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| learn:courses:real-analog-chapter-3:start [2017/04/20 18:38] – [Example 3.1] Martha | learn:courses:real-analog-chapter-3:start [2022/10/04 11:38] (current) – external edit 127.0.0.1 | ||
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| ====== Real Analog: Chapter 3 ====== | ====== Real Analog: Chapter 3 ====== | ||
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| ====== 3. Introduction and Chapter Objectives ====== | ====== 3. Introduction and Chapter Objectives ====== | ||
| - | In Chapters [[https:// | + | In Chapters [[/ |
| * Ohm's Law | * Ohm's Law | ||
| * Kirchhoff' | * Kirchhoff' | ||
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| ==== Supernodes ==== | ==== Supernodes ==== | ||
| - | In the previous examples, we identified __dependent__ nodes and determined __constrained__ voltages. Kirchhoff' | + | In the previous examples, we identified __dependent__ nodes and determined __constrained__ voltages. Kirchhoff' |
| Example: for the circuit below, determine the voltage difference, V, across the 2mA source. | Example: for the circuit below, determine the voltage difference, V, across the 2mA source. | ||
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| * Use Ohm's law to write KCL at each independent node, in terms of the node voltages. This will result in N equations in N unknowns, where N is the number of node voltages. Independent " | * Use Ohm's law to write KCL at each independent node, in terms of the node voltages. This will result in N equations in N unknowns, where N is the number of node voltages. Independent " | ||
| * Solve the equations of step 4 to determine the node voltages. | * Solve the equations of step 4 to determine the node voltages. | ||
| - | * Use the node voltages to determine any other esired | + | * Use the node voltages to determine any other desired |
| * Modifications to the above approach are allowed. For example, it is not necessary to define supernodes in step 4 above. Once can define unknown voltages at either terminal of a voltage source and write KCL at each of these nodes. However, the unknown current through the voltage source must be accounted for when writing KCL - this introduces an additional unknown into the governing equations. This added unknown requires an additional equation. This equation is obtained by explicitly writing a constraint equation relating the voltages at the two terminals of the voltage source. | * Modifications to the above approach are allowed. For example, it is not necessary to define supernodes in step 4 above. Once can define unknown voltages at either terminal of a voltage source and write KCL at each of these nodes. However, the unknown current through the voltage source must be accounted for when writing KCL - this introduces an additional unknown into the governing equations. This added unknown requires an additional equation. This equation is obtained by explicitly writing a constraint equation relating the voltages at the two terminals of the voltage source. | ||
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| + | From the above figure, the voltage drops across the resistor $R_1$ can then be determined as: | ||
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| + | $$V_1 = R_1 i_1$$ | ||
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| + | Since only mesh current $i_1$ passes through the resistor $R_1$. Likewise, the voltage drop for the resistor $R_4$ is | ||
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| + | Since mesh currents $i_1$ and $i_4$ both pass through $R_4$ and the current $i_2$ is in the opposite direction to our assumed polarity for the voltage $V_4$. | ||
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| + | Using the above expressions for $V_1$ and $V_4$, we can write KVL for the first mesh loop as: | ||
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| + | $$-V_S + R_1 i_1 + R_4 (i_1 - i_2) =0$$ | ||
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| + | When we write KVL for the mesh current $i_2$ in our example, we choose voltage polarities for resistors $R_4$, $R_2$, and $R_5$ as shown in the figure below – these polarities agree with the passive sign convention for voltages relative to the direction of the mesh current $i_2$. | ||
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| Using the above sign conventions, | Using the above sign conventions, | ||
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| - | Define three mesh currents for each of the three meshes in the above circuit and define unknown voltages $V_1$ abd $V_3$ across the two current sources as shown below. | + | Define three mesh currents for each of the three meshes in the above circuit and define unknown voltages $V_1$ and $V_3$ across the two current sources as shown below. |
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| - | === //Step 1//: Define mesh loops === | + | === Step 1: Define mesh loops === |
| Replacing the two current sources with open circuits and the two voltage sources with short circuits results in a single mesh current, $i_1$, as shown below. | Replacing the two current sources with open circuits and the two voltage sources with short circuits results in a single mesh current, $i_1$, as shown below. | ||
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| - | === //Step 2//: Constrained loops, version 1 === | + | === Step 2: Constrained loops, version 1 === |
| Initially, we choose the constrained loops shown below. Note that each loop passes through only one source and has the magnitude and direction imposed by the source. | Initially, we choose the constrained loops shown below. Note that each loop passes through only one source and has the magnitude and direction imposed by the source. | ||
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| - | ===//Step 3//: Write KVL around the mesh loops === | + | ===Step 3: Write KVL around the mesh loops === |
| Our example has only one mesh current, so only one KVL equation is required. This equation is: | Our example has only one mesh current, so only one KVL equation is required. This equation is: | ||
| $$-8V + 2\Omega \left( i_1 + 1A - 2A \right) + 4\Omega \left( i_1 - 2A \right) + 10V + 6\Omega \left( i_1 \right) = 0$$ | $$-8V + 2\Omega \left( i_1 + 1A - 2A \right) + 4\Omega \left( i_1 - 2A \right) + 10V + 6\Omega \left( i_1 \right) = 0$$ | ||
| - | === //Step 4//: Solve the system of equations to determine the mesh currents of the circuit === | + | === Step 4: Solve the system of equations to determine the mesh currents of the circuit === |
| Solving the above equation results in $i_1 = 0.667A$. The current through the $4\Omega$ resistor is then, accounting for the 2A constrained loop passing through the resistor, $i = i_1 -2A = -1.333A$. | Solving the above equation results in $i_1 = 0.667A$. The current through the $4\Omega$ resistor is then, accounting for the 2A constrained loop passing through the resistor, $i = i_1 -2A = -1.333A$. | ||
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