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| ====== Real Analog: Chapter 11 ====== | ====== Real Analog: Chapter 11 ====== | ||
| - | ====== 11. Introduction and Chapter | + | [[{}/ |
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| + | |||
| + | <-- | ||
| + | |||
| + | ===== 11. Introduction and Chapter Objectives | ||
| In section 10.6, we saw that a system’s frequency response provided a steady-state input-output relationship for a system, as a function of frequency. We could apply this frequency response to the phasor representation of the input signal in order to determine the system’s steady-state sinusoidal response – we simply evaluated the frequency response at the appropriate frequencies to determine the effect of the system on the input sinusoids. This approach had the potential for simplifying our analysis considerably, | In section 10.6, we saw that a system’s frequency response provided a steady-state input-output relationship for a system, as a function of frequency. We could apply this frequency response to the phasor representation of the input signal in order to determine the system’s steady-state sinusoidal response – we simply evaluated the frequency response at the appropriate frequencies to determine the effect of the system on the input sinusoids. This approach had the potential for simplifying our analysis considerably, | ||
| - | In [[https:// | + | In [[/ |
| - // | - // | ||
| - //Designing systems to provide a desired frequency response//. Audio components in stereo systems are generally designed to produce a desired frequency response. A graphic equalizer, for example, can be used to boost (or amplify) some frequency ranges and attenuate other frequency ranges. When adjusting the settings on an equalizer, you are essentially directly adjusting the system’s frequency response to provide a desired system response. | - //Designing systems to provide a desired frequency response//. Audio components in stereo systems are generally designed to produce a desired frequency response. A graphic equalizer, for example, can be used to boost (or amplify) some frequency ranges and attenuate other frequency ranges. When adjusting the settings on an equalizer, you are essentially directly adjusting the system’s frequency response to provide a desired system response. | ||
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| In Figure 11.2, the output is determined by multiplying the phasor representation of the input by the system’s frequency response. It is important to keep in mind that the arguments of this multiplication are complex functions of frequency – both the input and the frequency response at any frequency are complex numbers, so the output at any frequency is also a complex number. We typically use polar form to represent these complex numbers, so the __amplitude of the output signal is the product of the amplitude of the input signal and the magnitude response of the system__ and the __phase of the output signal is the sum of the phase of the input and the phase response of the system__. Mathematically, | In Figure 11.2, the output is determined by multiplying the phasor representation of the input by the system’s frequency response. It is important to keep in mind that the arguments of this multiplication are complex functions of frequency – both the input and the frequency response at any frequency are complex numbers, so the output at any frequency is also a complex number. We typically use polar form to represent these complex numbers, so the __amplitude of the output signal is the product of the amplitude of the input signal and the magnitude response of the system__ and the __phase of the output signal is the sum of the phase of the input and the phase response of the system__. Mathematically, | ||
| - | $$|\underline{Y}(j \omega)|=|\underline{U}(j \omega)| \cdot |H (j \omega)| | + | $$|\underline{Y}(j \omega)|=|\underline{U}(j \omega)| \cdot |H (j \omega)| |
| And | And | ||
| Line 87: | Line 122: | ||
| $$\underline{U}_{j \omega} = \begin{cases} | $$\underline{U}_{j \omega} = \begin{cases} | ||
| - | j \omega, & 0 < \leq 1 \\ | + | j \omega, & 0 < \omega |
| j(2-\omega), | j(2-\omega), | ||
| 0, & \text{otherwise} | 0, & \text{otherwise} | ||
| Line 258: | Line 293: | ||
| And: | And: | ||
| - | $$\angle H(j \omega) = =\tan^{-1}(\frac{\omega}{2})$$ | + | $$\angle H(j \omega) = -\tan^{-1}(\frac{\omega}{2})$$ |
| Plotting these functions results in the graphical frequency response shown below: | Plotting these functions results in the graphical frequency response shown below: | ||
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| The circuit frequency response, $H(j \omega)$, is the ratio of the output phasor to the input phasor: | The circuit frequency response, $H(j \omega)$, is the ratio of the output phasor to the input phasor: | ||
| - | $$H(j \omega) = \frac{\underline{V}{underline{V}_S} = \frac{j2 \omega}{1+j2 \omega}$$ | + | $$H(j \omega) = \frac{\underline{V}}{\underline{V}_S} = \frac{j2 \omega}{1+j2 \omega}$$ |
| So that the magnitude and phase response of the circuit are: | So that the magnitude and phase response of the circuit are: | ||
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| $$\angle H(j \omega) = 90^{\circ} - \tan^{-1}(2 \omega)$$ | $$\angle H(j \omega) = 90^{\circ} - \tan^{-1}(2 \omega)$$ | ||
| + | Plots of these functions are shown below. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Checking the Frequency Response ==== | ||
| + | A circuit’s __amplitude response__ is relatively easy to determine at low and high frequencies. For very low ($\omega \rightarrow 0$) and very high ($\omega \rightarrow \infty$) frequencies, | ||
| + | * __Capacitors at low and high frequencies__: | ||
| + | * __Inductors at low and high frequencies__: | ||
| + | |||
| + | Please note that the above statements are relative only to the amplitude response. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Example 11.8 === | ||
| + | Use the circuits’ low and high frequency behavior to check the amplitude response plots for example 11.6. | ||
| + | |||
| + | The circuit of interest is shown below. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | * At low frequencies, | ||
| + | * At high frequencies, | ||
| + | |||
| + | The expected behavior at low and high frequencies agrees with the amplitude response of example 11.6; the amplitude response is one at $\omega = 0$, and approaches zero as frequency increases. | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Example 11.9 ==== | ||
| + | Use the circuits’ low and high frequency behavior to check the amplitude response plots for example 11.7. The circuit of interest is shown below: | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | * At low frequencies, | ||
| + | * At high frequencies, | ||
| + | |||
| + | The expected behavior at low and high frequencies agrees with the amplitude response of example 11.7; the amplitude response is zero at $\omega =0$, and approaches one as frequency increases. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Example 11.10 ==== | ||
| + | What is the gain of the circuit shown below as $\omega \rightarrow 0$ and $\omega \rightarrow \infty$? The input is $v_{in}(t)$ and the output is $v_{out}(t)$. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | As $\omega \rightarrow 0$, the capacitor becomes an open circuit, and the overall circuit becomes as shown to the left below. As $\omega \rightarrow \infty$, the capacitor is replaced by a short circuit and the overall circuit becomes as shown to the right below. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | As $\omega \rightarrow 0$, the circuit becomes a voltage divider, and $v_{out} = \frac{3k \Omega}{2k \Omega + 3k \Omega} \cdot v_{in}$, so that the circuit' | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Graphical Representation of System Response ==== | ||
| + | We can use graphical depictions of the input signal spectrum and the frequency response of the system to obtain a graphical representation of a system’s response. This representation can be especially useful in interpreting and understanding the effect of a system on an input signal. This qualitative interpretation of a system’s response can be an invaluable aid in system design. | ||
| + | |||
| + | The frequency-domain representation of a system’s response is shown in Figure 11.4. The frequency-domain output is simply a point-by-point multiplication between the input signal and the system’s frequency response. Thus, the amplitude spectrum of the output is simply the point-by-point product of the input signal’s amplitude spectrum with the system’s magnitude response. The phase spectrum of the output is the sum of the input signal’s phase spectrum and the system’s phase response. An example of this process and its use in interpreting a system’s response is provided in the example below. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Example 11.11 ==== | ||
| + | Use graphical methods to interpret the response of the circuit of example 11.6 to an input voltage $v_{in}(t) = 3 \cos(t+20^{\circ}) + 8 \cos(8t-60^{\circ})$. The input voltage time-domain signal is shown below. The signal consists of a relatively small amplitude, low frequency sinusoid superimposed with a large amplitude, higher frequency signal. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | The frequency response of the circuit was determined in example 11.6; the frequency response plots are repeated below: | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | The input signal spectrum, $\underline{V}(j \omega)$ is shown below. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | We can obtain the spectrum of the output signal from a frequency-by-frequency product of the input amplitude and the circuit’s amplitude response and a frequency-by-frequency sum of the input phase and the circuit’s phase response. Keep in mind that when the input amplitude is zero, the output amplitude is also zero and the phase is undetermined. | ||
| + | |||
| + | This process results in the output spectrum shown below. Notice that the circuit attenuates the magnitude of the high frequency component of the signal relative to the low frequency component. The low frequency component is said to be “passed through” the circuit, while the higher frequency component is “stopped” by the circuit. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | The time-domain signal of the output signal is shown below, superimposed over the input signal. It is clear that the amplitudes of the higher frequency component of the output signal has been reduced relative to the amplitude of the lower frequency component. This circuit is said to be //frequency selective// – it selects low frequencies to pass and high frequencies to stop. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Section Summary ==== | ||
| + | * //Signal spectra// are a primarily a graphical representation of the frequency content of a __signal__. The spectrum consists, in general, of two plots: amplitude as a function of frequency and phase as a function of frequency. | ||
| + | * The signal spectrum is useful, since many signals of interest can be expressed as a superposition of sinusoidal components. This allows signals which do not appear “sinusoidal” to be analyzed using steady-state sinusoidal analysis techniques. | ||
| + | * //Frequency response plots// are a graphical representation of the frequency response of a __system__. Frequency response plots also consist of two parts: the system’s magnitude response as a function of frequency and the system’s phase response as a function of frequency. | ||
| + | * The combination of the signal spectrum and the frequency response plots can provide valuable insight into a system’s operation. The amplitude of the output signal is the __point-by-point product__ of the input signal’s amplitude and the system’s magnitude response. The phase of the output signal is the __point-by-point sum__ of the input signal’s phase and the system’s phase response. | ||
| + | * Perhaps even more importantly, | ||
| + | * When calculating a system’s frequency response, it is always important to check your result relative to the expected behavior of the system. This can be done by determining the magnitude response at very low ($\omega \rightarrow 0$) and very high ($\omega \rightarrow \infty$) frequencies. In these extreme conditions the circuit can be approximated as a purely resistive circuit, since energy storage elements can be replaced by either short-circuits or open-circuits as follows: | ||
| + | * __Capacitors at low and high frequencies__: | ||
| + | * __Inductors at low and high frequencies__: | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Exercises ==== | ||
| + | **1.** The input to a system is $u(t) = 3 \cos(2t-60^{\circ}) + 4 \cos(4t+30^{\circ}) + 7 \cos(6t+45^{\circ})$. The system frequency response is given by the piecewise function: | ||
| + | |||
| + | $$H(j \omega) = \begin{cases} | ||
| + | 0, & 0< \omega < 3 \\ | ||
| + | 1, & 3 \leq \omega \leq 5 \\ | ||
| + | 0, & \omega > 5 | ||
| + | \end{cases} | ||
| + | $$ | ||
| + | |||
| + | Sketch: | ||
| + | * The spectrum of the input signal (magnitude and phase) | ||
| + | * The frequency response of the system (magnitude response and phase response) | ||
| + | * And the output signal spectrum. | ||
| + | |||
| + | Compare your sketches to your results of exercise 2 in section 1.11. | ||
| + | |||
| + | **2.** Sketch the spectrum for the signal $v(t)=4+3 \cos(2t-45^{\circ}) + 2 \cos(4t+45^{\circ})$. | ||
| + | |||
| + | **3.** Use the circuit behavior at high and low frequencies to check the frequency response you calculated for the circuit of exercise 1 in section 11.1. | ||
| + | |||
| + | **4.** Determine the steady-state response of the output $y(t)$ for the circuit below at very low and very high frequencies. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ===== 11.3: Frequency Selective Circuits and Filters ===== | ||
| + | It is common to categorize circuits by the overall “shape” of their magnitude response. As we saw in example 11.11, in some frequency ranges the output amplitude may be high relative to the input amplitude, while in other frequency ranges the output amplitude will be low relative to the input amplitude. If the output amplitude at some frequency is high relative to the input amplitude, the magnitude response at that frequency is “large” and that frequency is said to be //passed// by the circuit. Conversely, if the output amplitude at some frequency is low relative to the input amplitude, the magnitude response at that frequency is “small” and that frequency is said to be //stopped// by the circuit. | ||
| + | |||
| + | Circuits which select certain frequencies to pass and other frequencies to stop are called //frequency selective// circuits or //filters// (since they tend to “filter out” certain frequency ranges of the input signal). The range of frequencies which are passed are called the // | ||
| + | * //Low-pass filters// pass low frequencies and stop high frequencies | ||
| + | * //High-pass filters// pass high frequencies and stop low frequencies | ||
| + | * //Band-pass filters// pass a range of frequencies between two ranges of stopped frequencies | ||
| + | * // | ||
| + | |||
| + | Filters are also categorized by their //order//. The order of the filter is simply the order of the differential equation governing the filter. Thus a first-order filter is governed by a first-order differential equation, a second-order filter is governed by a second-order differential equation, and so on. Low-pass and high-pass filters can be any order, while band-pass and band-stop filters must be __at least__ second order. | ||
| + | |||
| + | In this chapter, we restrict out attention to first order filters, so we will consider __only__ low pass and high pass filters. | ||
| + | |||
| + | ==== Ideal Low-pass and High-pass Filters ==== | ||
| + | We will first introduce the basic concepts relative to first order filters in the context of ideal filters. It must be clearly understood that ideal filters are not // | ||
| + | |||
| + | An ideal low pass filter has a magnitude response as shown in Figure 11.5. The passband is shown as the shaded area under the magnitude response. The magnitude response is discontinuous – it goes from one to zero instantaneously. The //cutoff frequency//, | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | An ideal high-pass filter has a magnitude response as shown in Figure 11.6. The passband is again shown as the shaded area under the magnitude response. The magnitude response is discontinuous – it goes from zero to one instantaneously. The //cutoff frequency//, | ||
| + | |||
| + | As previously noted, it is impossible to physically implement an ideal filter. Thus, all electrical circuits implement non-ideal filters. Non-ideal filters do not provide an instantaneous transition between the pass band and the stop band. Non-ideal first order filters are discussed in the following subsections. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ==== First Order Low-pass Filters ==== | ||
| + | The form of the governing differential equation for a first order low pass filter can be written as: | ||
| + | |||
| + | $$\frac{dy(t)}{dt} + \omega_c y(t) = K \cdot u(t) (Eq. 11.5)$$ | ||
| + | |||
| + | The frequency response of the filter can be determined to be: | ||
| + | |||
| + | $$H(j \omega) = \frac{K}{j \omega + \omega_c} | ||
| + | |||
| + | The magnitude response of the filter is thus: | ||
| + | |||
| + | $$|H(j \omega)| = \frac{K}{\sqrt{\omega^2 + \omega^2_c}} (Eq. 11.7)$$ | ||
| + | |||
| + | The maximum magnitude response is $\frac{K}{\omega_c}$ when $\omega = 0$ and the magnitude response is zero as $\omega \rightarrow \infty$. Thus, the filter is passing low frequencies and stopping high frequencies. __The differential equation describes a low-pass filter__. | ||
| + | |||
| + | The magnitude response of the filter is shown in Figure 11.7. The frequency response is a smooth curve, rather than the discontinuous function shown in Figure 11.5. There is no single frequency that obviously separates the passband from the stopband, so we must choose a relatively arbitrary point to define the boundary between the passband and the stopband. By consensus, the cutoff frequency for a low-pass filter is defined as the frequency at which the magnitude response is $\frac{1}{\sqrt{2}}$ times the magnitude response at $\omega = 0$. For the magnitude response given by equation (11.7), the cutoff frequency is $\omega = \omega_c$. This point is indicated on Figure 11.7. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ==== First Order High-pass Filters ==== | ||
| + | The form of the governing differential equation for a first order low pass filter can be written as: | ||
| + | |||
| + | $$\frac{dy(t)}{dt} + \omega_c y(t) = K \cdot \frac{du(t)}{dt} | ||
| + | |||
| + | The frequency response of the filter can be determined to be: | ||
| + | |||
| + | $$H(j \omega) = \frac{jK \omega}{j \omega + \omega_c} | ||
| + | |||
| + | The magnitude response of the filter is thus: | ||
| + | |||
| + | $$|H(j \omega)| = \frac{K \omega}{\sqrt{\omega^2 + \omega^2_c}} (Eq. 11.10)$$ | ||
| + | |||
| + | The maximum magnitude response is approximately //K// when $\omega \rightarrow \infty$ and the magnitude response is zero at $\omega = 0$. Thus, the filter is passing high frequencies and stopping low frequencies. __The differential equation describes a high-pass filter__. | ||
| + | |||
| + | The magnitude response of the filter is shown in Figure 11.8. As with the non-ideal low-pass filter, the frequency response is a smooth curve, rather than the discontinuous function shown in Figure 11.6. Again, there is no single frequency that obviously separates the passband from the stopband, so we must choose a relatively arbitrary point to define the boundary between the passband and the stopband. Consistent with our choice of cutoff frequency for the low-pass filter, the cutoff frequency for a high-pass filter is defined as the frequency at which the magnitude response is $\frac{1}{\sqrt{2}}$ times the magnitude response at $\omega \rightarrow \infty$. For the magnitude response given by equation (1.10), the cutoff frequency is $\omega = \omega_c$. This point is indicated on Figure 11.8. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | === Notes: === | ||
| + | * The cutoff frequency is also called the corner frequency, the 3dB frequency, or the half-power point. | ||
| + | * The cutoff frequency for __both__ low-pass and high-pass filters is defined as the frequency at which the magnitude is $\frac{1}{\sqrt{2}}$ times the __maximum__ value of the magnitude response. | ||
| + | * It can be seen from examples in section 11.2 that the phase response of a first order low-pass filter is $0^{\circ}$ at $\omega = 0$ and decreases to $-90^{\circ}$ as $\omega \rightarrow \infty$. The phase response is $-45^{\circ}$ at the cutoff frequency. | ||
| + | * It can be seen from examples in section 11.2 that the phase response of a first order high-pass filter is $90^{\circ}$ at $\omega = 0$ and decreases to $0^{\circ}$ as $\omega \rightarrow \infty$. The phase response is $45^{\circ}$ at the cutoff frequency. | ||
| + | * For both low-pass and high-pass filters, the cutoff frequency is the inverse of the time constant for the circuit, so that $\omega_c = \frac{1}{\tau}$. | ||
| + | * The circuit’s response at zero frequency is generally an important parameter to consider. This is called the DC gain, and is the ratio of the output amplitude to the input amplitude for a constant input. A constant input corresponds to a cosine with zero frequency. Low pass filters have a relatively high DC gain and a correspondingly large response to a constant input. High pass filters have a low DC gain; they have little or no response to constant inputs. | ||
| + | |||
| + | We conclude this section with examples of circuits from section 10.2 which implement low-pass and high-pass filter operations. | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Example 11.12: First Order Low-pass Filter ==== | ||
| + | The circuit below is the circuit from example 11.6. The input is $v_{in}(t)$ and the output is $v_{out}(t)$. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | In example 11.6, the frequency response was determined to be: | ||
| + | |||
| + | $$H(j \omega) = \frac{2}{2+ j \omega}$$ | ||
| + | |||
| + | The maximum value of the magnitude response is one at a frequency of zero radians/ | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Example 11.13: First Order High-pass Filter ==== | ||
| + | The circuit below is the circuit from example 11.7. The input is $v_s(t)$ and the output is $v(t)$. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | The frequency response of the circuit was previously determined to be: | ||
| + | |||
| + | $$H(j \omega) = \frac{\underline{V}}{\underline{V}_S} = \frac{j2 \omega}{1+ j2 \omega}$$ | ||
| + | |||
| + | The maximum value of the magnitude response is one as $\omega \rightarrow \infty$ and goes to zero at a frequency of zero radians/ | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Section Summary ==== | ||
| + | * Filters are frequency-selective systems. These systems provide a desired relationship between the input signal spectrum and the output signal spectrum. The filter does this by //passing// certain frequencies from the input to the output and // | ||
| + | * The range of frequencies which are passed is called the // | ||
| + | * The range of frequencies which are stopped is called the // | ||
| + | * Filters are broadly categorized as follows: | ||
| + | * //Low-pass filters// pass low frequencies and stop high frequencies | ||
| + | * //High-pass filters// pass high frequencies and stop low frequencies | ||
| + | * //Band-pass filters// pass a range of frequencies between two ranges of stopped frequencies | ||
| + | * // | ||
| + | * In this chapter, we were concerned only with first-order low-pass and high-pass filters. These filters are primarily characterized by the following parameters: | ||
| + | * Cutoff frequency: the cutoff is defined as the frequency at which the magnitude is $\frac{1}{\sqrt{2}}$ times the __maximum__ value of the magnitude response. | ||
| + | * //DC gain//: the DC gain is the ratio of the output amplitude to the input amplitude for a __constant__ input (a cosine function with zero frequency). Low pass filters have a relatively high DC gain and a correspondingly large response to a constant input. High pass filters have a low DC gain; they have little or no response to constant inputs. | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Exercises ==== | ||
| + | **1.** What is the cutoff frequency of the circuit below? (You may want to use your results from exercise 1 of section 11.1.). | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | **2.** Use the circuit behavior at high and low frequencies and your time constant calculated in exercise 1 above to sketch the frequency response of the circuit of exercise 1. Label your frequency response to include DC gain and cutoff frequency. Is the circuit a high-pass or low-pass filter? | ||
| + | |||
| + | **3.** Calculate the time constant and the cutoff frequency for the circuit below, if u(t) is the input and $y(t)$ is the output. Verify that the cutoff frequency is the inverse of the time constant. Use the circuit behavior at high and low frequencies to determine whether the circuit is a high-pass or low-pass filter. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | **4.** Calculate the frequency response of the circuit of exercise 3. Compare your frequency response to your results of exercise 3. Resolve any differences between the two. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ===== 11.4: Introduction to Bode Plots ===== | ||
| + | Plotting a systems’ frequency response on a linear scale, as done in sections 11.3 and 11.4, has a number of drawbacks, especially for higher-order systems((Higher order systems are often modeled as a series of lower-order systems (or in the technical parlance, // | ||
| + | * Use of logarithms converts the operation of multiplication and division to addition and subtraction. This can simplify the creation of frequency response plots for higher order systems. | ||
| + | * Frequencies and amplitudes of interest commonly span many orders of magnitude. Logarithmic scales improve the presentation of this type of data. | ||
| + | * Human senses are fundamentally logarithmic. The use of logarithmic scales is therefore more “natural”. (This is the reason for use of the Richter scale in measuring earthquake intensity, and the decibel scale in measuring sound levels. It is also the reason that increasing a musical tone by one octave corresponds to doubling its frequency.) | ||
| + | |||
| + | ==== Properties of Logarithms ==== | ||
| + | Since Bode plots employ logarithms extensively, | ||
| + | |||
| + | A plot of $log_{10}(x)$ vs. $x$ is shown in Figure 11.9 below. A few important features to note are: | ||
| + | |||
| + | * $log_{10}(x)$ is a real number only for positive values of x. | ||
| + | * $log_{10}(x)$ asymptotically approaches $-\infty$ as $x \rightarrow 0$. The slope of $log_{10}(x)$ becomes very large as $x \rightarrow 0$. | ||
| + | * The slope of $log_{10}(x)$ becomes small as $x \rightarrow \infty$. | ||
| + | * From the comments above relative to the slope of $log_{10}(x)$, | ||
| + | * $log_{10}(1) = 0$ | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | The basic defining property of a base-10 logarithm is that $x=10^y$, then $y = log_{10} x$. This property leads to the following rules governing logarithmic operations: | ||
| + | |||
| + | **1.** Logarithms convert multiplication and division to addition and subtraction, | ||
| + | |||
| + | $$log_{10}(xy) = log_{10}x+log_{10}y$$ | ||
| + | |||
| + | And: | ||
| + | |||
| + | $$log_{10} \left( \frac{x}{y} \right) = log_{10}x-log_{10}y$$ | ||
| + | |||
| + | This property is especially useful for us, since determining the spectrum of an output signal results from the product of an input signal’s spectrum with the frequency response. Thus, the output spectrum on a logarithmic scale can be obtained from a simple addition. | ||
| + | |||
| + | **2.** Logarithms convert exponentiation to multiplication by the exponent, so that: | ||
| + | |||
| + | $$log_{10}(x^n)=nlog_{10}x$$ | ||
| + | |||
| + | |||
| + | ==== Decibel Scales ==== | ||
| + | Magnitude responses are often presented in terms of decibels (abbreviated dB). Decibels are a logarithmic scale. A magnitude response is presented in units of decibels according to the following conversion: | ||
| + | |||
| + | $$|H(j \omega)|_{dB} = 20log_{10} (|H(j \omega)|) | ||
| + | |||
| + | Strictly speaking, magnitudes in decibels are only appropriate if the amplitude response is unitless (e.g. the units of the input and output must be the same in order for the logarithm to be a mathematically appropriate operation). However, in practice, magnitude responses are often presented in decibels regardless of the relative units of the input and output – thus, magnitude responses are provided in decibels even if the input is voltage and the output is current or vice-versa. | ||
| + | |||
| + | === Brief Historical Note === | ||
| + | Decibel units are related to the unit “bel”, which are named after Alexander Graham Bell. Units of bels are, strictly speaking, applicable only to power. Power in bels is expressed as $log_{10}( \frac{P}{P}_{ref} )$, where $P_{ref}$ is a “reference” power. Bels are an inconveniently large unit, so these were converted to decibels, or tenths of a bel. Thus, power in decibels is $10log_{10}(\frac{P}{P}_{ref})$. Since the units of interest to electrical engineers are generally voltages or currents, which must be squared to obtain power, we obtain $20log_{10}(|H(j \omega)|)$. The significant aspect of the decibel unit for us is not, however, the multiplicative factor of “20”), but the fact that the unit is __logarithmic__. | ||
| + | |||
| + | We conclude this subsection with a table of common values for $|H(j \omega)|$ and their associated decibel values. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | |||
| + | ==== Bode Plots ==== | ||
| + | Bode plots are simply plots of the magnitude and phase response of a system using a particular set of axes. For Bode plots, | ||
| + | * Units of frequency are on a base-10 logarithm scale. | ||
| + | * Amplitudes (or magnitudes) are in decibels (dB) | ||
| + | * Phases are presented on a linear scale | ||
| + | |||
| + | === Notes === | ||
| + | * Since frequencies are on a logarithmic scale, frequencies separated by the same multiplicative factor are evenly separated on a logarithmic scale. Some of these multiplicative factors have special names. For example, frequencies separated by a factor of two are said to be separated by //octaves// on a logarithmic scale and frequencies separated by a factor of 10 are said to be separated by // | ||
| + | * Since decibels are intrinsically a logarithmic scale, magnitudes which are separated by the same multiplicative factor are evenly separated on a decibel scale. For example, magnitudes which are separated by a factor of 10 are separated by 20dB on a decibel scale. | ||
| + | |||
| + | One convenient aspect of the presentation of frequency responses in terms of Bode plots is the ability to generate a reasonable sketch of a frequency response very easily. In general, this approach consists of approximating the Bode plot of a system by its asymptotic behavior as a set of straight lines. This is called a “straight line approximation” of the Bode plot; the approach is illustrated for a typical low-pass filter in the following subsection. | ||
| + | |||
| + | ==== Bode Plots for First Order Low-pass Filters ==== | ||
| + | The frequency response of a general first order low-pass filter is provided in section 11.3 as: | ||
| + | |||
| + | $$H(j \omega) = \frac{K}{j \omega + \omega_c} | ||
| + | |||
| + | Thus, magnitude response of the circuit is: | ||
| + | |||
| + | $$|H(j \omega)| = \frac{K}{\sqrt{\omega^2 + \omega^2_c}} (Eq. 11.13)$$ | ||
| + | |||
| + | And the phase response of the circuit is: | ||
| + | |||
| + | $$\angle H(j \omega) = -\tan^{-1} \left( \frac{\omega}{\omega_c} \right) | ||
| + | |||
| + | To estimate the asymptotic behavior of the frequency response, we consider the behavior of equations (11.13) and (11.14) for the low frequency and high frequency cases. In general, we consider “low” frequencies to be frequencies which are __less than a factor of 10 below the cutoff frequency__ (i.e. $\omega < \frac{\omega_c}{10}$, | ||
| + | |||
| + | * __Low frequencies__: | ||
| + | * The magnitude response given by equation (2) is $|H(j \omega)| = \frac{K}{\sqrt{\omega^2 + \omega^2_c}}$. If $\omega << \omega$, the denominator is approximately $\sqrt{\omega^2_c}=\omega_c$ and the magnitude response $|H(j \omega)| \approx \frac{K}{\omega_c}$. If $\omega << \omega_c$, $\frac{\omega}{\omega_c} \approx 0$ and the phase response is approximately $\angle H(j \omega) \approx -\tan^{-1}(0) = 0^{\circ}$. | ||
| + | * __High frequencies__: | ||
| + | * If $\omega >> \omega_c$, the denominator of the amplitude response is $\sqrt{\omega^2 + \omega^2_c} \approx \sqrt{\omega^2} = \omega$. Therefore, for high frequencies, | ||
| + | |||
| + | === Summary: Low-pass Filter Straight-line Bode Plot Approximations === | ||
| + | The straight line approximation to the magnitude response is constant below the cutoff frequency, with a value (in decibels) of $20log_{10} \left( \frac{K}{\omega_c} \right)$. Above the cutoff frequency, the Bode plot straight-line approximation has a constant slope of -20 dB/decade. | ||
| + | |||
| + | The straight-line approximation to the phase response is zero degrees up to a frequency of $\frac{\omega_c}{10}$ and is $-90^{\circ}$ above a frequency of $10 \omega_c$. A straight line is used ot connect the $\frac{\omega_c}{10}$ and $10 \omega_c$ frequencies. | ||
| + | |||
| + | A straight-line approximation to the Bode plot for a typical low-pass circuit, with $K = \omega_c$ (so that the frequency response is $H(j \omega) = \frac{\omega_c}{j \omega + \omega_c}$ and the DC gain is 1, or 0dB) along with an exact curve is provided below in Figure 11.10. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | We conclude this section with a numerical example of the straight-line approximation to a Bode plot for a specific circuit. | ||
| + | |||
| + | ==== Example 11.14 ==== | ||
| + | Sketch a straight-line approximation to the Bode plot for the circuit below. The input is $v_{in}(t)$ and the output is $v_{out}(t)$. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | The frequency response for this circuit is $H(j \omega) = \frac{1}{j \omega + 2}$. Therefore, the cutoff frequency is $\omega_c=2$ rad/sec and the gain in decibels at low frequencies is $|H(j0)|_{dB} = 20log_{10} \left( \frac{1}{2} \right) \approx -6dB$. Thus, the straight-line magnitude response is -6dB below the cutoff frequency and decreases by 20dB/decade above the cutoff frequency. The straight-line phase response is $0^{\circ}$ below 0.2rad/sec, $-90^{\circ}$ above 20 rad/sec and a straight line between these frequencies. The associated plots are shown below. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Section Summary ==== | ||
| + | * Bode plots are a very useful format for plotting frequency responses. Bode plots provide magnitude responses and phase responses in the following format: | ||
| + | * Units of frequency are on a base-10 logarithm scale. | ||
| + | * Amplitudes (or magnitudes) are in decibels (dB) | ||
| + | * Phases are presented on a linear scale. | ||
| + | * Magnitude responses in decibels are calculated according to: | ||
| + | $$|H(j \omega)|_{dB} = 20log_{10}(|H(j \omega)|)$$ | ||
| + | * The use of logarithmic scales in Bode plots has a number of advantages. Logarithms convert multiplication and division into addition and subtraction, | ||
| + | * Bode plots also have the advantage of being approximated fairly well by straight-line approximations. This allows the engineer to sketch a fairly accurate frequency response plot with only a minimal number of calculations. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Exercises ==== | ||
| + | **1.** Sketch a Bode plot (straight-line approximation) for the circuit below. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | **2.** Sketch a Bode plot (straight-line approximation) for the circuit below. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | |||
| + | [[{}/ | ||
| + | [[{}/ | ||