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learn:courses:real-analog-chapter-2:start [2016/12/12 23:59] – Marthalearn:courses:real-analog-chapter-2:start [2022/10/04 11:38] (current) – external edit 127.0.0.1
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 ====== Real Analog: Chapter 2 ====== ====== Real Analog: Chapter 2 ======
 +[[{}/learn/courses/real-analog-chapter-1/start|Back to Chapter 1]]
 +--> Chapter 2 Materials#  
 +  * Lecture Material:
 +    * {{ :learn:courses:real-analog-chapter-2:lecture5.ppt |Lecture 5 PowerPoint Slides}}: Circuit reduction
 +    * {{ :learn:courses:real-analog-chapter-2:lecture6.ppt |Lecture 6 PowerPoint Slides}}: Circuit reduction examples, practical application: temperature measurement
 +    * [[http://www.youtube.com/watch?v=AMszE8OLHRk&list=PLDEC730F6A8CDE318&index=5&feature=plpp_video| Lecture 5 Video]]
 +    * [[http://www.youtube.com/watch?v=Ns-IrK1gobc&list=PLDEC730F6A8CDE318&index=6&feature=plpp_video|Lecture 6 Video]]
 +  * Chapter 2 Videos:
 +    * [[http://www.youtube.com/watch?v=OaMhItvzeKg&list=PL170A01159D42313D&index=5&feature=plpp_video| Lab 2 Video 2]]: Potentiometers: Variable resistors
 +    * [[http://www.youtube.com/watch?v=lGcZVwMSFCc&list=PL170A01159D42313D&index=6&feature=plpp_video| Lab 2 Video 2]]: Resistors 2: Resistance of networks of resistors. Using time-varying voltage sources to plot voltage-current characteristics of resistors.
 +    * [[http://www.youtube.com/watch?v=XDa7gaOQcuw&list=PL170A01159D42313D&index=7&feature=plpp_video| Lab 2 Video 3]]: Non-ideal effects: Non-ideal voltage sources and voltage measurements. Analog Discovery power supply limitations.
 +  * {{ :learn:courses:real-analog-chapter-2:real-analog-chapter-2.pdf | Chapter 2 Complete PDF}}
 +    * {{ :learn:courses:real-analog-chapter-2:lab2p1p1.pdf |Lab 2.1.1}}
 +      * {{ :learn:courses:real-analog-chapter-2:lab2p1p1_worksheet.docx |Worksheet 2.1.1}}
 +    * {{ :learn:courses:real-analog-chapter-2:lab2p3p1.pdf |Lab 2.3.1}}
 +      * {{ :learn:courses:real-analog-chapter-2:lab2p3p1_worksheet.docx |Worksheet 2.3.1}}
 +    * {{ :learn:courses:real-analog-chapter-2:lab2p3p2.pdf |Lab 2.3.2}}
 +      * {{ :learn:courses:real-analog-chapter-2:lab2p3p2_worksheet.docx |Worksheet 2.3.2}}
 +    * {{ :learn:courses:real-analog-chapter-2:lab2p4.pdf |Lab 2.4}}
 +      * {{ :learn:courses:real-analog-chapter-2:lab2p4_worksheet.docx |Worksheet 2.4}}
 +    * {{ :learn:courses:real-analog-chapter-2:lab2p5.pdf |Lab 2.5}}
 +      * {{ :learn:courses:real-analog-chapter-2:lab2p5_worksheet.docx |Worksheet 2.5}}
 +  * {{ :learn:courses:real-analog-chapter-2:realanalog-exercisesolutions-chapter2.pdf |Exercise Solutions}}: Chapter 2 exercise solutions
 +  * {{ :learn:courses:real-analog-chapter-2:homework2.docx |Homework}}: Chapter 2 homework problems
 +
 +<--
 +
 ====== 2. Introduction and Chapter Objectives ====== ====== 2. Introduction and Chapter Objectives ======
-In [[https://reference.digilentinc.com/learn/courses/real-analog-chapter-1/start| Chapter 1]], we presented Kirchhoff's laws (which govern the interaction between circuit elements) and Ohm's law (which governs the voltage-current relationships for resistors). These analytical tools provide us with the ability to analyze any circuit containing only resistors and ideal power supplies. However, we also saw in Chapter 1 that a circuit analysis, which relies strictly on a brute-force application of these tools can become complex rapidly - we essentially must use as our unknowns the voltage differences across __all__ resistors and the currents through __all__ resistors. This generally results in a large number of unknowns and a correspondingly large number of equations, which must be written and solved in order to analyze any but the simplest circuit. +In [[/learn/courses/real-analog-chapter-1/start| Chapter 1]], we presented Kirchhoff's laws (which govern the interaction between circuit elements) and Ohm's law (which governs the voltage-current relationships for resistors). These analytical tools provide us with the ability to analyze any circuit containing only resistors and ideal power supplies. However, we also saw in Chapter 1 that a circuit analysis, which relies strictly on a brute-force application of these tools can become complex rapidly - we essentially must use as our unknowns the voltage differences across __all__ resistors and the currents through __all__ resistors. This generally results in a large number of unknowns and a correspondingly large number of equations, which must be written and solved in order to analyze any but the simplest circuit. 
  
 In the next few chapters, we will still apply Kirchhoff's laws and Ohm's law in our circuit analysis, but we will focus on improving the efficiency of our analyses. Typically, this improvement in efficiency is achieved by reducing the number of unknowns in the circuit, which reduces the number of equations, which must be written to describe the circuit's operation.  In the next few chapters, we will still apply Kirchhoff's laws and Ohm's law in our circuit analysis, but we will focus on improving the efficiency of our analyses. Typically, this improvement in efficiency is achieved by reducing the number of unknowns in the circuit, which reduces the number of equations, which must be written to describe the circuit's operation. 
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 or or
  
-$$i_1=i_2     (Eq. 2.1)$$+$$i_1=i_2   \qquad  (Eq. 2.1)$$
  
-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.+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 current 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.
  
 {{ :learn:courses:real-analog-chapter-2:chapter2a.png |Figure 2.1.}} {{ :learn:courses:real-analog-chapter-2:chapter2a.png |Figure 2.1.}}
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 Applying KVL around the loop: Applying KVL around the loop:
  
-$$ -v+v_1+v_2=0 \Rightarrow v=v_1+v_2     (Eq. 2.3)$$+$$ -v+v_1+v_2=0 \Rightarrow v=v_1+v_2   \qquad  (Eq. 2.3)$$
  
 Substituting equations (2.2) into equation (2.3) and solving for the current //i// results in: Substituting equations (2.2) into equation (2.3) and solving for the current //i// results in:
  
-$$ i = \frac{v}{R_1+R_2}   (Eq. 2.4) $$+$$ i = \frac{v}{R_1+R_2}  \qquad (Eq. 2.4) $$
  
-Now cinsider the circuit of Fig. 2.2(b). Application of Ohm's law to this circuit and solution for the current //i// gives:+Now consider the circuit of Fig. 2.2(b). Application of Ohm's law to this circuit and solution for the current //i// gives:
  
-$$ i = \frac{v}{R_{eq}}   (Eq. 2.5) $$+$$ i = \frac{v}{R_{eq}}  \qquad (Eq. 2.5) $$
  
 {{ :learn:courses:real-analog-chapter-2:chapter2b.png |Figure 2.2.}} {{ :learn:courses:real-analog-chapter-2:chapter2b.png |Figure 2.2.}}
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 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: 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:
  
-$$R_{eq}=R_1+R_2    (Eq. 2.6)$$+$$R_{eq}=R_1+R_2  \qquad  (Eq. 2.6)$$
  
 Figures 2.2(a) and 2.2(b) are called //equivalent circuits// if the equivalent resistance of Fig. 2.2(b) is chosen as shown in equation (2.6). R<sub>eq</sub> of equation (2.6) is called the equivalent resistance of the series combination of resistors R<sub>1</sub> and R<sub>2</sub>.  Figures 2.2(a) and 2.2(b) are called //equivalent circuits// if the equivalent resistance of Fig. 2.2(b) is chosen as shown in equation (2.6). R<sub>eq</sub> of equation (2.6) is called the equivalent resistance of the series combination of resistors R<sub>1</sub> and R<sub>2</sub>. 
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 Combining equations (2.2) with equation (2.4) results in the following expressions for //V<sub>i</sub>// and //v<sub>2</sub>//: Combining equations (2.2) with equation (2.4) results in the following expressions for //V<sub>i</sub>// and //v<sub>2</sub>//:
  
-$$v_1= \frac{R_1}{R_1+R_2}v  (Eq. 2.7)$$+$$v_1= \frac{R_1}{R_1+R_2}v \qquad (Eq. 2.7)$$
  
-$$v_2= \frac{R_2}{R_1+R_2}v  (Eq. 2.8)$$+$$v_2= \frac{R_2}{R_1+R_2}v \qquad (Eq. 2.8)$$
  
 These results are commonly called //voltage divider// relationships, because they state that the total voltage drop across a series combination of resistors is divided among the individual resistors in the combination. The ratio of each individual resistor's voltage drop to the overall voltage drop is the same as the ratio of the individual resistance to the total resistance.  These results are commonly called //voltage divider// relationships, because they state that the total voltage drop across a series combination of resistors is divided among the individual resistors in the combination. The ratio of each individual resistor's voltage drop to the overall voltage drop is the same as the ratio of the individual resistance to the total resistance. 
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 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 //k<sup>th</sup>// resistance in a series combination of resistors given by: 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 //k<sup>th</sup>// resistance in a series combination of resistors given by:
  
-$$ v_k={R_k}{R_1+R_2+ \cdots +R_N}v     (Eq. 2.9)$$+$$ v_k=\frac{R_k}{R_1+R_2+ \cdots +R_N}v  \qquad   (Eq. 2.9)$$
  
  
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 Circuit elements are said to be connected in //parallel// if all of the elements share the same pair of nodes. An example of two circuit elements connected in parallel is shown in Fig. 2.3. Applying KVL around the loop of Fig. 2.3 results in: Circuit elements are said to be connected in //parallel// if all of the elements share the same pair of nodes. An example of two circuit elements connected in parallel is shown in Fig. 2.3. Applying KVL around the loop of Fig. 2.3 results in:
  
-$$ v_1=v_2      (Eq. 2.10)$$+$$ v_1=v_2   \qquad   (Eq. 2.10)$$
  
 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. 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.
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 {{ :learn:courses:real-analog-chapter-2:chapter2e.png |Figure 2.3}} {{ :learn:courses:real-analog-chapter-2:chapter2e.png |Figure 2.3}}
  
-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:+We can simplify circuits, which consist of resistors connected in parallel. Consider the resistive circuit shown in Fig. 2.4(a). The resistors 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_1= \frac{v}{R_1} \\
 i_2= \frac{v}{R_2} \\ i_2= \frac{v}{R_2} \\
-(Eq. 2.11) $$+\qquad (Eq. 2.11) $$
  
 Applying KCL at node a: Applying KCL at node a:
  
-$$i=i_1 + i_2      (Eq. 2.12)$$+$$i=i_1 + i_2  \qquad    (Eq. 2.12)$$
  
 Substituting equations (2.11) into equation (2.12): Substituting equations (2.11) into equation (2.12):
  
-$$ i = \left[ \frac{1}{R_1} + \frac{1}{R_2} \right] v       (Eq. 2.13)$$+$$ i = \left[ \frac{1}{R_1} + \frac{1}{R_2} \right] v   \qquad    (Eq. 2.13)$$
  
 or  or 
  
-$$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}\cdot i        (Eq. 2.14)$$    +$$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}\cdot i   \qquad     (Eq. 2.14)$$    
  
 If we set $ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}$, we can draw Fig. 2.4(b) as being equivalent to Fig. 2.4(b).  If we set $ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}$, we can draw Fig. 2.4(b) as being equivalent to Fig. 2.4(b). 
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 A parallel combination of //N// resistors $R_1, R_2, \cdots, R_N$ can be replaced with a single equivalent resistance: A parallel combination of //N// resistors $R_1, R_2, \cdots, R_N$ can be replaced with a single equivalent resistance:
  
-$$ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}+ \cdots \frac{1}{R_N}}     (Eq. 2.15)$$+$$ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}+ \cdots \frac{1}{R_N}}  \qquad   (Eq. 2.15)$$
  
 {{ :learn:courses:real-analog-chapter-2:chapter2f.png |Figure 2.4}} {{ :learn:courses:real-analog-chapter-2:chapter2f.png |Figure 2.4}}
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 For the special case of two parallel resistances, $R_1$ and $R_2$, the equivalent resistance is commonly written as: For the special case of two parallel resistances, $R_1$ and $R_2$, the equivalent resistance is commonly written as:
  
-$$R_{eq}= \frac{R_1R_2}{R_1+R_2}        (Eq. 2.16)$$+$$R_{eq}= \frac{R_1R_2}{R_1+R_2}   \qquad     (Eq. 2.16)$$
  
 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.  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. 
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 Substituting equation (2.14) into equations (2.11) results in: Substituting equation (2.14) into equations (2.11) results in:
  
-$$i_1 = \frac{1}{R_1}\cdot \frac{i}{ \frac{1}{R_1} + \frac{1}{R_2}}    (Eq. 2.17)$$+$$i_1 = \frac{1}{R_1}\cdot \frac{i}{ \frac{1}{R_1} + \frac{1}{R_2}}  \qquad  (Eq. 2.17)$$
  
 Simplifying: Simplifying:
  
-$$i_1=\frac{R_2}{R_1+R_2}      (Eq. 2.18)$$+$$i_1=\frac{R_2}{R_1+R_2}  \qquad    (Eq. 2.18)$$
  
 Likewise, for the current $i_2$: Likewise, for the current $i_2$:
  
-$$i_2 = \frac{R_1}{R_1+R_2}    (Eq. 2.19)$$+$$i_2 = \frac{R_1}{R_1+R_2}  \qquad  (Eq. 2.19)$$
  
 Equations (2.18) and (2.19) are the current //divider relationships// for two parallel resistances, so called because the current into the parallel resistance combination is divided between the two resistors. The ratio of one resistor's current to the overall current in the same as the ratio of the __other__ resistance to the total resistance. Equations (2.18) and (2.19) are the current //divider relationships// for two parallel resistances, so called because the current into the parallel resistance combination is divided between the two resistors. The ratio of one resistor's current to the overall current in the same as the ratio of the __other__ resistance to the total resistance.
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 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: 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:
  
-$$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2} + \cdots \frac{1}{R_N}} \cdot i       (Eq. 2.20)$$+$$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2} + \cdots \frac{1}{R_N}} \cdot i     \qquad  (Eq. 2.20)$$
  
 Since the voltage difference across all resistors is the same, the current through the k<sup>th</sup> resistor is, by Ohm's law: Since the voltage difference across all resistors is the same, the current through the k<sup>th</sup> resistor is, by Ohm's law:
  
-$$i_k=\frac{v}{R_k}     (Eq. 2.21)$$+$$i_k=\frac{v}{R_k}   \qquad  (Eq. 2.21)$$
  
 Where $R_k$ is the resistance of the k<sup>th</sup> resistor. Combining equations (2.20) and (2.21) gives: Where $R_k$ is the resistance of the k<sup>th</sup> resistor. Combining equations (2.20) and (2.21) gives:
  
-$$i_k= \frac{\frac{1}{R_k}}{\frac{1}{R_1}+\frac{1}{R_2}+ \cdots \frac{1}{R_N}} \cdot i       (Eq. 2.22)$$+$$i_k= \frac{\frac{1}{R_k}}{\frac{1}{R_1}+\frac{1}{R_2}+ \cdots \frac{1}{R_N}} \cdot i    \qquad   (Eq. 2.22)$$
  
 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: 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:
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 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 //k<sup>th</sup>// resistance in a parallel combination of resistors is given by: 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 //k<sup>th</sup>// resistance in a parallel combination of resistors is given by:
  
-$$i_k=\frac{G_k}{G_1+G_2+ \cdots + G_N}i     (Eq. 2.23)$$+$$i_k=\frac{G_k}{G_1+G_2+ \cdots + G_N}i  \qquad  (Eq. 2.23)$$
  
 Where //i// is the total current through the parallel combination of resistors.  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, in lieu of equation (2.16). +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 $R_1$ 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, in lieu of equation (2.16). 
  
 Double-checking results for parallel resistances: Double-checking results for parallel resistances:
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 By voltage division, the voltages $v_b$ and $v_a$ (relative to ground) are: By voltage division, the voltages $v_b$ and $v_a$ (relative to ground) are:
  
-$$v_b=\frac{R+\Delta R}{2R+\Delta R}V_s$$+$$v_b=\frac{(R+\Delta R)}{2R+\Delta R}V_s$$
  
 and and
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 We will model a real or //practical// voltage source as a series connection of an ideal voltage source and an //internal resistance//. This model is depicted schematically in Fig. 2.5, in which the non-ideal voltage source contains an ideal voltage source providing voltage $V_s$ and an internal resistance $R_s$. The non-ideal voltage source delivers a voltage V and a current //i//, where: We will model a real or //practical// voltage source as a series connection of an ideal voltage source and an //internal resistance//. This model is depicted schematically in Fig. 2.5, in which the non-ideal voltage source contains an ideal voltage source providing voltage $V_s$ and an internal resistance $R_s$. The non-ideal voltage source delivers a voltage V and a current //i//, where:
  
-$$V=V_s-i \codt R_s     (Eq. 2.24)$$+$$V=V_s-i \cdot R_s   \qquad  (Eq. 2.24)$$
  
 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.  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. 
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 ==== Exercises ==== ==== 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?+1. A voltage source with an internal resistance of 2‎Ω as shown below is used to apply power to a 3Ω resistor. What voltage would you measure across the 3Ω resistor?
  
 {{ :learn:courses:real-analog-chapter-2:chapter2z.png |Exercise image 1.}} {{ :learn:courses:real-analog-chapter-2:chapter2z.png |Exercise image 1.}}
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 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.  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's voltage difference should be as small as possible - an ammter, therefore, should have an extremely small internal resistance. +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's voltage difference should be as small as possible - an ammeter, therefore, should have an extremely small internal resistance. 
  
 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 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. 
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 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: 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:
  
-$$i=i_s \frac{R_M}{R+R_M}    (Eq. 2.26)$$+$$i=i_s \frac{R_M}{R+R_M}  \qquad  (Eq. 2.26)$$
  
 The voltage across the resistor R is then, by Ohm's law: The voltage across the resistor R is then, by Ohm's law:
  
-$$V=i_s \frac{R \cdot R_M}{R+R_M}      (Eq. 2.27)$$+$$V=i_s \frac{R \cdot R_M}{R+R_M}   \qquad   (Eq. 2.27)$$
  
 If $R_M >>R$, this expression simplifies to: If $R_M >>R$, this expression simplifies to:
  
-$$V \approx i_s \frac{R \cdot R_M}{R_M} = R \cdot i_s        (Eq. 2.28)$$+$$V \approx i_s \frac{R \cdot R_M}{R_M} = R \cdot i_s    \qquad    (Eq. 2.28)$$
  
-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. +And negligible error is introduced 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. 
  
 {{ :learn:courses:real-analog-chapter-2:chapter2bb.png |Figure 2.8}} {{ :learn:courses:real-analog-chapter-2:chapter2bb.png |Figure 2.8}}
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 {{ :learn:courses:real-analog-chapter-2:chapter2cc.png |Figure 2.9.}} {{ :learn:courses:real-analog-chapter-2:chapter2cc.png |Figure 2.9.}}
 +
 +----
 +
 +==== 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. 
 +
 +{{ :learn:courses:real-analog-chapter-2:chapter2dd.png |Caution example image.}}
 +
 +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's behavior - that is, our measurement will always change the parameter being measured. One goal when measuring a voltage or current is to ensure that the measurement effects are negligible. 
 +  * 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?
 +
 +{{ :learn:courses:real-analog-chapter-2:chapter2ee.png |Exercise image 1.}}
 +
 +----
 +
 +[[{}/learn/courses/real-analog-chapter-1/start|Back to Chapter 1]]
 +[[{}/learn/courses/real-analog-chapter-3/start|Go to Chapter 3]]
 +