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| learn:courses:real-analog-chapter-1:start [2016/12/08 20:02] – [Independent Current Sources] Martha | learn:courses:real-analog-chapter-1:start [2023/02/09 15:12] (current) – external edit 127.0.0.1 | ||
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| ====== Real Analog: Chapter 1 ====== | ====== Real Analog: Chapter 1 ====== | ||
| - | ====== 1. Introduction and Chapter Objectives | + | [[{}/ |
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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. | 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. | ||
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| ==== After Completing this Chapter, You Should be Able to: ==== | ==== After Completing this Chapter, You Should be Able to: ==== | ||
| - | * Define voltage and current in terms of electrical charge | + | * Define voltage and current in terms of electrical charge |
| - | * State common prefixes and the symbols used in scientific notation | + | * State common prefixes and the symbols used in scientific notation |
| - | * State the passive sign convention from memory | + | * 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 | + | * 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 | + | * Write symbols for independent voltage and current sources |
| - | * State from memory the function of independent voltage and current sources | + | * State from memory the function of independent voltage and current sources |
| - | * Write symbols for dependent voltage and current sources | + | * Write symbols for dependent voltage and current sources |
| - | * State governing equations for the four types of dependent sources | + | * State governing equations for the four types of dependent sources |
| - | * State Ohm’s Law from memory | + | * State Ohm’s Law from memory |
| - | * Use Ohm’s Law to perform voltage and current calculations for resistive circuit elements | + | * Use Ohm’s Law to perform voltage and current calculations for resistive circuit elements |
| - | * Identify nodes in an electrical circuit | + | * Identify nodes in an electrical circuit |
| - | * Identify loops 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 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 | + | * 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 | + | * Apply Kirchhoff’s voltage and current laws to electrical circuits |
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| Mathematically, | Mathematically, | ||
| - | $$ i=\frac{dq}{dw}\ (Eq. 1.2) $$ | + | $$ i=\frac{dq}{dt}\ (Eq. 1.2) $$ |
| 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 // | 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 // | ||
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| - | === Note === | + | **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 waste of time - it is best to arbitrarily assume __either__ a voltage polarity of current direction for each circuit element. | |
| - | 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. | 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. | ||
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| 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. | 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 ==== | ==== Voltage Subscript and Sign Conventions ==== | ||
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| === Example 1.5: === | === 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. | + | 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. |
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| 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. | 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. | |
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| - | ==== Note ==== | + | |
| - | The voltage-current relationship for resistors always agrees with the passive sign convention. Resistors always absorb power. | + | |
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| 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 // | 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 range of voltages and currents. | ||
| - | ==== 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. | ||
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| - | ===== Conductance | + | ==== Conductance ==== |
| - | // | + | // |
| $$ G=\frac{1}{R} | $$ G=\frac{1}{R} | ||
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| - | ---- | + | **Note:** |
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| - | ==== 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/ | 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/ | ||
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| - | ===== Power Dissipation | + | ==== Power Dissipation ==== |
| Instantaneous power was defined by equation (1.3) in section 1.1 as: | Instantaneous power was defined by equation (1.3) in section 1.1 as: | ||
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| $$ P(t)= \frac{i^2(t)}{G}=Gv^2(t) | $$ P(t)= \frac{i^2(t)}{G}=Gv^2(t) | ||
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| - | ==== Note: ==== | + | **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. | 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. | ||
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| - | ===== Practical Resistors | + | ==== 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. | 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. | ||
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| - | ===== Section Summary ==== | + | ==== 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. | * 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. | * 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. | ||
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| - | ====== Kirchhoff' | + | ==== Kirchhoff' |
| This section provides some basic definitions and background information for two important circuit analysis tools: Kirchhoff' | This section provides some basic definitions and background information for two important circuit analysis tools: Kirchhoff' | ||
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| - | ====Kirchhoff' | + | ==== Kirchhoff' |
| Kirchhoff' | Kirchhoff' | ||
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| - | ===== Kirchhoff' | + | ==== Kirchhoff' |
| Kirchhoff' | Kirchhoff' | ||
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| Where // | Where // | ||
| - | === Note === | + | **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. | 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. | ||
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| - | ===== Application | + | ==== Application |
| 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. | 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. | ||
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| - | ===== Example 1.12 ==== | + | ==== Example 1.12 ==== |
| For the circuit below, determine // | For the circuit below, determine // | ||
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| - | ===== Example 1.13 ===== | + | ==== Example 1.13 ==== |
| Determine // | Determine // | ||
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| - | ===== Section Summary | + | ==== 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 (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. | * 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. | ||
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