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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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| + | <-- | ||
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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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| + | === Summary: Series and Parallel Capacitors === | ||
| + | * The equivalent capacitance of a __series combination of capacitors__ $C_1$, $C_2$, $\dots$, $C_N$ is governed by a relation which is analogous to that providing the equivalent resistance of a __parallel__ combination of resistors: | ||
| + | |||
| + | $$\frac{1}{C_{eq}} = \sum_{K=l}^{N} \frac{1}{C_k}$$ | ||
| + | |||
| + | * The equivalent of a __parallel combination of capacitors__ $C_1$, $C_2$, $\dots$, $C_N$ is governed by a relation which is analogous to that providing the equivalent resistance of a __series combination of resistors__: | ||
| + | |||
| + | $$C_{eq} = \sum_{k=l}^{N} C_k$$ | ||
| + | |||
| + | ==== Practical Capacitors ==== | ||
| + | Commercially available capacitors are manufactured in a wide range of both conductor and dielectric materials and are available in a wide range of capacitances and voltage ratings. The voltage rating of the device is the maximum voltage, which can be safely applied to the capacitor; using voltages higher than the rated value will damage the capacitor. The capacitance of commercially available capacitors is commonly measured in microfarads ($\mu F$; one microfarad is $10^{-6}$ of a Farad) or pico-farads (pF; one picofarad is $10^{-12}$ of a Farad). Large capacitors are available, but are relatively infrequently used. These are generally called “super-capacitors” or “ultra-capacitors” and are available in capacitances up to tens of Farads. For most applications, | ||
| + | |||
| + | Several approaches are used for labeling a capacitor with its capacitance value. Large capacitors often have their value printed plainly on them, such as "$10 \mu F$" (for 10 microfarads). Smaller capacitors, appearing as small disks or wafers, often have their values printed on them in an encoded manner. For these capacitors, a three digit number indicates the capacitor value in pico-farads. The first two digits provides the " | ||
| + | |||
| + | Capacitors are generally classified according to the dielectric material used. Common capacitor types include mica, ceramic, Mylar, paper, Teflon and polystyrene. An important class of capacitors which require special mention are // | ||
| + | |||
| + | Practical capacitors, unlike ideal capacitors, will dissipate some power. This power loss is primarily due to //leakage currents//. These currents are due to the fact that real dielectric materials are not perfect insulators – some small current will tend to flow through them. The overall effect is comparable to placing a high resistance in parallel with an ideal capacitor, as shown in Figure 6.27. Different types of capacitors have different leakage currents. Mica capacitors tend to have low leakage currents, the leakage currents of ceramic capacitors vary according to the type of capacitor, and electrolytic capacitors have high leakage currents. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ==== Section Summary ==== | ||
| + | * Capacitors store electrical energy. This energy is stored in an electric field between two conductive elements, separated by an insulating material. | ||
| + | * Capacitor energy storage is dependent upon the voltage across the capacitor, if the capacitor voltage is known, the energy in the capacitor is known. | ||
| + | * The voltage-current relationship for a capacitor is: | ||
| + | $$i(t) = C \frac{dv(t)}{dt}$$ | ||
| + | |||
| + | Where C is the capacitance of the capacitor. Units of capacitance are Farads (abbreviated F). The capacitance of a capacitor, very roughly speaking, gives an indication of how much energy it can store | ||
| + | |||
| + | * The above voltage-current relation results in the following important properties of capacitors: | ||
| + | * If the capacitor voltage is constant, the current through the capacitor is zero. Thus, if the capacitor voltage is constant, the capacitor can be modeled as an open circuit. | ||
| + | * Changing the capacitor voltage instantaneously requires infinite power. Thus (for now, anyway) we will assume that capacitors cannot instantaneously change their voltage. | ||
| + | * Capacitors placed in series or parallel with one another can be modeled as a single equivalent capacitance. Thus, capacitors in series or in parallel are not “independent” energy storage elements. | ||
| + | |||
| + | ==== Exercises ==== | ||
| + | - Determine the maximum and minimum capacitances that can be obtained from four $1 \mu F$ capacitors. Sketch the circuit schematics that provide these capacitances. | ||
| + | - 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 ===== | ||
| + | |||
| + | We continue our study of energy storage elements with a discussion of // | ||
| + | |||
| + | In this section, we describe physical properties of inductors and provide a mathematical model for an __ideal__ inductor. Using this ideal inductor model, we will develop mathematical relationships for the energy stored in an inductor and governing relations for series and parallel connections of inductors. The section concludes with a brief discussion of practical (non-ideal) inductors. | ||
| + | |||
| + | ==== Inductors ==== | ||
| + | Passing a current through a conductive wire will create a //magnetic field// around the wire. This magnetic field is generally thought of in terms of as forming closed loops of //magnetic flux// around the current-carrying element. This physical process is used to create // | ||
| + | |||
| + | The amount of inductance an inductor has is governed by the geometry of the inductor and the properties of the core material. These effects can be complex; rather than attempt a comprehensive discussion of these effects, we will simply claim that, in general, inductance is dependent upon the following parameters: | ||
| + | |||
| + | * The number of times the wire is wrapped around the core. More coils of wire results in a higher inductance. | ||
| + | * The core material’s type and shape. Core materials are commonly ferromagnetic materials, since they result in higher magnetic flux and correspondingly higher energy storage. Air, however, is a fairly commonly used core material – presumably because of its ready availability. | ||
| + | * The spacing between turns of the wire around the core. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | 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//: | ||
| + | |||
| + | $$\psi(t) = Li(t) (Eq. 6.26)$$ | ||
| + | |||
| + | Voltage is the time rate of change of magnetic flux, so: | ||
| + | |||
| + | $$v(t) = \frac{d \psi (t)}{dt} | ||
| + | |||
| + | Combining equations (6.26) and (6.27) results in the voltage-current relationship for an ideal inductor: | ||
| + | |||
| + | $$v(t) = L \frac{di(t)}{dt} | ||
| + | |||
| + | The circuit symbol for an inductor is shown in Figure 6.29, along with the sign conventions for the voltage- current relationship of equation (6.28). The passive sign convention is used in the voltage-current relationship, | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | Integrating both sides of equation (6.28) results in the following form for the inductor’s voltage-current relationship: | ||
| + | |||
| + | $$i(t) = \frac{1}{L} \int_{t_0}^t \! v(\xi)d\xi + i(t_0) | ||
| + | |||
| + | In equation (6.29), $i(t_0)$ is a known current at some initial time $t_0$ and $\xi$ is used as a dummy variable of integration to emphasize that the only “$t$” which survives the integration process is the upper limit of the integral. | ||
| + | |||
| + | === Important Result === | ||
| + | The voltage-current relationship for an ideal inductor can be stated in either differential or integral form, as follows: | ||
| + | |||
| + | * $v(t) = L \frac{di(t)}{dt}$ | ||
| + | * $i(t) = \frac{1}{L} \int_{t_0}^t \! v(\xi)d\xi + i(t_0)$ | ||
| ---- | ---- | ||
| + | ==== Example 6.6 ==== | ||
| + | A circuit contains a 100mH inductor. The current as a function of time through the inductor is measured and shown below. Plot the voltage across the inductor as a function of time. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | * In the time range 0< | ||
| + | * In the time range $1ms< | ||
| + | * In the time range $2ms< | ||
| + | * In the time range $3ms< | ||
| + | |||
| + | The plot of voltage vs. time is shown below: | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | Power is the product of voltage and current. If the signs of voltage and current are the same according to the passive sign convention, the circuit element __absorbs__ power. If the signs of voltage and current are not the same, the circuit element __generates__ power. From the above voltage and current curves, the inductor is absorbing power from the circuit during the times 0< | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | Example 2: If the voltage as a function of time across an inductor with inductance L = 10 mH is as shown below, determine the current as a function of time through the capacitor. Assume that the current through the capacitor is 0A at time $t = 0$. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | * At time $t = 0$, the current is given to be 0A. | ||
| + | * 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 $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. | ||
| + | |||
| + | A plot of the current through the inductor as a function of time is shown below. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ---- | ||
| + | |||
| + | It is often useful, when analyzing circuits containing inductors, to examine the circuit’s response to constant operating conditions and to instantaneous changes in operating condition. We examine the inductor’s response to each of these operating conditions below: | ||
| + | |||
| + | * //Inductor response to constant current:// | ||
| + | * If the current through the inductor is constant, equation (6.28) indicates that the voltage across the inductor is zero. __Thus, if the current through the inductor is constant, the inductor is equivalent to a short circuit__. | ||
| + | * //Inductor response to instantaneous current changes:// | ||
| + | * If the current through the inductor changes instantaneously, | ||
| + | * Any circuit that allows an instantaneous change in the current through an ideal inductor is not physically realizable. We may sometimes assume, for mathematical convenience, | ||
| + | |||
| + | === Important Inductor Properties === | ||
| + | * Inductors can be replaced by short-circuits, | ||
| + | * Currents through inductors cannot change instantaneously. No such requirement is placed on voltages. | ||
| + | |||
| + | ==== Energy Storage ==== | ||
| + | The instantaneous power dissipated by an electrical circuit element is the product of the voltage and current: | ||
| + | |||
| + | $$p(t) = v(t) \cdot i(t) (Eq. 6.30)$$ | ||
| + | |||
| + | Using equation (6.28) to write the voltage in equation (6.30) in terms of the inductor’s current: | ||
| + | |||
| + | $$p(t) = L \cdot i(t) \frac{di(t)}{dt} | ||
| + | |||
| + | As was previously done for capacitors, we integrate the power with respect to time to get the energy stored in the inductor: | ||
| + | |||
| + | $$W_L(t) = \sum_{-\infty}^{t} Li(\xi)\frac{di(\xi)}{dt}dt$$ | ||
| + | |||
| + | Which, after some manipulation (comparable to the approach taken when we calculated energy storage in capacitors), | ||
| + | |||
| + | $$W_L(t) = \frac{1}{2} Li^2(t) | ||
| + | |||
| + | ==== Inductors in Series ==== | ||
| + | Consider the series connection of //N// inductors shown in Figure 6.30. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | Applying Kirchhoff’s voltage law around the loop results in: | ||
| + | |||
| + | $$v(t) = v_1(t) + v_2(t) + \dots v_N(t) | ||
| + | |||
| + | Using equation (6.28) to write the inductor voltage drops in terms of the current through the loop gives: | ||
| + | |||
| + | $$v(t) = L_1 \frac{di(t)}{dt} + L_2 \frac{di(t)}{dt} + \dots + L_N \frac{di(t)}{dt} \\ | ||
| + | |||
| + | = (L_1 + L_2 + \dots L_N) \frac{di(t)}{dt}$$ | ||
| + | |||
| + | Using summation notation results in: | ||
| + | |||
| + | $$v(t) = \left( \sum_{k=l}^{N} L_k \right) \frac{di(t)}{dt} | ||
| + | |||
| + | This is the same equation that governs the circuit of Figure 6.31, if: | ||
| + | |||
| + | $$L_{eq} = \sum_{k=l}^{N} L_k (Eq. 6.35)$$ | ||
| + | |||
| + | Thus, the equivalent inductance of a series combination of inductors is simply the sum of the individual inductances. This result is analogous to the equations which provide the equivalent resistance of a __series__ combination of resistors. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ==== Inductors in Parallel ==== | ||
| + | Consider the parallel combination of N inductors, as shown in Figure 6.32. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | Applying Kirchhoff’s current law at the upper node results in: | ||
| + | |||
| + | $$i(t) = i_1(t) + i_2(t) + \dots i_N (t) (Eq. 6.36)$$ | ||
| + | |||
| + | Using equation (6.29) to write the inductor currents in terms of their voltage drops gives: | ||
| + | |||
| + | $$i(t) = \left[ \frac{1}{L_1} \int_{t_0}^t v(\xi)d\xi + i_1 (t_0) \right] + \left[ \frac{1}{L_2} \int_{t_0}^t v(\xi) d\xi + i_2 (t_0) \right] + \dots + \left[ \frac{1}{L_N} \int_{t_0}^{t} v(\xi) d \xi + i_N (t_0) \right] \\ | ||
| + | |||
| + | = \left[ \frac{1}{L_1} \int_{t_0}^{t} v(\xi)d\xi + \frac{1}{L_2} \int_{t_0}^{t} v(\xi)d\xi + \dots \frac{1}{L_N} \int_{t_0}^{t} v(\xi)d\xi \right] + \left[ i_1(t_0) + i_2(t_0) + \dots + i_N(t_0) \right] \\ | ||
| + | |||
| + | = \left( \frac{1}{L_1} + \frac{1}{L_2} + \dots + \frac{1}{L_N} \right) \int_{t_0}^{t} v(\xi)d\xi + i(t_0)$$ | ||
| + | |||
| + | This can be re-written using summation notation as: | ||
| + | |||
| + | $$i(t) = \left( \sum_{k=1}^{N} \frac{1}{L_k} \right) \int_{t_0}^{t} v(\xi)d\xi + i(t_0) | ||
| + | |||
| + | This is the same equation that governs the circuit of Figure 6.31, if: | ||
| + | |||
| + | $$\frac{1}{L_{eq}} = \sum_{k=1}^{N} \frac{1}{L_k} | ||
| + | |||
| + | For the special case of two inductors $L_1$ and $L_2$ in series, equation (13) simplifies to: | ||
| + | |||
| + | $$L_{eq} = \frac{L_1L_2}{L_1 + L_2} (Eq. 6.39)$$ | ||
| + | |||
| + | Equations (6.38) and (6.39) are analogous to the equations which provide the equivalent resistance of __parallel__ combinations of resistors. | ||
| + | |||
| + | === Summary: Series and Parallel Inductors === | ||
| + | * The equivalent inductance of a __series combination of inductors__ $L_1$, $L_2$, $\dots$, $L_N$ is governed by a relation which is analogous to that providing the equivalent resistance of a __series combination of resistors__: | ||
| + | |||
| + | $$L_{eq} = \sum_{k=1}^{N} L_k$$ | ||
| + | |||
| + | * The equivalent inductance of a __parallel combination of inductors__ $L_1$, $L_2$, $\dots$, $L_N$ is governed by a relation which is analogous to that providing the equivalent resistance of a __parallel__ combination of resistors: | ||
| + | |||
| + | $$\frac{1}{L_{eq}} = \sum_{k=1}^{N} \frac{1}{L_k}$$ | ||
| + | |||
| + | ==== Practical Inductors ==== | ||
| + | Most commercially available inductors are manufactured by winding wire in various coil configurations around a core. Cores can be a variety of shapes; Figure 6.28 in this chapter shows a core, which is basically a cylindrical bar. Toroidal cores are also fairly common – a closely wound toroidal core has the advantage that the magnetic field is confined nearly entirely to the space inside the winding. | ||
| + | |||
| + | Inductors are available with values from less than 1 micro-Henry ($1 \mu H = 10^{-6}$ Henries) up to tens of Henries. A 1H inductor is very large; inductances of most commercially available inductors are measured in millihenries ($1mH = 10^{-3}$ Henries) or microhenries. Larger inductors are generally used for low-frequency applications (in which the signals vary slowly with time). | ||
| + | |||
| + | Attempts at creating inductors in integrated-circuit form have been largely unsuccessful; | ||
| + | |||
| + | Practical inductors, unlike the ideal inductors discussed in this chapter, dissipate power. An equivalent circuit model for a practical inductor is generally created by placing a resistance in series with an ideal inductor, as shown in Figure 6.33. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ==== Section Summary ==== | ||
| + | * Inductors store magnetic energy. This energy is stored in a magnetic field (typically) generated by a coiled wire wrapped around a core material. | ||
| + | * Inductor energy storage is dependent upon the current through the inductor, if the inductor current is known, the energy in the inductor is known. | ||
| + | * The voltage-current relationship for an inductor is: | ||
| + | |||
| + | $$v(t) = L \frac{di(t)}{dt}$$ | ||
| + | |||
| + | 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. | ||
| + | |||
| + | * 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. | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Exercises ==== | ||
| + | - Determine the equivalent inductance of the network below: | ||
| + | {{ : | ||
| + | |||
| + | |||
| + | [[{}/ | ||
| + | [[{}/ | ||