
| - | In this chapter, we introduce analysis methods based on //circuit reduction// | ||
| - | |||
| - | We begin our discussion of circuit reduction techniques by presenting two specific, but very useful, concepts: //Series// and // | ||
| - | consequences of a voltage or current measurement. | ||
| - | |||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== After Completing this Chapter, You Should be Able to: ==== | ||
| - | * Identify series and parallel combinations of circuit elements | ||
| - | * Determine the equivalent resistance of series resistor combinations | ||
| - | * Determine the equivalent resistance of parallel resistor combinations | ||
| - | * State voltage and current divider relationships from memory | ||
| - | * Determine the equivalent resistance of electrical circuits consisting of series and parallel combinations of resistors | ||
| - | * Sketch equivalent circuits for non-ideal voltage and current meters | ||
| - | * Analyze circuits containing non-ideal voltage or current sources | ||
| - | * Determine the effect of non-ideal meters on the parameter being measured | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ===== 2.1 Series Circuit Elements and Voltage Division ===== | ||
| - | There are a number of common circuit element combinations that are quite easily analyzed. These “special cases” are worth noting since many complicated circuits contain these circuit combinations as sub-circuits. Recognizing these sub-circuits and analyzing them appropriately can significantly simplify the analysis of a circuit. | ||
| - | |||
| - | This chapter emphasizes two important circuit element combinations: | ||
| - | |||
| - | ==== Series Connections ==== | ||
| - | Circuit elements are said to be connected in //series// if all of the elements carry the same current. An example of two circuit elements connected in series is shown in Fig. 2.1. Applying KCL at node a and taking currents out of the node as positive we see that: | ||
| - | |||
| - | $$-i_1+i_2=0$$ | ||
| - | |||
| - | or | ||
| - | |||
| - | $$i_1=i_2 | ||
| - | |||
| - | Equation (2.1) is a direct outcome of the fact that the (single) node a in Fig. 2.1 interconnects only two elements - there are no other elements connected to this node through which curren can be diverted. This observation is so apparent (in many cases((If there is any doubt whether the elements are in series, apply KCL! Assuming elements are in series which are not in series can have disastrous consequences.))) that equation (2.1) is generally written by inspection for series elements such as those shown in Fig. 2.1 __without__ explicitly writing KCL. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | When resistors are connected in series, a simplification of the circuit is possible. Consider the resistive circuit shown in Fig. 2.2(a). Since the resistors are in series, they both carry the same current. Ohm's law gives: | ||
| - | |||
| - | \begin{align*} | ||
| - | v_1 = R_1i \\ | ||
| - | v_2=R_2i \\ | ||
| - | (Eq.2.2) | ||
| - | \end{align*} | ||
| - | |||
| - | Applying KVL around the loop: | ||
| - | |||
| - | $$ -v+v_1+v_2=0 \Rightarrow v=v_1+v_2 | ||
| - | |||
| - | Substituting equations (2.2) into equation (2.3) and solving for the current //i// results in: | ||
| - | |||
| - | $$ i = \frac{v}{R_1+R_2} | ||
| - | |||
| - | Now cinsider the circuit of Fig. 2.2(b). Application of Ohm's law to this circuit and solution for the current //i// gives: | ||
| - | |||
| - | $$ i = \frac{v}{R_{eq}} | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | Comparing equation (2.4) with equation (2.5), we can see that the circuits of Figs. 2.2(a) and 2.2(b) are indistinguishable if we select: | ||
| - | |||
| - | $$R_{eq}=R_1+R_2 | ||
| - | |||
| - | Figures 2.2(a) and 2.2(b) are called // | ||
| - | |||
| - | This result can be generalized to a series combination of //N// resistances as follows: | ||
| - | |||
| - | A series combination of //N// resistors R< | ||
| - | |||
| - | ==== Voltage Division ==== | ||
| - | Combining equations (2.2) with equation (2.4) results in the following expressions for // | ||
| - | |||
| - | $$v_1= \frac{R_1}{R_1+R_2}v | ||
| - | |||
| - | $$v_2= \frac{R_2}{R_1+R_2}v | ||
| - | |||
| - | These results are commonly called //voltage divider// relationships, | ||
| - | |||
| - | The above results can be generalized for a series combination of //N// resistance as follows: | ||
| - | |||
| - | The voltage drop across any resistor in a series combination of N resistances is proportional to the total voltage drop across the combination of resistors. The constant of proportionality is the same as the ratio of the individual resistor value to the total resistance of the series combination. For example, the voltage drop of the // | ||
| - | |||
| - | $$ v_k={R_k}{R_1+R_2+ \cdots +R_N}v | ||
| - | |||
| - | |||
| - | ---- | ||
| - | |||
| - | ====Example 2.1 ==== | ||
| - | For the circuit below, determine the voltage across the 5Ω resistor, //v//, the current supplied by the source, //i//, and the power supplied by the source. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | The voltage across the 5Ω resistor can be determined from our voltage divider relationship: | ||
| - | |||
| - | $$ v = [\frac{5Ω}{5Ω + 15Ω + 10Ω}] \cdot 15V= \frac{5}{30} \cdot 15V = 2.5V $$ | ||
| - | |||
| - | The current supplied by the source can be determined by dividing the total voltage by the equivalent resistance: | ||
| - | |||
| - | $$i= \frac{15V}{R_{eq}}= \frac{15V}{5Ω+15Ω+10Ω}= \frac{15V}{30Ω}=0.5A$$ | ||
| - | |||
| - | The power supplied by the source is the product of the source voltage and the source current: | ||
| - | |||
| - | $$P=iv=(0.5A)(15V)=7.5W$$ | ||
| - | |||
| - | We can double-check the consistency between the voltage //v// and the current //i// with Ohm's law. Applying Ohm's law to the 5Ω resistor, with a 0.5A current, results in $v=(5Ω)(0.5A)=2.5V$, | ||
| - | |||
| - | ==== Section Summary ==== | ||
| - | * If only two elements connect at a single node, the two elements are in //series//. A more general definition, however, is that circuit elements in series all share the same current - this definition allows us to determine series combinations that contain more than two elements. Identification of series circuit elements allows us to simplify our analysis, since there is a reduction in the number of unknowns: there is only a single unknown current for all series elements. | ||
| - | * A series combination of resistors can be replaced by a single // | ||
| - | * If the total voltage difference across a set of series is known, the voltage differences across any individual resistor can be determined by the concept of //voltage division//. The term voltage division comes from the fact that the voltage drop across a series combination of resistors is divide among the individual resistors. The ratio between the voltage difference across a particular resistor and the total voltage difference is the same as the ratio between the resistance of that resistor and the total resistance of the combination. If // | ||
| - | |||
| - | $$ \frac{v_k}{v_{TOT}} = \frac{R_k}{R_{TOT}} $$ | ||
| - | |||
| - | ==== Exercises ==== | ||
| - | 1. Determine the voltage V< | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ===== 2.2 Parallel Circuit Elements and Current Division ===== | ||
| - | Circuit elements are said to be connected in // | ||
| - | |||
| - | $$ v_1=v_2 | ||
| - | |||
| - | This result is so common that equation (2.10) is generally written by inspection fro parallel elements such as those shown in Fig. 2.3 __without__ explicitly writing KVL. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | We can simplify circuits, which consist of resistors connected in parallel. Consider the resistive circuit shown in Fig. 2.4(a). The ressitors are connected in parallel, so both resistors have a voltage difference of //v//. Ohm's law applied to each resistor results in: | ||
| - | |||
| - | $$ i_1= \frac{v}{R_1} \\ | ||
| - | i_2= \frac{v}{R_2} \\ | ||
| - | (Eq. 2.11) $$ | ||
| - | |||
| - | Applying KCL at node a: | ||
| - | |||
| - | $$i=i_1 + i_2 (Eq. 2.12)$$ | ||
| - | |||
| - | Substituting equations (2.11) into equation (2.12): | ||
| - | |||
| - | $$ i = \left[ \frac{1}{R_1} + \frac{1}{R_2} \right] v (Eq. 2.13)$$ | ||
| - | |||
| - | or | ||
| - | |||
| - | $$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}\cdot i (Eq. 2.14)$$ | ||
| - | |||
| - | If we set $ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}$, | ||
| - | |||
| - | We can generalize this result for //N// parallel resistances: | ||
| - | |||
| - | A parallel combination of //N// resistors $R_1, R_2, \cdots, R_N$ can be replaced with a single equivalent resistance: | ||
| - | |||
| - | $$ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}+ \cdots \frac{1}{R_N}} | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | For the special case of two parallel resistances, | ||
| - | |||
| - | $$R_{eq}= \frac{R_1R_2}{R_1+R_2} | ||
| - | |||
| - | This alternative way to calculate $R_{eq}$ can be also used to calculate $R_{eq}$ for larger numbers of parallel resistors since any number of resistors could be combined two at a time. | ||
| - | |||
| - | ==== Current Division ==== | ||
| - | Substituting equation (2.14) into equations (2.11) results in: | ||
| - | |||
| - | $$i_1 = \frac{1}{R_1}\cdot \frac{i}{ \frac{1}{R_1} + \frac{1}{R_2}} | ||
| - | |||
| - | Simplifying: | ||
| - | |||
| - | $$i_1=\frac{R_2}{R_1+R_2} | ||
| - | |||
| - | Likewise, for the current $i_2$: | ||
| - | |||
| - | $$i_2 = \frac{R_1}{R_1+R_2} | ||
| - | |||
| - | Equations (2.18) and (2.19) are the current //divider relationships// | ||
| - | |||
| - | The above results can be generalized for a series combination of //N// resistances. By Ohm's law, $v = R_{eq} i$. Substituting our previous result for the equivalent resistance for a parallel combination of //N// resistors results in: | ||
| - | |||
| - | $$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2} + \cdots \frac{1}{R_N}} \cdot i (Eq. 2.20)$$ | ||
| - | |||
| - | Since the voltage difference across all resistors is the same, the current through the k< | ||
| - | |||
| - | $$i_k=\frac{v}{R_k} | ||
| - | |||
| - | Where $R_k$ is the resistance of the k< | ||
| - | |||
| - | $$i_k= \frac{\frac{1}{R_k}}{\frac{1}{R_1}+\frac{1}{R_2}+ \cdots \frac{1}{R_N}} \cdot i (Eq. 2.22)$$ | ||
| - | |||
| - | It is often more convenient to provide the generalized result of equation (2.20) in terms of the conductance of the individual resistors. Recall that the conductance is the reciprocal of the resistance, $G=\frac{1}{R}$. Thus, equation (2.22) can be re-expressed as follows: | ||
| - | |||
| - | The Current through any resistor in a parallel combination of //N// resistances is proportional to the total current into the combination of resistors. The constant of proportionality is the same as the ratio of the conductance of the individual resistor value to the total conductance of the parallel combination. For example, the current through the // | ||
| - | |||
| - | $$i_k=\frac{G_k}{G_1+G_2+ \cdots + G_N}i (Eq. 2.23)$$ | ||
| - | |||
| - | Where //i// is the total current through the parallel combination of resistors. | ||
| - | |||
| - | One final comment about notation: two parallel bars are commonly used as shorthand notation to indicate that two circuit elements are in parallel. For example, the notation $R_1 \parallel R_2$ indicates that the resistors $R_2$ and $R_2$ are in parallel. The notation $R_1 \parallel R_2$ is often used as shorthand notation for the __equivalent resistance__ of the parallel resistance combination, | ||
| - | |||
| - | Double-checking results for parallel resistances: | ||
| - | * The equivalent resistance for a parallel combination of //N// resistors will always be less than the smallest resistance in the combination. In fact, the equivalent resistance will always obey the following inequalities: | ||
| - | |||
| - | $$\frac{R_{min}}{N} \leq R_{eq} \leq R_{min}$$ | ||
| - | |||
| - | * Where $R_{min}$ is the smallest resistance value in the parallel combination. | ||
| - | * In a parallel combination of resistances, | ||
| - | |||
| - | ==== Section Summary ==== | ||
| - | * If several elements interconnect the same two nodes, the two elements are in // | ||
| - | * A parallel combination of resistors can be replaced by a single // | ||
| - | |||
| - | $$R_{eq} = \frac{1}{\frac{1}{R_1}+\frac{1}{R_2}+\cdots\frac{1}{R_N}}$$ | ||
| - | * If the total current through a set of parallel resistors is known, the current through any individual resistor can be determined by the concept of //current division//. The term current division comes form the fact that the current through a parallel combination of resistors is divided among the individual resistors. The ratio between the current through a particular resistor and the total current is the same as the ratio between the conductance of that resistor and the total conductance of the combination. If $i_k$ is the voltage across the k< | ||
| - | |||
| - | $$\frac{v_k}{i_{TOT}}=\frac{G_k}{G_{TOT}}$$ | ||
| - | |||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Exercises ==== | ||
| - | 1. Determine the value of //I// in the circuit below. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | 2. Determine the value of //R// in the circuit below which makes I=2mA. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ===== 2.3 Circuit Reduction and Analysis ===== | ||
| - | The previous results give us an ability to potentially simplify the analysis of some circuits. This simplification results if we can use //circuit reduction// techniques to convert a complicated circuit to a simpler, but equivalent, circuit which we can use to perform the necessary analysis. Circuit reduction is not always possible, but when it is applicable it can significantly simplify the analysis of a circuit. | ||
| - | |||
| - | Circuit reduction relies upon identification of parallel and series combinations of circuit elements. The parallel and series elements are then combined into equivalent elements and the resulting //reduced// circuit is analyzed. The principles of circuit reduction are illustrated below in a series of examples. | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Example 2.2 ==== | ||
| - | Determine the equivalent resistance seen by the terminals of the resistive network shown below. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | The sequence of operations performed is illustrated below. The 6Ω and 3Ω resistances are combined in parallel to obtain an equivalent 2Ω resistance. This 2Ω resistance and the remaining 6Ω resistance are in series, these are combined into an equivalent 8Ω resistance. Finally, this 8Ω resistor and the 24Ω resistor are combined in parallel to obtain an equivalent 6Ω resistance. Thus, the equivalent resistance of the overall network is 6Ω. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Example 2.3 ==== | ||
| - | In the circuit below, determine the power delivered by the source. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | In order to determine power delivery, we need to determine the total current provided by the source to the rest of the circuit. We can determine current easily if we convert the resistor network to a single, equivalent, resistance. A set of steps for doing this are outlined below. | ||
| - | |||
| - | Step 1: The four ohm and two ohm resistors, highlighted on the figure to the left below in blue, are in series. Series resistances add directly, so these can be replaced with a single six ohm resistor, as shown on the figure to the right below. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | Step 2: The tree ohm resistor and the two six ohm resistors are now all in parallel, as indicated on the figure to the left below. These resistances can be combined into a single equivalent resistor $R_{eq}=\frac{1}{\frac{1}{3}+\frac{1}{6}+\frac{1}{6}}=1.5\Omega$. The resulting equivalent circuit is shown to the right below. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | The current out of the source can now be readily determined from the figure to the right above. The voltage drop across the 1.5Ω resistor is 6V, so Ohm's law gives $i=\frac{6V}{1.5\Omega}=4A$. Thus, the power delivered by the source is $P=(4A)(6V)=24W$. Since the sign of the current relative to the current does __not__ agree with the passive sign convention, the power is __generated__ by the source. | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Example 2.4 ==== | ||
| - | For the circuit shown below, determine the voltage, $v_s$, across the 2A source. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | The two 1Ω resistors and the two 2Ω resistors are in series with one another, as indicated on the figure to the left below. These can be combined by simply adding the series resistances, | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | The three remaining resistors are all in parallel (they all share the same nodes) so they can be combined using the relation $R_{eq}=\frac{1}{\frac{1}{2}+\frac{1}{4}+\frac{1}{4}}=1\Omega$. Note that it is not necessary to combine all three simultaneously, | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | The voltage across the source can now be determined from Ohm's law: $v_s=(1\Omega)(2A)=2V$. The assumed polarity of the source voltage is correct. | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Example 2.5: Wheatstone Bridge ==== | ||
| - | A Wheatstone bridge circuit is shown below. The bridge is generally presented as shown in the figure to the left; we will generally use the equivalent circuit shown to the right. A Wheatstone bridge is commonly used to convert a variation in resistance to a variation in voltage. A constant supply voltage $V_s$ is applies to the circuit. The resistors in the circuit all have a nominal resistance of R; the variable resistor has a variation $\Delta R$ from this nominal value. The output voltage $v_{ab}$ indicates the variation $\Delta R$ in the variable resistor. The variable resistor in the network is often a transducer whose resistance varies dependent upon some external variable such as temperature. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | By voltage division, the voltages $v_b$ and $v_a$ (relative to ground) are: | ||
| - | |||
| - | $$v_b=\frac{R+\Delta R}{2R+\Delta R}V_s$$ | ||
| - | |||
| - | and | ||
| - | |||
| - | $$v_a=Ri_2=\frac{V_s \cdot R}{2R}=\frac{V_s}{2}$$ | ||
| - | |||
| - | The voltage $v_{ab}$ is then: | ||
| - | |||
| - | $$v_{ab}=v_a-v_b= \left( \frac{1}{2} - \frac{R+ \Delta R}{2R + \Delta R} \right) V_s = \left( \frac{(2R+ \Delta R) - 2(R+ \Delta R)}{2(2R+ \Delta R)} \right) V_s = - \frac{\Delta R}{2(2R+ \Delta R)} \cdot V_s $$ | ||
| - | |||
| - | For the case in which $\Delta R << 2R$, this simplifies to: | ||
| - | |||
| - | $$v_{ab} \approx - \frac{V_s}{4R}\Delta R$$ | ||
| - | |||
| - | And the output voltage is proportional to the change in resistance of the variable resistor. | ||
| - | |||
| - | === Practical Applications: | ||
| - | A number of common sensors result in a resistance variation resulting from some external influence. // | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Section Summary ==== | ||
| - | * In a circuit, which contains obvious series and/or parallel combinations of resistors, analysis can be simplified by combining these resistances into equivalent resistances. The reduction in the overall number of resistances reduces the number of unknowns in the circuit, with a corresponding reduction in the number of governing equations. Reducing the number of equations and unknowns typically simplifies the analysis of the circuit. | ||
| - | * Not all circuits are reducible. | ||
| - | |||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Exercises ==== | ||
| - | - For the circuit shown, determine: | ||
| - | - $R_{eq}$ (the equivalent resistance seen by the source) | ||
| - | - The currents $I_1$ and $I_2$ | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ===== 2.4 Non-ideal Power Supplies ===== | ||
| - | In section 1.2, we discussed ideal power sources. In that section, an ideal voltage supply was characterized as providing a specified | ||
| - | |||
| - | In this section, we present simple models for voltage and current sources which incorporate more realistic assumptions as to the behavior of these devices. | ||
| - | |||
| - | ==== Non-ideal Voltage Sources ==== | ||
| - | An ideal voltage source was defined in section 1.2 as providing a specified voltage, regardless of the current flow out of the device. For example, an __ideal__ 12V battery will provide 12V across its terminals, regardless of the load connected to the terminals. A real 12V battery, however, provides 12V across its terminals only when its terminals are open-circuited. As we draw current from the terminals, the battery will provide less than 12V - the voltage will decrease as more and more current is drawn from the battery. The real battery thus appears to have an internal voltage drop which increases with increased current. | ||
| - | |||
| - | We will model a real or // | ||
| - | |||
| - | $$V=V_s-i \cdot R_s (Eq. 2.24)$$ | ||
| - | |||
| - | Equation (2.24) indicates that the voltage delivered by our non-ideal voltage source model decreases as the current out of the voltage source increases, which agrees with expectations. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Example 2.6 ==== | ||
| - | Consider the case in which we connect a resistive load to the non-ideal voltage source. The figure below provides a schematic of the overall system; $R_L$ is the load resistance, $V_L$ is the voltage delivered to the load, and $i_L$ is the current delivered to the load. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | In the case above, the current delivered to the load is $i=\frac{V_s}{R_s+R_L}$ and the load voltage is $V_L=V_s \frac{R_L}{R_s+R_L}$. Thus, if the load resistance is infinite (the load is an open circuit), $V_L=V_S$, but the power supply delivers no current and hence no power to the load. If the load resistance is zero (the load is a short circuit), $V_L=0$ and the power supply delivers current $i_L=\frac{V_s}{R_s}$ to the load; the power delivered to the load, however, is still zero. | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Example 2.7: Charging a Battery ==== | ||
| - | We have a " | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | If we attempt to analyze this circuit by applying KVL around the loop, we obtain 12V=4V. This is obviously incorrect and we cannot proceed with our analysis - our model disagrees with reality! | ||
| - | |||
| - | To resolve this issue, we will include the internal resistance of the batteries. Assuming a 3Ω internal resistance in each battery, we obtain the following model for the system: | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | Applying KVL around the loop, and using Ohm's law to write the voltages across the battery internal resistances in terms of the current between the batteries results in: | ||
| - | |||
| - | $$-12V+(3\Omega)i+(3\Omega)i+4V=0$$ | ||
| - | |||
| - | Which can be solved for the current //i// to obtain: | ||
| - | |||
| - | $$i= \frac{12V-4V}{6\Omega}=1.33A$$ | ||
| - | |||
| - | Notice that as the voltage of the " | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Non-ideal Current Sources ==== | ||
| - | An ideal current source was defined in section 1.2 as providing a specified current, regardless of the voltage difference across the device. This model suffers from the same basic drawback as our ideal voltage source model - the model can deliver infinite power, which is inconsistent with the capabilities of a real current source. | ||
| - | |||
| - | We will use the circuit shown schematically in Fig. 2.6 to model a non-ideal current source. The non-ideal model consists of an ideal current source, $i_s$, placed in parallel with an internal resistance, $R_s$. The source delivers a voltage //V// and current //i//. The output current is given by: | ||
| - | |||
| - | $$i=i_S-\frac{V}{R_S} | ||
| - | |||
| - | Equation (2.25) shows that the current delivered by the source decreases as the delivered voltage increases. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Example 2.8 ==== | ||
| - | Consider the case in which we connect a resistive load to the non-ideal current source. The figure below provides a schematic of the overall system; $R_L$ is the load resistance, $V_L$ is the voltage delivered to the load, and $i_L$ is the current delivered to the load. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | In the case above, the current delivered to the load can be determined from a current divider relation as $i_l=i_s \cdot \frac{R_s}{R_s+R_L}$ and the load voltage, by Ohm's law, is $V_L=i_L R_L = i_s \frac{R_S R_L}{R_s + R_L}$. If the load resistance is zero (the load is a short circuit), $i_L=i_s$, but the power supply delivers no voltage and hence no power to the load. In the case of infinite load resistance (the load is open circuit), $i_L = 0$. In this case, we can neglect $R_s$ in the denominator of the load voltage equation to obtain $V_L \approx i_s \frac{R_S R_L}{R_L}$ so that $V_L \approx i_s R_S$. Since the current is zero, however, the power delivered to the load is still zero. | ||
| - | |||
| - | If we explicitly calculate the power delivered to the load, we obtain $V_L = i_{s}^{2} \frac{R_S R_L}{R_s + R_L} \cdot \frac{R_s}{R_s+R_L}$. A plot of the power delivered to the load as a function of the load resistance is shown below; a logarithmic scale is used on the horizontal axis to make the plot more readable. As expected, the power is zero for high and low load resistances. The peak of the curve occurs when the load resistance is equal to the source resistance, $R_L=R_s$. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Section Summary ==== | ||
| - | * In many cases, power supplies can be modeled as ideal power supplies, as presented in section 1.2. However, in some cases representation as a power supply as ideal results in unacceptable errors. For example, ideal power supplies can deliver infinite power, which is obviously unrealistic. | ||
| - | * In this section, we present a simple model for a non-ideal power supply. | ||
| - | * Our non-ideal voltage source consists of an ideal voltage source in series with a resistance which is internal to the power supply. | ||
| - | * Our non-ideal current supply consists of an ideal current source in parallel with a resistance which is internal to the power supply. | ||
| - | * Voltage and current divider formulas allow us to easily quantify the effects of the internal resistances of the non-ideal power supplies. Our analysis indicates that the non-ideal effects are negligible, as long as the resistance of the load is large relative to the internal resistance of the power supply. | ||
| - | |||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Exercises ==== | ||
| - | 1. A voltage source with an internal resistance of 2Ω as shown below is used ot apply power to a 3Ω resisotr. What voltage would you measure across the 3Ω resistor? | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | 2. The voltage source of exercise 1 above is used to apply power to a 2kΩ resistor. What voltage would you measure across the 2kΩ resistor? | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ===== 2.5 Practical Voltage and Current Measurement ===== | ||
| - | The process of measuring a physical parameter will almost invariably change the parameter being measures. This effect is both undesirable and, in general, unavoidable. One goal of any measurement is to effect the parameter being measured __as little as possible__. | ||
| - | |||
| - | The above statement is true of voltage and current measurements. An __ideal__ voltmeter, connected in parallel with some circuit element, will measure the voltage across the element without affecting the current flowing through the element. Unfortunately, | ||
| - | |||
| - | In this section, we introduce some effects of measuring voltages and currents with practical meters. | ||
| - | |||
| - | ==== Voltmeter and Ammeter Models ==== | ||
| - | We will model both voltmeters and ammeters as having some internal resistance and a method for displaying the measured voltage difference or current. Fig. 2.7 shows schematic representations of voltmeters and ammeters. | ||
| - | |||
| - | The ammeter in Fig. 2.7(a) has an internal resistance $R_M$; the current through the ammeter is $i_A$ and the voltage difference across the ammeter is $V_M$. The ammeter' | ||
| - | |||
| - | The voltmeter in Fig. 2.7(b) is also represented as having an internal resistance $R_M$; the current through the meter is $i_v$ and the voltage difference across the meter is $V_v$. The current through the voltmeter should be as small as possible - the voltmeter should have an extremely high internal resistance. | ||
| - | |||
| - | The effects of non-zero ammeter voltages and non-zero voltmeter currents are explored in more detail in the following subsections. | ||
| - | |||
| - | ==== Voltage Measurement ==== | ||
| - | Consider the circuit shown in Fig. 2.8(a). A current source, $i_s$, provides current to a circuit element with resistance, R. We want to measure the voltage drop, V, across the circuit element. We do this by attaching a voltmeter across the circuit element as shown in Fig. 2.8(b). | ||
| - | |||
| - | In Fig. 2.8(b) the voltmeter resistance is in parallel to the circuit element we wish to measure the voltage across and the combination of the circuit element and the voltmeter becomes a current divider. The current through the resistor R then becomes: | ||
| - | |||
| - | $$i=i_s \frac{R_M}{R+R_M} | ||
| - | |||
| - | The voltage across the resistor R is then, by Ohm's law: | ||
| - | |||
| - | $$V=i_s \frac{R \cdot R_M}{R+R_M} | ||
| - | |||
| - | If $R_M >>R$, this expression simplifies to: | ||
| - | |||
| - | $$V \approx i_s \frac{R \cdot R_M}{R_M} = R \cdot i_s (Eq. 2.28)$$ | ||
| - | |||
| - | And negligible error is introduces into the measurement - the measured voltage is approximately the same as the voltage without the voltmeter. If, however, the voltmeter resistance is comparable to the resistance R, the simplification of equation (2.28) is not appropriate and significant changes are made to the system by the presence of the voltmeter. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | ==== Current Measurement ==== | ||
| - | Consider the circuit shown in Fig. 2.9(a). A voltage source, $V_s$, provides power to a circuit element with resistance, R. We want to measure the current, $i$, through the circuit element. We do this by attaching an ammeter in series with the circuit element as shown in Fig. 2.9(b). | ||
| - | |||
| - | In Fig. 2.9(b) the series combination of the ammeter resistance and the circuit element whose current we wish to measure creates a voltage divider. KVL around the single circuit loop provides: | ||
| - | |||
| - | $$V_s=i(R_M+R) | ||
| - | |||
| - | Solving for the current results in: | ||
| - | |||
| - | $$i= \frac{V_s}{R_M+R} | ||
| - | |||
| - | If $R_M << R$, this simplifies to: | ||
| - | |||
| - | $$i \approx \frac{V_s}{R} | ||
| - | |||
| - | And the measured current is a good approximation to current in the circuit of Fig. 2.9(a). However, if the ammeter resistance is not small compared to the resistance R, the approximation of equation (2.31) is not appropriate and the measured current is no longer representative of the circuit' | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Caution ==== | ||
| - | Incorrect connections of ammeters or voltmeters can cause damage to the meter. For example, consider the connection of an ammeter in __parallel__ with a relatively large resistance, as shown below. | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | In this configuration the ammeter current, $i_M= \frac{V_S}{R_M}$. Since the ammeter resistance is typically very small, this can result in high currents being provided to the ammeter. This, in turn, may result in excessive power being provided to the ammeter and resulting damage to the device. | ||
| - | |||
| - | Ammeters are generally intended to be connected in __series__ with the circuit element(s) whose current is being measured. Voltmeters are generally intended to be connected in __parallel__ with the circuit element(s) whose voltage is being measured. Alternate connections can result in damage to the meter. | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Section Summary ==== | ||
| - | * Measurement of voltage and/or current in a circuit will always result in some effect on the circuit' | ||
| - | * In this section, we present simple models for voltmeters and ammeters (voltage and current measurement devices, respectively). | ||
| - | * Our non-ideal voltmeter consists of an ideal voltmeter (which had infinite resistance, and thus draws no current from the circuit) in parallel with a resistance which is internal to the voltmeter. This model replicates the finite current which is necessarily drawn from the circuit by a real voltmeter. | ||
| - | * Our non-ideal ammeter consists of an ideal ammeter (which has zero resistance, and thus introduces no voltage drop in the circuit) in series with a resistance which is internal to the ammeter. This resistance allows us to model the finite voltage drop which is introduced into the circuit by a real current measurement. | ||
| - | * Voltage and current divider formulas allow us to easily quantify the effects of the internal resistances of voltage and current meters. Our analysis indicates that the non-ideal effects are negligible, as long as: | ||
| - | * The resistance of the voltmeter is large relative to the resistance across which the voltage measurement is being made. | ||
| - | * The resistance of the ammeter is small compared to the overall circuit resistance. | ||
| - | |||
| - | ---- | ||
| - | |||
| - | ==== Exercises ==== | ||
| - | - A voltmeter with an internal resistance of 10MΩ is used to measure the voltage $v_{ab}$ in the circuit below. What is the measured voltage? What voltage measurement would you expect from an ideal voltmeter? | ||
| - | |||
| - | {{ : | ||
| - | |||
| - | ---- | ||
| - | |||
| - | --> | ||