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| ====== Real Analog: Chapter 6 ====== | ====== Real Analog: Chapter 6 ====== | ||
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| + | * Lecture Material: | ||
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| + | * Chapter 6 Videos: | ||
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| ====== 6. Introduction and Chapter Objectives ====== | ====== 6. Introduction and Chapter Objectives ====== | ||
| So far, we have considered circuits that have been governed by algebraic relations. These circuits have, in general, contained only power sources and resistive elements. All elements in these circuits, therefore, have either __supplied power__ from external sources or __dissipated__ power. For these resistive circuits, we can apply either time-varying or constant signals to the circuit without really affecting our analysis approach. Ohm’s law, for example, is equally applicable to time-varying or constant voltages and currents: | So far, we have considered circuits that have been governed by algebraic relations. These circuits have, in general, contained only power sources and resistive elements. All elements in these circuits, therefore, have either __supplied power__ from external sources or __dissipated__ power. For these resistive circuits, we can apply either time-varying or constant signals to the circuit without really affecting our analysis approach. Ohm’s law, for example, is equally applicable to time-varying or constant voltages and currents: | ||
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| $${f} \left( t \right) = Ae^{-at}$$ | $${f} \left( t \right) = Ae^{-at}$$ | ||
| - | Where $a>0$. The function ${f}(t)$ is illustrated in Figure 6.18. The value of the function is //A// at $t = 0$ and decreases to zero as $t \rightarrow \infty$. As $t \rightarrow -infty$ the function increases without bound. The constant a dictates the rate at which the function decreases as time increases. | + | Where $a>0$. The function ${f}(t)$ is illustrated in Figure 6.18. The value of the function is //A// at $t = 0$ and decreases to zero as $t \rightarrow \infty$. As $t \rightarrow -\infty$ the function increases without bound. The constant a dictates the rate at which the function decreases as time increases. |
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| Or, using the unit step function to limit the function to positive values of time: | Or, using the unit step function to limit the function to positive values of time: | ||
| - | $${f} \left( t \right) = Ae\frac{1t}{\tau} \cdot u_o \left( t \right) | + | $${f} \left( t \right) = Ae\frac{-t}{\tau} \cdot u_o \left( t \right) |
| The function of equations (6.10) and (6.11) is illustrated in Figure 6.19. The time constant, $\tau$, is a positive number which dictates the rate at which the function will decay with time. When the time $t = \tau, {f} \left( t \right) Ae^{-1} = 0.368A$ and the function has decayed to 36.8% of its original value. In fact, __the function decreases by 36.8% every $\tau$ seconds__. Therefore, a signal with a small time constant decays more rapidly than a signal with a large time constant, as illustrated in Figure 6.20. | The function of equations (6.10) and (6.11) is illustrated in Figure 6.19. The time constant, $\tau$, is a positive number which dictates the rate at which the function will decay with time. When the time $t = \tau, {f} \left( t \right) Ae^{-1} = 0.368A$ and the function has decayed to 36.8% of its original value. In fact, __the function decreases by 36.8% every $\tau$ seconds__. Therefore, a signal with a small time constant decays more rapidly than a signal with a large time constant, as illustrated in Figure 6.20. | ||
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| * An exponential function, defined for $t>0$, is mathematically defined as: | * An exponential function, defined for $t>0$, is mathematically defined as: | ||
| - | $${f} \left( t \right) = AE^\frac{-t}{\tau} \cdot u_0 \left( t \right)$$ | + | $${f} \left( t \right) = Ae^\frac{-t}{\tau} \cdot u_0 \left( t \right)$$ |
| The function has an initial value, $A$, and a time constant, $\tau$. The time constant indicates how quickly the function decays; the value of the function decreases by 63.2% every $\tau$ seconds. Exponential functions are important to use because the solutions of linear, constant coefficient, | The function has an initial value, $A$, and a time constant, $\tau$. The time constant indicates how quickly the function decays; the value of the function decreases by 63.2% every $\tau$ seconds. Exponential functions are important to use because the solutions of linear, constant coefficient, | ||
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| ==== Example 6.3 ==== | ==== Example 6.3 ==== | ||
| - | If the voltage as a function of time across a capacitor with capacitance $C = 1\muF$ is as shown below, determine the current as a function of time through the capacitor. | + | If the voltage as a function of time across a capacitor with capacitance $C = 1 \mu F$ is as shown below, determine the current as a function of time through the capacitor. |
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| - Determine voltage divider relationships to provide $v_1$ and $v_2$ for the two uncharged series capacitors shown below. Use your result to determine $v_2$ if $C_1 = C_2 = 10 \mu F$. | - Determine voltage divider relationships to provide $v_1$ and $v_2$ for the two uncharged series capacitors shown below. Use your result to determine $v_2$ if $C_1 = C_2 = 10 \mu F$. | ||
| - | ==== 6.4: Inductors ==== | + | ===== 6.4: Inductors |
| We continue our study of energy storage elements with a discussion of // | We continue our study of energy storage elements with a discussion of // | ||
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| - | We will denote the total magnetic flux created by the inductor by , as shown in Figure 6.28. For a linear inductor, the flux is proportional to the current passing through the wound wires. The constant of proportionality is the inductance, //L//: | + | We will denote the total magnetic flux created by the inductor by $\psi$, as shown in Figure 6.28. For a linear inductor, the flux is proportional to the current passing through the wound wires. The constant of proportionality is the inductance, //L//: |
| $$\psi(t) = Li(t) (Eq. 6.26)$$ | $$\psi(t) = Li(t) (Eq. 6.26)$$ | ||
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| * In the time period 0<t<1 msec, the voltage is constant and positive so the current will increase linearly. The total current change during this time period is the area under the voltage curve curve, divided by the inductance: $\frac{1}{0.01}(10V)(1x10^{-3} sec) = 1A$ | * In the time period 0<t<1 msec, the voltage is constant and positive so the current will increase linearly. The total current change during this time period is the area under the voltage curve curve, divided by the inductance: $\frac{1}{0.01}(10V)(1x10^{-3} sec) = 1A$ | ||
| * In the time period $1< | * In the time period $1< | ||
| - | * In the time period $2< | + | * In the time period $2< |
| * In the time period 3<t<4 seconds, the voltage is zero. The integral of zero over any time period is zero, so there is no change in current during this time range and the current remains constant at 0A. | * In the time period 3<t<4 seconds, the voltage is zero. The integral of zero over any time period is zero, so there is no change in current during this time range and the current remains constant at 0A. | ||
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| As was previously done for capacitors, we integrate the power with respect to time to get the energy stored in the inductor: | As was previously done for capacitors, we integrate the power with respect to time to get the energy stored in the inductor: | ||
| - | $$W_L(t) = \sum_{-\infty}^{t}Li(\xi}\frac{di(\xi}{dt}dt$$ | + | $$W_L(t) = \sum_{-\infty}^{t} Li(\xi)\frac{di(\xi)}{dt}dt$$ |
| Which, after some manipulation (comparable to the approach taken when we calculated energy storage in capacitors), | Which, after some manipulation (comparable to the approach taken when we calculated energy storage in capacitors), | ||
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| $$\frac{1}{L_{eq}} = \sum_{k=1}^{N} \frac{1}{L_k}$$ | $$\frac{1}{L_{eq}} = \sum_{k=1}^{N} \frac{1}{L_k}$$ | ||
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| + | ==== Practical Inductors ==== | ||
| + | Most commercially available inductors are manufactured by winding wire in various coil configurations around a core. Cores can be a variety of shapes; Figure 6.28 in this chapter shows a core, which is basically a cylindrical bar. Toroidal cores are also fairly common – a closely wound toroidal core has the advantage that the magnetic field is confined nearly entirely to the space inside the winding. | ||
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| + | Inductors are available with values from less than 1 micro-Henry ($1 \mu H = 10^{-6}$ Henries) up to tens of Henries. A 1H inductor is very large; inductances of most commercially available inductors are measured in millihenries ($1mH = 10^{-3}$ Henries) or microhenries. Larger inductors are generally used for low-frequency applications (in which the signals vary slowly with time). | ||
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| + | Attempts at creating inductors in integrated-circuit form have been largely unsuccessful; | ||
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| + | Practical inductors, unlike the ideal inductors discussed in this chapter, dissipate power. An equivalent circuit model for a practical inductor is generally created by placing a resistance in series with an ideal inductor, as shown in Figure 6.33. | ||
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| + | ==== Section Summary ==== | ||
| + | * Inductors store magnetic energy. This energy is stored in a magnetic field (typically) generated by a coiled wire wrapped around a core material. | ||
| + | * Inductor energy storage is dependent upon the current through the inductor, if the inductor current is known, the energy in the inductor is known. | ||
| + | * The voltage-current relationship for an inductor is: | ||
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| + | $$v(t) = L \frac{di(t)}{dt}$$ | ||
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| + | Where L is the inductance of the inductor. Units of inductance are Henries (abbreviated H). The inductance of an inductor, very roughly speaking, gives an indication of how much energy it can store. | ||
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| + | * The above voltage-current relation results in the following important properties of inductors: | ||
| + | * If the inductor current is constant, the voltage across the inductor is zero. Thus, if the inductor current is constant, the inductor can be modeled as a short circuit. | ||
| + | * Changing the inductor current instantaneously requires infinite power. Thus (for now, anyway) we will assume that inductors cannot instantaneously change their current. | ||
| + | * Inductors placed in series or parallel with one another can be modeled as a single equivalent inductance. Thus, inductors in series or in parallel are not “independent” energy storage elements. | ||
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| + | ---- | ||
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| + | ==== Exercises ==== | ||
| + | - Determine the equivalent inductance of the network below: | ||
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