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| learn:courses:real-analog-chapter-2:start [2017/04/19 17:40] – [Series Connections] Martha | learn:courses:real-analog-chapter-2:start [2023/02/12 05:57] (current) – external edit 127.0.0.1 | ||
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| - | ====== Real Analog: Chapter 2 ====== | + | ===== Real Analog: Chapter 2 ====== |
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| - | --> Chapter 2 Downloads# | + | --> Chapter 2 Materials# |
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| - | ====== 2. Introduction and Chapter Objectives | + | ===== 2. Introduction and Chapter Objectives ===== |
| - | In [[https:// | + | In [[/ |
| In the next few chapters, we will still apply Kirchhoff' | In the next few chapters, we will still apply Kirchhoff' | ||
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| or | or | ||
| - | $$i_1=i_2 | + | $$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 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. | 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. | ||
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| Applying KVL around the loop: | Applying KVL around the loop: | ||
| - | $$ -v+v_1+v_2=0 \Rightarrow v=v_1+v_2 | + | $$ -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: | Substituting equations (2.2) into equation (2.3) and solving for the current //i// results in: | ||
| - | $$ i = \frac{v}{R_1+R_2} | + | $$ i = \frac{v}{R_1+R_2} |
| - | Now cinsider | + | Now consider |
| - | $$ i = \frac{v}{R_{eq}} | + | $$ i = \frac{v}{R_{eq}} |
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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: | ||
| - | $$R_{eq}=R_1+R_2 | + | $$R_{eq}=R_1+R_2 |
| Figures 2.2(a) and 2.2(b) are called // | Figures 2.2(a) and 2.2(b) are called // | ||
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| Combining equations (2.2) with equation (2.4) results in the following expressions for // | 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_1= \frac{R_1}{R_1+R_2}v |
| - | $$v_2= \frac{R_2}{R_1+R_2}v | + | $$v_2= \frac{R_2}{R_1+R_2}v |
| These results are commonly called //voltage divider// relationships, | These results are commonly called //voltage divider// relationships, | ||
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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 // | 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 // | ||
| - | $$ v_k={R_k}{R_1+R_2+ \cdots +R_N}v | + | $$ v_k=\frac{R_k}{R_1+R_2+ \cdots +R_N}v |
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| Circuit elements are said to be connected in // | Circuit elements are said to be connected in // | ||
| - | $$ v_1=v_2 | + | $$ 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. | 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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| - | We can simplify circuits, which consist of resistors connected in parallel. Consider the resistive circuit shown in Fig. 2.4(a). The ressitors | + | We can simplify circuits, which consist of resistors connected in parallel. Consider the resistive circuit shown in Fig. 2.4(a). The resistors |
| $$ i_1= \frac{v}{R_1} \\ | $$ i_1= \frac{v}{R_1} \\ | ||
| i_2= \frac{v}{R_2} \\ | i_2= \frac{v}{R_2} \\ | ||
| - | (Eq. 2.11) $$ | + | \qquad |
| Applying KCL at node a: | Applying KCL at node a: | ||
| - | $$i=i_1 + i_2 (Eq. 2.12)$$ | + | $$i=i_1 + i_2 |
| Substituting equations (2.11) into equation (2.12): | Substituting equations (2.11) into equation (2.12): | ||
| - | $$ i = \left[ \frac{1}{R_1} + \frac{1}{R_2} \right] v | + | $$ i = \left[ \frac{1}{R_1} + \frac{1}{R_2} \right] v \qquad |
| or | or | ||
| - | $$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}\cdot i (Eq. 2.14)$$ | + | $$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2}}\cdot i \qquad |
| If we set $ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}$, | If we set $ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}$, | ||
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| A parallel combination of //N// resistors $R_1, R_2, \cdots, R_N$ can be replaced with a single equivalent resistance: | A parallel combination of //N// resistors $R_1, R_2, \cdots, R_N$ can be replaced with a single equivalent resistance: | ||
| - | $$ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}+ \cdots \frac{1}{R_N}} | + | $$ R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}+ \cdots \frac{1}{R_N}} |
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| For the special case of two parallel resistances, | For the special case of two parallel resistances, | ||
| - | $$R_{eq}= \frac{R_1R_2}{R_1+R_2} | + | $$R_{eq}= \frac{R_1R_2}{R_1+R_2} |
| This alternative way to calculate $R_{eq}$ can be also used to calculate $R_{eq}$ for larger numbers of parallel resistors since any number of resistors could be combined two at a time. | This alternative way to calculate $R_{eq}$ can be also used to calculate $R_{eq}$ for larger numbers of parallel resistors since any number of resistors could be combined two at a time. | ||
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| Substituting equation (2.14) into equations (2.11) results in: | Substituting equation (2.14) into equations (2.11) results in: | ||
| - | $$i_1 = \frac{1}{R_1}\cdot \frac{i}{ \frac{1}{R_1} + \frac{1}{R_2}} | + | $$i_1 = \frac{1}{R_1}\cdot \frac{i}{ \frac{1}{R_1} + \frac{1}{R_2}} |
| Simplifying: | Simplifying: | ||
| - | $$i_1=\frac{R_2}{R_1+R_2} | + | $$i_1=\frac{R_2}{R_1+R_2} |
| Likewise, for the current $i_2$: | Likewise, for the current $i_2$: | ||
| - | $$i_2 = \frac{R_1}{R_1+R_2} | + | $$i_2 = \frac{R_1}{R_1+R_2} |
| Equations (2.18) and (2.19) are the current //divider relationships// | Equations (2.18) and (2.19) are the current //divider relationships// | ||
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| The above results can be generalized for a series combination of //N// resistances. By Ohm's law, $v = R_{eq} i$. Substituting our previous result for the equivalent resistance for a parallel combination of //N// resistors results in: | The above results can be generalized for a series combination of //N// resistances. By Ohm's law, $v = R_{eq} i$. Substituting our previous result for the equivalent resistance for a parallel combination of //N// resistors results in: | ||
| - | $$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2} + \cdots \frac{1}{R_N}} \cdot i | + | $$v= \frac{1}{\frac{1}{R_1}+\frac{1}{R_2} + \cdots \frac{1}{R_N}} \cdot i \qquad |
| Since the voltage difference across all resistors is the same, the current through the k< | Since the voltage difference across all resistors is the same, the current through the k< | ||
| - | $$i_k=\frac{v}{R_k} | + | $$i_k=\frac{v}{R_k} |
| Where $R_k$ is the resistance of the k< | Where $R_k$ is the resistance of the k< | ||
| - | $$i_k= \frac{\frac{1}{R_k}}{\frac{1}{R_1}+\frac{1}{R_2}+ \cdots \frac{1}{R_N}} \cdot i | + | $$i_k= \frac{\frac{1}{R_k}}{\frac{1}{R_1}+\frac{1}{R_2}+ \cdots \frac{1}{R_N}} \cdot i |
| It is often more convenient to provide the generalized result of equation (2.20) in terms of the conductance of the individual resistors. Recall that the conductance is the reciprocal of the resistance, $G=\frac{1}{R}$. Thus, equation (2.22) can be re-expressed as follows: | It is often more convenient to provide the generalized result of equation (2.20) in terms of the conductance of the individual resistors. Recall that the conductance is the reciprocal of the resistance, $G=\frac{1}{R}$. Thus, equation (2.22) can be re-expressed as follows: | ||
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| The Current through any resistor in a parallel combination of //N// resistances is proportional to the total current into the combination of resistors. The constant of proportionality is the same as the ratio of the conductance of the individual resistor value to the total conductance of the parallel combination. For example, the current through the // | The Current through any resistor in a parallel combination of //N// resistances is proportional to the total current into the combination of resistors. The constant of proportionality is the same as the ratio of the conductance of the individual resistor value to the total conductance of the parallel combination. For example, the current through the // | ||
| - | $$i_k=\frac{G_k}{G_1+G_2+ \cdots + G_N}i | + | $$i_k=\frac{G_k}{G_1+G_2+ \cdots + G_N}i |
| Where //i// is the total current through the parallel combination of resistors. | Where //i// is the total current through the parallel combination of resistors. | ||
| - | One final comment about notation: two parallel bars are commonly used as shorthand notation to indicate that two circuit elements are in parallel. For example, the notation $R_1 \parallel R_2$ indicates that the resistors $R_2$ and $R_2$ are in parallel. The notation $R_1 \parallel R_2$ is often used as shorthand notation for the __equivalent resistance__ of the parallel resistance combination, | + | One final comment about notation: two parallel bars are commonly used as shorthand notation to indicate that two circuit elements are in parallel. For example, the notation $R_1 \parallel R_2$ indicates that the resistors $R_1$ and $R_2$ are in parallel. The notation $R_1 \parallel R_2$ is often used as shorthand notation for the __equivalent resistance__ of the parallel resistance combination, |
| Double-checking results for parallel resistances: | Double-checking results for parallel resistances: | ||
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| By voltage division, the voltages $v_b$ and $v_a$ (relative to ground) are: | By voltage division, the voltages $v_b$ and $v_a$ (relative to ground) are: | ||
| - | $$v_b=\frac{R+\Delta R}{2R+\Delta R}V_s$$ | + | $$v_b=\frac{(R+\Delta R)}{2R+\Delta R}V_s$$ |
| and | and | ||
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| We will model a real or // | We will model a real or // | ||
| - | $$V=V_s-i \cdot R_s | + | $$V=V_s-i \cdot R_s \qquad |
| 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? |
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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' | + | 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 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} | + | $$i=i_s \frac{R_M}{R+R_M} |
| The voltage across the resistor R is then, by Ohm's law: | The voltage across the resistor R is then, by Ohm's law: | ||
| - | $$V=i_s \frac{R \cdot R_M}{R+R_M} | + | $$V=i_s \frac{R \cdot R_M}{R+R_M} |
| If $R_M >>R$, this expression simplifies to: | If $R_M >>R$, this expression simplifies to: | ||
| - | $$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 |
| - | And negligible error is introduces | + | And negligible error is introduced |
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