Differences
This shows you the differences between two versions of the page.
| Both sides previous revisionPrevious revisionNext revision | Previous revision | ||
| learn:courses:real-analog-chapter-4:start [2017/04/21 20:50] – [4.2: Linear Systems] Martha | learn:courses:real-analog-chapter-4:start [2022/10/04 11:38] (current) – external edit 127.0.0.1 | ||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ====== Real Analog: Chapter 4 ====== | ====== Real Analog: Chapter 4 ====== | ||
| - | --> | + | [[{}/ |
| - | * {{ : | + | --> |
| - | * {{ : | + | |
| - | * {{ : | + | |
| - | * {{ : | + | * {{ : |
| - | * {{ : | + | * {{ : |
| - | * {{ : | + | * [[http:// |
| - | * {{ : | + | * [[http:// |
| - | * {{ : | + | * [[http:// |
| - | * {{ : | + | * Chapter 4 Videos: |
| - | * {{ : | + | * [[http:// |
| - | * {{ : | + | * [[http:// |
| + | * [[http:// | ||
| + | * {{ : | ||
| + | * {{ : | ||
| + | * {{ : | ||
| + | * {{ : | ||
| + | * {{ : | ||
| + | * {{ : | ||
| + | * {{ : | ||
| + | * {{ : | ||
| + | * {{ : | ||
| + | * {{ : | ||
| + | * {{ : | ||
| * {{ : | * {{ : | ||
| * {{ : | * {{ : | ||
| Line 100: | Line 111: | ||
| {{ : | {{ : | ||
| - | We previously wrote mesh equations for this circuit (for a specific value of $V_{in}$) in [[https:// | + | We previously wrote mesh equations for this circuit (for a specific value of $V_{in}$) in [[/ |
| {{ : | {{ : | ||
| Line 114: | Line 125: | ||
| $$V= 2 \left( i_1 - 3V_x \right)$$ | $$V= 2 \left( i_1 - 3V_x \right)$$ | ||
| - | The above four equations provide an inpuot-output description of the circuit. If desired, they can be combined to eliminate all variables except $V_{in}$ and $V$ and re-written in the form $V = f \{ V_{in} \}$ per equation (1). __Note that all information about the original system, except the relationship between the input and output signals, is lost once we do this__. | + | The above four equations provide an inppot-output description of the circuit. If desired, they can be combined to eliminate all variables except $V_{in}$ and $V$ and re-written in the form $V = f \{ V_{in} \}$ per equation (1). __Note that all information about the original system, except the relationship between the input and output signals, is lost once we do this__. |
| The system-level block diagram for the circuit might then be drawn as: | The system-level block diagram for the circuit might then be drawn as: | ||
| Line 288: | Line 299: | ||
| {{ : | {{ : | ||
| - | The circuit above can be consider | + | The circuit above can be considered |
| {{ : | {{ : | ||
| Line 456: | Line 467: | ||
| First, we will kill all sources in circuit A and determine the voltage resulting from an applied current, as shown in Figure 4.8 below. With the sources killed, circuit A will look strictly like an equivalent resistance to any external circuitry. This equivalent resistance is designated as $R_{TH}$ in Figure 4.8. The voltage resulting from an applied current, with circuit A dead is: | First, we will kill all sources in circuit A and determine the voltage resulting from an applied current, as shown in Figure 4.8 below. With the sources killed, circuit A will look strictly like an equivalent resistance to any external circuitry. This equivalent resistance is designated as $R_{TH}$ in Figure 4.8. The voltage resulting from an applied current, with circuit A dead is: | ||
| - | $$v_1 = R_{TH} \cdot i | + | $$v_1 = R_{TH} \cdot i |
| {{ : | {{ : | ||
| Line 466: | Line 477: | ||
| Superimposing the two voltages above results in: | Superimposing the two voltages above results in: | ||
| - | $$v=v_1 + v_{OC} | + | $$v=v_1 + v_{OC} |
| Or | Or | ||
| - | $$v= R_{TH} \cdot i + v_{OC} | + | $$v= R_{TH} \cdot i + v_{OC} |
| Equation (4.5) is Thévenin’s theorem. It indicates that the voltage-current characteristic of any linear circuit (with the exception noted below) can be duplicated by an independent voltage source in series with a resistance $R_{TH}$, known as the //Thévenin resistance// | Equation (4.5) is Thévenin’s theorem. It indicates that the voltage-current characteristic of any linear circuit (with the exception noted below) can be duplicated by an independent voltage source in series with a resistance $R_{TH}$, known as the //Thévenin resistance// | ||
| Line 498: | Line 509: | ||
| {{ : | {{ : | ||
| - | Applying KCL at $v_0$, we obtain: $-2A + \frac{v_0 - 6V}{6\Omega} + \frac{v_0}{3|Omega} = 0 \Rightarrow v_0 = v_{OC} = 6V$. Thus, the Thévenin equivalent circuit is on the left below. Re-introducing the load resistance, as shown on the right below, allows us to easily analyze the overall circuit. | + | Applying KCL at $v_0$, we obtain: $-2A + \frac{v_0 - 6V}{6\Omega} + \frac{v_0}{3\Omega} = 0 \Rightarrow v_0 = v_{OC} = 6V$. Thus, the Thévenin equivalent circuit is on the left below. Re-introducing the load resistance, as shown on the right below, allows us to easily analyze the overall circuit. |
| {{ : | {{ : | ||
| Line 509: | Line 520: | ||
| Once again, we kill all sources in circuit A, as shown in Figure 4.8 above but this time we determine the current resulting from an applied voltage. With the sources killed, circuit A still looks like an equivalent resistance to any external circuitry. This equivalent resistance is designated as $R_{TH}$ in Figure 4.8. The current resulting from an applied voltage, with circuit A dead is: | Once again, we kill all sources in circuit A, as shown in Figure 4.8 above but this time we determine the current resulting from an applied voltage. With the sources killed, circuit A still looks like an equivalent resistance to any external circuitry. This equivalent resistance is designated as $R_{TH}$ in Figure 4.8. The current resulting from an applied voltage, with circuit A dead is: | ||
| - | $$i_1 = \frac{v}{R_{TH}} | + | $$i_1 = \frac{v}{R_{TH}} |
| Notice that equation (4.6) can be obtained by rearranging equation (4.3). | Notice that equation (4.6) can be obtained by rearranging equation (4.3). | ||
| Line 519: | Line 530: | ||
| Employing superposition, | Employing superposition, | ||
| - | $$i = i_1 - i_{SC} | + | $$i = i_1 - i_{SC} |
| So | So | ||
| - | $$i = \frac{v}{R_{TH}} - i_{SC} | + | $$i = \frac{v}{R_{TH}} - i_{SC} |
| Equation (4.8) is Norton’s theorem. It indicates that the voltage-current characteristic of any linear circuit (with the exception noted below) can be duplicated by an independent current source in parallel with a resistance. The current source has the magnitude $i_{SC}$ and the resistance is $R_{TH}$, where $i_{SC}$ is the current seen at the circuit’s terminals if the terminals are short-circuited and $R_{TH}$ is the equivalent resistance of the circuit seen from the two terminals, with all independent sources in the circuit killed. The equivalent Norton circuit is shown in Fig. 4.12. | Equation (4.8) is Norton’s theorem. It indicates that the voltage-current characteristic of any linear circuit (with the exception noted below) can be duplicated by an independent current source in parallel with a resistance. The current source has the magnitude $i_{SC}$ and the resistance is $R_{TH}$, where $i_{SC}$ is the current seen at the circuit’s terminals if the terminals are short-circuited and $R_{TH}$ is the equivalent resistance of the circuit seen from the two terminals, with all independent sources in the circuit killed. The equivalent Norton circuit is shown in Fig. 4.12. | ||
| Line 530: | Line 541: | ||
| === Procedure for Determining Norton Equivalent Circuit: === | === Procedure for Determining Norton Equivalent Circuit: === | ||
| - | - Ordered List ItemIdentify | + | - Identify |
| - Determine the equivalent resistance $R_{TH}$ of the circuit. The approach for determining $R_{TH}$ is the same for Norton circuits as Thévenin circuits. | - Determine the equivalent resistance $R_{TH}$ of the circuit. The approach for determining $R_{TH}$ is the same for Norton circuits as Thévenin circuits. | ||
| - Re-activate the sources and determine the short-circuit current $i_{SC}$ across the circuit terminals. Use any analysis approach you choose to determine the short-circuit current. | - Re-activate the sources and determine the short-circuit current $i_{SC}$ across the circuit terminals. Use any analysis approach you choose to determine the short-circuit current. | ||
| Line 751: | Line 762: | ||
| + | [[{}/ | ||
| + | [[{}/ | ||