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| learn:courses:real-analog-chapter-11:start [2017/01/18 21:37] – [Bode Plots for First Order Low-pass Filters] Martha | learn:courses:real-analog-chapter-11:start [2023/02/08 20:37] (current) – external edit 127.0.0.1 | ||
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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 [[/ |
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| - //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 | ||
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| $$\underline{U}_{j \omega} = \begin{cases} | $$\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} | ||
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| 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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| ==== Checking the Frequency Response ==== | ==== 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, | 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__: | + | * __Capacitors at low and high frequencies__: |
| - | * __Inductors at low and high frequencies__: | + | * __Inductors at low and high frequencies__: |
| Please note that the above statements are relative only to the amplitude response. | Please note that the above statements are relative only to the amplitude response. | ||
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| - | 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;s gain is $\frac{3}{5}$. As $\omega \rightarrow \infty$, $v_{out}$ is measured across a short circuit, and the gain is zero. | + | 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's gain is $\frac{3}{5}$. As $\omega \rightarrow \infty$, $v_{out}$ is measured across a short circuit, and the gain is zero. |
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| 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 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. | + | 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. |
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| * The cutoff frequency is also called the corner frequency, the 3dB frequency, or the half-power point. | * 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. | * 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 $\omaga = 0$ 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 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. | * 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 $\omega_c = \frac{1}{\tau}$. | * For both low-pass and high-pass filters, the cutoff frequency is the inverse of the time constant for the circuit, so that $\omega_c = \frac{1}{\tau}$. | ||
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| 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: | 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: | ||
| - | $$|H(j \omega)|_{dB} = 20log_{10} (|H(j \omega)|) | + | $$|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. | 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 === | === 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 $log_{10}(\frac{P}{P}_{ref})$, | + | 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. | We conclude this subsection with a table of common values for $|H(j \omega)|$ and their associated decibel values. | ||
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| - | The frequency response for this circuit is $H(j \omega) = \frac{1}{j \omega + 2}$. Therefore, the cuttof | + | The frequency response for this circuit is $H(j \omega) = \frac{1}{j \omega + 2}$. Therefore, the cutoff |
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