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| learn:courses:real-analog-chapter-1:start [2016/12/07 16:26] – [Example 1.10] Martha | learn:courses:real-analog-chapter-1:start [2023/02/09 07:12] (current) – external edit 127.0.0.1 | ||
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| ====== Real Analog: Chapter 1 ====== | ====== Real Analog: Chapter 1 ====== | ||
| - | ====== 1. Introduction and Chapter Objectives | + | [[{}/ |
| + | --> Chapter 1 Materials# | ||
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| + | <-- | ||
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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. | ||
| - | ---- | ||
| ==== 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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| ---- | ---- | ||
| - | ==== Exercises | + | === Exercises === |
| 1. Assign reference voltage and current directions to the circuit elements represented by the shaded boxes in the circuits below. | 1. Assign reference voltage and current directions to the circuit elements represented by the shaded boxes in the circuits below. | ||
| {{ : | {{ : | ||
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| In this section we consider some very important active circuit elements: voltage and current sources. We will discuss two basic types of sources: // | 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 | + | ==== 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 // | An independent voltage source maintains a specified voltage across its terminals. The symbol used to indicate a voltage source delivering a voltage // | ||
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| {{ : | {{ : | ||
| - | ===== Independent Current Sources | + | ==== 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 // | 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 // | ||
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| {{ : | {{ : | ||
| - | ===== Dependent Sources | + | ==== 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: | 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) | * Voltage-controlled voltage source (VCVS) | ||
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| {{ : | {{ : | ||
| - | ===== Section Summary | + | ==== 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. | * 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. | * 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. | ||
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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. | |
| - | ---- | + | |
| - | ==== 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. | ||
| - | |||
| - | ---- | ||
| {{ : | {{ : | ||
| - | ===== Conductance | + | ==== Conductance ==== |
| - | // | + | // |
| $$ G=\frac{1}{R} | $$ G=\frac{1}{R} | ||
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| - | ---- | + | **Note:** |
| - | + | ||
| - | ==== 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/ | ||
| - | ---- | ||
| - | ===== 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) | ||
| - | ---- | ||
| - | ==== 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. | ||
| - | ---- | ||
| - | ===== 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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| ---- | ---- | ||
| + | |||
| + | 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 | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | [[{}/ | ||
| + | [[{}/ | ||
| + | |||