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SECTION 2 First Order Filters analysis.pdf

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first order circuit analysis,first order filter transfer function

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ENGR 202 – Electrical Fundamentals II SECTION 2: FIRST-ORDER FILTERS
K. Webb ENGR 202 Introduction 2
K. Webb ENGR 202 3 Filters  We are all familiar with water and air filters  Basis for operation is size selectivity  Small particles (e.g. air or water molecules) are allowed to pass  Larger particles (e.g. dust, sediment) are not  Unwanted components are filtered out of the flow.  Electrical filters are similar  Basis for operation is frequency selectivity  Signal components in certain frequency ranges are filtered out  Signal components at other frequencies are allowed to pass
K. Webb ENGR 202 4 Noise  All real-world electrical signals are noisy  You’ve seen this in the lab  Zoom in closely on a low-amplitude sinusoid with the scope (even one supplied directly from the function generator) – it won’t look like a perfectly clean sinusoid
K. Webb ENGR 202 5 Noise  We will use the term noise to mean any electrical signal that interferes with or corrupts a signal we are trying to measure.  Noise has many sources:  Measurement instruments themselves  60Hz power line interference  Electrical components – resistors, transistors, etc.  Wireless LAN, fluorescent lights, computers, etc.  We’d like to be able to remove, or filter out, this noise  Improve the accuracy of measurements  Often possible, if we know the frequency characteristics of the signal and the noise
K. Webb ENGR 202 6 Filtering Noise  We’ll learn how to design filters to remove noise Filter Noisy Signal Filtered Signal  First, we must introduce two important concepts:  Frequency-domain representation of electrical signals  What is meant by “frequency characteristics” of an electrical signal?  Frequency response of linear systems  How does a linear system (e.g. a filter) behave as a function of frequency?
K. Webb ENGR 202 Frequency Spectrum 7
K. Webb ENGR 202 8 Frequency Domain  We are accustomed to looking at electrical signals in the time domain  Amplitude plotted as function of time  Can also be represented in the frequency domain  Amplitude plotted as a function of frequency  Frequency spectrum  Describes the frequency content of a signal  Can think of signals as a sum of different frequency sinusoids  What frequencies (sinusoids) are present
K. Webb ENGR 202 9 Frequency Spectrum  Frequency spectrum  An amplitude vs. frequency plot  X-axis is frequency – not time  Y-axis is amplitude  Amplitude units may be in decibels (dB)  Shows the relative amount of energy at each frequency  Time-domain plot and frequency spectrum are alternate representations of the same signal
K. Webb ENGR 202 10 Frequency Spectra – Examples  Single sinusoid: 𝑣𝑣 𝑡𝑡 = 1𝑉𝑉 cos 2𝜋𝜋 ⋅ 800𝐻𝐻𝐻𝐻 ⋅ 𝑡𝑡  Sum of three sinusoids: 𝑣𝑣 𝑡𝑡 = 1𝑉𝑉[cos 2𝜋𝜋 ⋅ 800𝐻𝐻𝐻𝐻 ⋅ 𝑡𝑡 + cos 2𝜋𝜋 ⋅ 1200𝐻𝐻𝐻𝐻 ⋅ 𝑡𝑡 + cos 2𝜋𝜋 ⋅ 2000𝐻𝐻𝐻𝐻 ⋅ 𝑡𝑡 Time-domain Frequency-domain Time-domain Frequency-domain
K. Webb ENGR 202 11 Frequency Spectra – Examples  White noise:  Band-limited (colored) noise: Time-domain Frequency-domain Time-domain Frequency-domain
K. Webb ENGR 202 Frequency Spectra - Examples  Consider the following scenario  Measuring a sensor output in the lab  Know the signal is roughly sinusoidal  Suspected frequency: ~1 kHz  Same signal in the frequency domain: 12  Measured signal corrupted by noise/interference  Difficult to identify the interfering signal from the time-domain plot  Three interfering tones  All near 100 kHz  Can now design a filter to remove the noise:
K. Webb ENGR 202 13 Fourier Transform  Fourier transform  Transforms a time-domain representation to a frequency spectrum 𝑉𝑉 𝜔𝜔 = � −∞ ∞ 𝑣𝑣 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗𝑑𝑑𝑑𝑑  Inverse Fourier transform  Transforms from the frequency domain to the time domain 𝑣𝑣 𝑡𝑡 = 1 2𝜋𝜋 � −∞ ∞ 𝑉𝑉 𝜔𝜔 𝑒𝑒𝑗𝑗𝑗𝑗𝑗𝑗𝑑𝑑𝑑𝑑  A mathematical transform  Two different ways of looking at the same signal  A change in perspective not a change of the signal itself
K. Webb ENGR 202 Frequency Response of Linear Systems 14
K. Webb ENGR 202 15 Frequency Response Function  Linear systems (electrical, mechanical, etc.) can be described by their frequency responses  Frequency response  Ratio of the system output phasor to the system input phasor  In general, a complex function of frequency 𝐻𝐻 𝜔𝜔 = 𝐘𝐘 𝐗𝐗 = 𝐘𝐘 𝜔𝜔 𝐗𝐗 𝜔𝜔  Complex-valued – has both magnitude and phase  Magnitude: ratio of output to input magnitudes  Gain of the system  Phase: difference in phase between output and input  Phase shift through the system
K. Webb ENGR 202 16 Frequency Response – Bode Plots  Frequency response  Description of system behavior as a function of frequency  Gain and phase  Represented graphically – formatted as a Bode plot  Magnitude plot on top, phase plot below  Logarithmic frequency axes  Magnitude usually has units of decibels (dB)  Phase has units of degrees
K. Webb ENGR 202 17 Bode Plots Logarithmic frequency axes Units of magnitude are dB Magnitude plot on top Phase plot below Units of phase are degrees
K. Webb ENGR 202 18 Decibels - dB  Frequency response gain most often expressed and plotted with units of decibels (dB)  A logarithmic scale  Provides detail of very large and very small values on the same plot  Commonly used for ratios of powers or amplitudes  Conversion from a linear scale to dB: 𝐻𝐻 𝜔𝜔 𝑑𝑑𝑑𝑑 = 20 ⋅ log10 𝐻𝐻 𝜔𝜔  Conversion from dB to a linear scale: 𝐻𝐻 𝜔𝜔 = 10 𝐻𝐻 𝜔𝜔 𝑑𝑑𝑑𝑑 20
K. Webb ENGR 202 19 Decibels – dB  Multiplying two gain values corresponds to adding their values in dB  E.g., the overall gain of cascaded systems 𝐻𝐻1 𝜔𝜔 ⋅ 𝐻𝐻2 𝜔𝜔 𝑑𝑑𝑑𝑑 = 𝐻𝐻1 𝜔𝜔 𝑑𝑑𝑑𝑑 + 𝐻𝐻2 𝜔𝜔 𝑑𝑑𝑑𝑑  Negative dB values corresponds to sub-unity gain  Positive dB values are gains greater than one dB Linear 60 1000 40 100 20 10 0 1 dB Linear 6 2 -3 1/√2 = 0.707 -6 0.5 -20 0.1
K. Webb ENGR 202 20 Value of Logarithmic Axes - dB  Gain axis is linear in dB  A logarithmic scale  Allows for displaying detail at very large and very small levels on the same plot  Gain plotted in dB  Two resonant peaks clearly visible  Linear gain scale  Smaller peak has disappeared
K. Webb ENGR 202 21 Value of Logarithmic Axes - Frequency  Frequency axis is logarithmic  Allows for displaying detail at very low and very high frequencies on the same plot  Log frequency axis  Can resolve frequency of both resonant peaks  Linear frequency axis  Lower resonant frequency is unclear
K. Webb ENGR 202 22 Interpreting Bode Plots Bode plots tell you the gain and phase shift at all frequencies: choose a frequency, read gain and phase values from the plot For a 10KHz sinusoidal input, the gain is 0dB (1) and the phase shift is 0°. For a 10MHz sinusoidal input, the gain is -32dB (0.025), and the phase shift is -176°.
K. Webb ENGR 202 23 Interpreting Bode Plots
K. Webb ENGR 202 Example Problems 24
K. Webb ENGR 202 A measured signal has the frequency spectrum shown here. Assuming the larger signal component has an amplitude of 500 mV, and that both signal components are in phase, write a time- domain expression for the measured signal.
K. Webb ENGR 202 Determine the frequency response function, 𝐻𝐻 𝜔𝜔 , for the following circuit. What are the circuit’s gain and phase at 200 kHz?
K. Webb ENGR 202
K. Webb ENGR 202 The input to a circuit with the following Bode plot is 𝑣𝑣𝑖𝑖 𝑡𝑡 = 1.2𝑉𝑉 ⋅ cos 2𝜋𝜋 ⋅ 10𝑘𝑘𝑘𝑘𝑘𝑘 ⋅ 𝑡𝑡 What is the output, 𝑣𝑣𝑜𝑜 𝑡𝑡 ?
K. Webb ENGR 202 Filters are classified by the ranges of frequencies they pass and those they filter out Types of Filters 29
K. Webb ENGR 202 30 Filter Operation  Frequency spectrum describes frequency content of electrical signals  Frequency response describes system (circuit) gain and phase at different frequencies  Can design circuits (i.e. filters) to have high gain at desirable frequencies and low gain at undesirable frequencies  Want to filter out high frequencies?  Design a filter with low gain at high frequencies and high gain at low frequencies.  Want to filter out all signals between 1 MHz and 10 MHz?  Design a filter with low gain in this range and high gain everywhere else.
K. Webb ENGR 202 31 Types of Filters  Filters are classified according to the ranges of frequencies they pass and those they filter out  Low pass filters: pass low frequencies, filter out high frequencies  High pass filters: pass high frequencies, filter out low frequencies  Band pass filters: pass only a range of frequencies, filter out everything else  Band stop (notch) filters: filter out only a certain range of frequencies, pass all others
K. Webb ENGR 202 32 Ideal Filters  Ideal filter gain characteristics:  Unity gain in the pass band  Range of frequencies to be passed  Zero gain in the stop band  Range of frequencies to be filtered out  Abrupt transition between pass band and stop band  Signals with frequency components in the pass band pass through the filter unaltered  Signals with frequency components in the stop band are completely filtered out – removed from the signal
K. Webb ENGR 202 33 Ideal Filters – Magnitude Response Ideal Low Pass Filter Ideal High Pass Filter Ideal Band Pass Filter Ideal Band Stop Filter pass band stop band stop band pass band pass band stop band stop band stop band pass band pass band  Note the use of a linear gain scale here  Stop band gain of zero translates to −∞ dB  Ideal filters often referred to as brick wall filters
K. Webb ENGR 202 34 Real Filters – Magnitude Response Magnitude response for a real low pass filter: pass band stop band Pass band edge is freq. at which gain is down by 3 dB – the -3 dB frequency. This is the filter’s bandwidth. Roll-off rate between pass band and stop band depends on the type of filter – particularly, the order of the filter.
K. Webb ENGR 202 First-order – only one energy-storage element Passive – contain only resistors and capacitors or inductors – no opamps or transistors First-Order Passive Filters 35
K. Webb ENGR 202 36 Filters as Voltage Dividers  Already familiar with resistive voltage dividers: 𝑣𝑣𝑜𝑜 = 𝑣𝑣𝑠𝑠 𝑅𝑅2 𝑅𝑅1 + 𝑅𝑅2  Frequency response function: 𝐻𝐻 𝜔𝜔 = 𝐕𝐕𝑜𝑜 𝐕𝐕𝑠𝑠 = 𝑅𝑅2 𝑅𝑅1 + 𝑅𝑅2  A real constant – independent of frequency  Same gain at all frequencies  No phase shift at any frequency  Now consider a circuit whose resistances have been replaced with impedances : 𝐻𝐻 𝜔𝜔 = 𝐕𝐕𝑜𝑜 𝐕𝐕𝑠𝑠 = 𝑍𝑍2 𝑍𝑍1 + 𝑍𝑍2  Frequency response is now a complex function of frequency  Gain and phase vary as a function of frequency  Basis for the design of first-order filters
K. Webb ENGR 202 RC Low Pass Filter 37
K. Webb ENGR 202 38 RC Low Pass Filter  Now, let 𝑍𝑍1 be resistive and 𝑍𝑍2 be capacitive  Frequency response: 𝐻𝐻 𝜔𝜔 = 𝑉𝑉 𝑜𝑜 𝑉𝑉 𝑠𝑠 = 𝑍𝑍2 𝑍𝑍1 + 𝑍𝑍2 = � 1 𝑗𝑗𝑗𝑗𝑗𝑗 𝑅𝑅 + � 1 𝑗𝑗𝑗𝑗𝑗𝑗 𝐻𝐻 𝜔𝜔 = 1 1 + 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗  Recall from ENGR 201 that the transient response of this same circuit is characterized by its time constant, 𝜏𝜏 = 𝑅𝑅𝑅𝑅  In the frequency domain, this is the corner frequency or break frequency 𝜔𝜔𝑐𝑐 = 1 𝜏𝜏 = 1 𝑅𝑅𝑅𝑅 and 𝑓𝑓𝑐𝑐 = 1 2𝜋𝜋𝜋𝜋𝜋𝜋  The frequency at which gain is down by 3 dB  The -3 dB frequency  Frequency at which the magnitude of R and C impedances are equal
K. Webb ENGR 202 39 RC Low Pass Filter  To gain insight into the behavior of this filter circuit, consider two limiting cases  As 𝑓𝑓 → 0,  Capacitor→ open circuit  𝑖𝑖 𝑡𝑡 → 0  𝑣𝑣𝑜𝑜 → 𝑣𝑣𝑠𝑠  Gain → unity  As 𝑓𝑓 → ∞  Capacitor → short circuit  𝑣𝑣𝑜𝑜 shorted to ground  Gain → zero
K. Webb ENGR 202 40 RC Low Pass Filter – Bode Plot
K. Webb ENGR 202 Gain is -3 dB at 𝑓𝑓𝑐𝑐, the bandwidth of the filter 41 RC Low Pass Filter – Magnitude Response Response rolls off at -20dB/decade, or -6dB/octave above 𝑓𝑓𝑐𝑐 Low-frequency asymptote at 0dB 𝑓𝑓 𝑐𝑐 = 200 𝑘𝑘𝑘𝑘𝑘𝑘
K. Webb ENGR 202 42 RC Low Pass Filter – Phase Response 𝑓𝑓 𝑐𝑐 = 200 𝑘𝑘𝑘𝑘𝑘𝑘 Phase approximation rolls off at -45°/decade around 𝑓𝑓𝑐𝑐 Phase is -45° at 𝑓𝑓𝑐𝑐 Low-frequency asymptote at 0° High-frequency asymptote at -90° One decade above 𝑓𝑓𝑐𝑐 One decade below 𝑓𝑓𝑐𝑐
K. Webb ENGR 202 43 RC Low Pass Filter – Magnitude Response  Known slope can be used to relate gains at different frequencies 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐻𝐻 𝑓𝑓2 𝑑𝑑𝑑𝑑 − 𝐻𝐻 𝑓𝑓1 𝑑𝑑𝑑𝑑 log10 𝑓𝑓2 − log10 𝑓𝑓1  For example: −20 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐻𝐻 7𝑀𝑀𝑀𝑀𝑀𝑀 − 𝐻𝐻 1𝑀𝑀𝑀𝑀𝑀𝑀 𝑑𝑑𝑑𝑑 log10 7𝑀𝑀𝑀𝑀𝑀𝑀 − log10 1𝑀𝑀𝑀𝑀𝑀𝑀 −20 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐻𝐻 7𝑀𝑀𝑀𝑀𝑀𝑀 − −14𝑑𝑑𝑑𝑑 6.845 − 6 𝐻𝐻 7𝑀𝑀𝑀𝑀𝑀𝑀 = −30.9𝑑𝑑𝑑𝑑 -20dB/dec 𝐻𝐻 1𝑀𝑀𝑀𝑀𝑀𝑀 = −14𝑑𝑑𝑑𝑑 𝐻𝐻 7𝑀𝑀𝑀𝑀𝑀𝑀 = ??
K. Webb ENGR 202 44 RC LP Filter – Application Example  Simple first-order RC low pass filters provide a quick and easy way to remove noise from electrical signals  Consider for example a dual-tone multi-frequency (DTMF) signal  Touch-tone telephone signal (key 5 in this example)  Tone is the sum of two sinusoids (key 5 = 1336Hz and 770Hz)  Pressing the “5” key generates the DTMF signal  Noise on the DTMF signal makes decoding impossible  Filter noise to enabling decoding
K. Webb ENGR 202 45 RC LP Filter – Application Example  Key number 5 is pressed  DTMF signal generated  Sum of 770 Hz and 1336 Hz sinusoids  Decoder at the receiving end decodes the DTMF signal and determines that a 5 was pressed
K. Webb ENGR 202 46 RC LP Filter – Application Example  Consider a more realistic scenario  DTMF signal corrupted by a significant amount of noise  The decoder is no longer able to determine that a 5 was pressed
K. Webb ENGR 202 47 RC LP Filter – Application Example  The goal is to filter the received signal so that the decoder can accurately interpret the DTMF signal  Designing the low pass filter  White noise  Flat frequency spectrum  Equal power at all frequencies  DTMF frequency range: 697 Hz – 1633 Hz  Want to attenuate as much noise as possible  Want to attenuate DTMF signals as little as possible  RC LPF with corner frequency at 10 kHz will limit DTMF-band attenuation to < 0.2 dB Filter
K. Webb ENGR 202 48 RC LP Filter – Application Example  RC LPF design  Need to select R and C to set the corner frequency 𝑓𝑓𝑐𝑐 = 1 2𝜋𝜋𝜋𝜋𝜋𝜋 = 10 𝑘𝑘𝑘𝑘𝑘𝑘  Say we have a 0.1 𝜇𝜇F capacitor available  Solve for R 𝑅𝑅 = 1 2𝜋𝜋𝑓𝑓𝐶𝐶𝐶𝐶 𝑅𝑅 = 1 2𝜋𝜋 ⋅ 10 𝑘𝑘𝑘𝑘𝑘𝑘 ⋅ 0.1𝜇𝜇𝜇𝜇 = 159 Ω 𝑅𝑅 = 159 Ω, 𝐶𝐶 = 0.1 𝜇𝜇𝜇𝜇
K. Webb ENGR 202 49 RC LP Filter – Application Example Bode plot of the resulting filter: noise attenuated DTMF signals lie in this range – passed through the filter with little attenuation
K. Webb ENGR 202 50 RC LP Filter – Application Example  Filter allows DTMF signal to pass mostly unaltered  Noise below 10 kHz is mostly passed through  Noise above 10 kHz is attenuated, but not removed  Received signal is not noiseless, but clean enough to be decoded
K. Webb ENGR 202 Example Problems 51
K. Webb ENGR 202 Design a filter to pass the desired 500 Hz signal and to attenuate the unwanted 100 kHz by 40 dB. What is the signal-to-noise ratio (SNR) at the output of the filter?
K. Webb ENGR 202
K. Webb ENGR 202
K. Webb ENGR 202 RC High Pass Filter 55
K. Webb ENGR 202 56 RC High Pass Filter  Now, swap the locations of the resistor and capacitor  Frequency response: 𝐻𝐻 𝜔𝜔 = 𝑉𝑉 𝑜𝑜 𝑉𝑉 𝑠𝑠 = 𝑍𝑍2 𝑍𝑍1 + 𝑍𝑍2 = 𝑅𝑅 𝑅𝑅 + � 1 𝑗𝑗𝑗𝑗𝑗𝑗 𝐻𝐻 𝜔𝜔 = 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 1 + 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗  Corner frequency is the same as for the low pass filter 𝜔𝜔𝑐𝑐 = 1 𝜏𝜏 = 1 𝑅𝑅𝑅𝑅 and 𝑓𝑓𝑐𝑐 = 1 2𝜋𝜋𝜋𝜋𝜋𝜋  The frequency at which gain is down by 3 dB  Frequency at which the capacitor impedance magnitude is equal to the resistor impedance magnitude  Now, gain is constant above 𝑓𝑓𝑐𝑐 and rolls off below 𝑓𝑓𝑐𝑐
K. Webb ENGR 202 57 RC High Pass Filter  To gain insight into the behavior of this filter circuit, consider two limiting cases  As 𝑓𝑓 → 0,  Capacitor→ open circuit  𝑖𝑖 𝑡𝑡 → 0  𝑣𝑣𝑜𝑜 → 0  Gain → zero  As 𝑓𝑓 → ∞  Capacitor → short circuit  𝑣𝑣𝑜𝑜 shorted to 𝑣𝑣𝑠𝑠  Gain → unity
K. Webb ENGR 202 58 RC High Pass Filter – Bode Plot
K. Webb ENGR 202 59 RC High Pass Filter – Magnitude Response 𝑓𝑓 𝑐𝑐 = 1 𝑘𝑘𝑘𝑘𝑘𝑘 Response rolls off at 20 dB/decade, or 6 dB/octave below 𝑓𝑓𝑐𝑐 Gain is -3dB at 𝑓𝑓𝑐𝑐 High-frequency asymptote at 0 dB
K. Webb ENGR 202 60 RC High Pass Filter – Phase Response 𝑓𝑓 𝑐𝑐 = 1 𝑘𝑘𝑘𝑘𝑘𝑘 Phase approximation rolls off at -45°/decade around 𝑓𝑓𝑐𝑐 Phase is +45° at 𝑓𝑓𝑐𝑐 Low-frequency asymptote at +90° High-frequency asymptote at 0° One decade above 𝑓𝑓𝑐𝑐 One decade below 𝑓𝑓𝑐𝑐
K. Webb ENGR 202 61 RC HP Filter – Application Example  High pass filters are useful for removing low-frequency content, including DC, from electrical signals.  For example, consider the following scenario:  Instrumented a flow loop in the lab  Pumps, temperature sensors, pressure sensors, and flow meters  Flow meter output seems to be erroneous every ~1 msec  Suspected cause: coupled through the +12V power supply from one of the pumps  Want to measure the flow meter’s +12V power supply with a channel on our data acquisition system (DAQ)  Dynamic range of DAQ input: ±5 V  Use a high-pass filter to remove the +12V DC component from the power supply voltage
K. Webb ENGR 202 62 RC HP Filter – Application Example  Want to a +12 V supply with a ±5 V DAQ input  High pass filter will remove the DC component of the supply voltage  High pass filter used to remove DC signal components  Couples only AC signal components to the DAQ input  AC coupling  Similar to the AC coupling setting on the scopes in the lab
K. Webb ENGR 202 63 RC HP Filter – Application Example  High pass filter design  Want to remove DC  Low corner frequency  High RC time constant  Large R and C  Arbitrarily set 𝑓𝑓𝑐𝑐 = 10 𝐻𝐻𝐻𝐻  DAQ system  Datasheet says 𝑅𝑅𝑖𝑖𝑖𝑖 = 10 𝑀𝑀Ω  Let 𝑅𝑅𝑖𝑖𝑖𝑖 be the filter resistance  Calculate C to get desired 𝑓𝑓𝑐𝑐 𝐶𝐶 = 1 2𝜋𝜋𝑓𝑓𝑐𝑐𝑅𝑅 = 1 2𝜋𝜋 ⋅ 10𝐻𝐻𝐻𝐻 ⋅ 10𝑀𝑀Ω 𝐶𝐶 = 15.9 𝑛𝑛𝑛𝑛  Or anything in that neighborhood  Not critical – just want to block DC
K. Webb ENGR 202 64 RC HP Filter – Application Example RC high pass filter: High pass filter Bode plot: The +12V DC component of the power supply voltage is completely removed.
K. Webb ENGR 202 65 RC HP Filter – Application Example The noisy +12V power supply at the malfunctioning flow meter: High pass filter output – AC coupled power supply voltage:  DC value of signal is +12 V  Outside ±5 V DAQ input dynamic range  DC value of signal is now 0 V  Within ±5 V DAQ input range  Glitches clearly measured with the DAQ
K. Webb ENGR 202 66 Oscilloscopes – AC Coupling  Scope inputs allow you to select between DC and AC coupling  Usually under the channel menu  DC coupling: input signal is terminated in 1MΩ and connected directly to the preamp and ADC in the scope  AC coupling: input signal is switched through a capacitor that forms a high pass filter with the 1MΩ input resistor  𝑓𝑓𝑐𝑐 ≈ 3.5 𝐻𝐻𝐻𝐻 – removes DC  Useful for looking at power supply ripple, etc.
K. Webb ENGR 202 67 Oscilloscopes – AC Coupling High-impedance scope front-end: Configured for DC coupling: Configured for AC coupling:
K. Webb ENGR 202 RL Filters 68
K. Webb ENGR 202 69 First-order RL filters  Can also use inductors to make RL low pass and high pass filters  Capacitors are usually preferable for simple first-order filters  Smaller  Cheaper  Draw no DC current RL low pass filter: RL high pass filter: Corner frequency: 𝑓𝑓𝑐𝑐 = 𝑅𝑅 2𝜋𝜋𝜋𝜋 Corner frequency: 𝑓𝑓𝑐𝑐 = 𝑅𝑅 2𝜋𝜋𝜋𝜋
K. Webb ENGR 202 70 RL Low Pass Filter  RL low pass filter  Frequency response: 𝐻𝐻 𝜔𝜔 = 𝑉𝑉 𝑜𝑜 𝑉𝑉 𝑠𝑠 = 𝑍𝑍2 𝑍𝑍1 + 𝑍𝑍2 = 𝑅𝑅 𝑅𝑅 + 𝑗𝑗𝑗𝑗𝑗𝑗 𝐻𝐻 𝜔𝜔 = 𝑅𝑅 𝑅𝑅 + 𝑗𝑗𝑗𝑗𝐿𝐿  Corner frequency is one over the time constant 𝜔𝜔𝑐𝑐 = 1 𝜏𝜏 = 𝑅𝑅 𝐿𝐿 and 𝑓𝑓𝑐𝑐 = 𝑅𝑅 2𝜋𝜋𝜋𝜋  The frequency at which gain is down by 3 dB  Frequency at which the inductor impedance magnitude is equal to the resistor impedance magnitude  Bode plot identical to that of the RC low pass filter  As it is for all first-order low pass systems
K. Webb ENGR 202 71 RL Low Pass Filter  Again consider the filter’s behavior for two limiting cases  As 𝑓𝑓 → 0,  Inductor → short circuit  𝑣𝑣𝑜𝑜 shorted to 𝑣𝑣𝑠𝑠  Gain → unity  As 𝑓𝑓 → ∞  Inductor → open circuit  𝑖𝑖 𝑡𝑡 → 0  𝑣𝑣𝑜𝑜 → 0  Gain → zero
K. Webb ENGR 202 72 RL High Pass Filter  Now, swap the locations of the resistor and inductor  Frequency response: 𝐻𝐻 𝜔𝜔 = 𝑉𝑉 𝑜𝑜 𝑉𝑉 𝑠𝑠 = 𝑍𝑍2 𝑍𝑍1 + 𝑍𝑍2 = 𝑗𝑗𝑗𝑗𝑗𝑗 𝑅𝑅 + 𝑗𝑗𝑗𝑗𝑗𝑗 𝐻𝐻 𝜔𝜔 = 𝑗𝑗𝑗𝑗𝑗𝑗 𝑅𝑅 + 𝑗𝑗𝑗𝑗𝐿𝐿  Corner frequency is the same as for the low pass filter 𝜔𝜔𝑐𝑐 = 1 𝜏𝜏 = 𝑅𝑅 𝐿𝐿 and 𝑓𝑓𝑐𝑐 = 𝑅𝑅 2𝜋𝜋𝜋𝜋  Bode plot is identical to that of the RC high pass filter  Gain is constant above 𝑓𝑓𝑐𝑐 and rolls off below 𝑓𝑓𝑐𝑐
K. Webb ENGR 202 73 RL High Pass Filter  Again, consider the two limiting frequency cases  As 𝑓𝑓 → 0,  Inductor → short circuit  𝑣𝑣𝑜𝑜 shorted to ground  Gain → zero  As 𝑓𝑓 → ∞  Inductor → open circuit  𝑖𝑖 𝑡𝑡 → 0  𝑣𝑣𝑜𝑜 → 𝑣𝑣𝑠𝑠  Gain → unity
K. Webb ENGR 202 Audio Filter Demo 74
K. Webb ENGR 202 75 Analog Discovery Instrument  2-chan. Scope  14-bit, 100MSa/s  5MHz bandwidth  2-chan. function generator  14-bit, 100MSa/s  5MHz bandwidth  2-chan. spectrum analyzer  Network analyzer  Voltmeter  ±5V power supplies  16-chan. logic analyzer  16-chan. digital pattern generator  USB connectivity
K. Webb ENGR 202 76 Analog Discovery – Audio Demo  Demo board plugs in to Analog Discovery module  Summation of multiple tones  Optional filtering of audio signal  3.5 mm audio output jack
K. Webb ENGR 202 77 Analog Discovery – Audio Demo

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SECTION 2 First Order Filters analysis.pdf

  • 1. ENGR 202 – Electrical Fundamentals II SECTION 2: FIRST-ORDER FILTERS
  • 2. K. Webb ENGR 202 Introduction 2
  • 3. K. Webb ENGR 202 3 Filters  We are all familiar with water and air filters  Basis for operation is size selectivity  Small particles (e.g. air or water molecules) are allowed to pass  Larger particles (e.g. dust, sediment) are not  Unwanted components are filtered out of the flow.  Electrical filters are similar  Basis for operation is frequency selectivity  Signal components in certain frequency ranges are filtered out  Signal components at other frequencies are allowed to pass
  • 4. K. Webb ENGR 202 4 Noise  All real-world electrical signals are noisy  You’ve seen this in the lab  Zoom in closely on a low-amplitude sinusoid with the scope (even one supplied directly from the function generator) – it won’t look like a perfectly clean sinusoid
  • 5. K. Webb ENGR 202 5 Noise  We will use the term noise to mean any electrical signal that interferes with or corrupts a signal we are trying to measure.  Noise has many sources:  Measurement instruments themselves  60Hz power line interference  Electrical components – resistors, transistors, etc.  Wireless LAN, fluorescent lights, computers, etc.  We’d like to be able to remove, or filter out, this noise  Improve the accuracy of measurements  Often possible, if we know the frequency characteristics of the signal and the noise
  • 6. K. Webb ENGR 202 6 Filtering Noise  We’ll learn how to design filters to remove noise Filter Noisy Signal Filtered Signal  First, we must introduce two important concepts:  Frequency-domain representation of electrical signals  What is meant by “frequency characteristics” of an electrical signal?  Frequency response of linear systems  How does a linear system (e.g. a filter) behave as a function of frequency?
  • 7. K. Webb ENGR 202 Frequency Spectrum 7
  • 8. K. Webb ENGR 202 8 Frequency Domain  We are accustomed to looking at electrical signals in the time domain  Amplitude plotted as function of time  Can also be represented in the frequency domain  Amplitude plotted as a function of frequency  Frequency spectrum  Describes the frequency content of a signal  Can think of signals as a sum of different frequency sinusoids  What frequencies (sinusoids) are present
  • 9. K. Webb ENGR 202 9 Frequency Spectrum  Frequency spectrum  An amplitude vs. frequency plot  X-axis is frequency – not time  Y-axis is amplitude  Amplitude units may be in decibels (dB)  Shows the relative amount of energy at each frequency  Time-domain plot and frequency spectrum are alternate representations of the same signal
  • 10. K. Webb ENGR 202 10 Frequency Spectra – Examples  Single sinusoid: 𝑣𝑣 𝑡𝑡 = 1𝑉𝑉 cos 2𝜋𝜋 ⋅ 800𝐻𝐻𝐻𝐻 ⋅ 𝑡𝑡  Sum of three sinusoids: 𝑣𝑣 𝑡𝑡 = 1𝑉𝑉[cos 2𝜋𝜋 ⋅ 800𝐻𝐻𝐻𝐻 ⋅ 𝑡𝑡 + cos 2𝜋𝜋 ⋅ 1200𝐻𝐻𝐻𝐻 ⋅ 𝑡𝑡 + cos 2𝜋𝜋 ⋅ 2000𝐻𝐻𝐻𝐻 ⋅ 𝑡𝑡 Time-domain Frequency-domain Time-domain Frequency-domain
  • 11. K. Webb ENGR 202 11 Frequency Spectra – Examples  White noise:  Band-limited (colored) noise: Time-domain Frequency-domain Time-domain Frequency-domain
  • 12. K. Webb ENGR 202 Frequency Spectra - Examples  Consider the following scenario  Measuring a sensor output in the lab  Know the signal is roughly sinusoidal  Suspected frequency: ~1 kHz  Same signal in the frequency domain: 12  Measured signal corrupted by noise/interference  Difficult to identify the interfering signal from the time-domain plot  Three interfering tones  All near 100 kHz  Can now design a filter to remove the noise:
  • 13. K. Webb ENGR 202 13 Fourier Transform  Fourier transform  Transforms a time-domain representation to a frequency spectrum 𝑉𝑉 𝜔𝜔 = � −∞ ∞ 𝑣𝑣 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑗𝑗𝑗𝑗𝑑𝑑𝑑𝑑  Inverse Fourier transform  Transforms from the frequency domain to the time domain 𝑣𝑣 𝑡𝑡 = 1 2𝜋𝜋 � −∞ ∞ 𝑉𝑉 𝜔𝜔 𝑒𝑒𝑗𝑗𝑗𝑗𝑗𝑗𝑑𝑑𝑑𝑑  A mathematical transform  Two different ways of looking at the same signal  A change in perspective not a change of the signal itself
  • 14. K. Webb ENGR 202 Frequency Response of Linear Systems 14
  • 15. K. Webb ENGR 202 15 Frequency Response Function  Linear systems (electrical, mechanical, etc.) can be described by their frequency responses  Frequency response  Ratio of the system output phasor to the system input phasor  In general, a complex function of frequency 𝐻𝐻 𝜔𝜔 = 𝐘𝐘 𝐗𝐗 = 𝐘𝐘 𝜔𝜔 𝐗𝐗 𝜔𝜔  Complex-valued – has both magnitude and phase  Magnitude: ratio of output to input magnitudes  Gain of the system  Phase: difference in phase between output and input  Phase shift through the system
  • 16. K. Webb ENGR 202 16 Frequency Response – Bode Plots  Frequency response  Description of system behavior as a function of frequency  Gain and phase  Represented graphically – formatted as a Bode plot  Magnitude plot on top, phase plot below  Logarithmic frequency axes  Magnitude usually has units of decibels (dB)  Phase has units of degrees
  • 17. K. Webb ENGR 202 17 Bode Plots Logarithmic frequency axes Units of magnitude are dB Magnitude plot on top Phase plot below Units of phase are degrees
  • 18. K. Webb ENGR 202 18 Decibels - dB  Frequency response gain most often expressed and plotted with units of decibels (dB)  A logarithmic scale  Provides detail of very large and very small values on the same plot  Commonly used for ratios of powers or amplitudes  Conversion from a linear scale to dB: 𝐻𝐻 𝜔𝜔 𝑑𝑑𝑑𝑑 = 20 ⋅ log10 𝐻𝐻 𝜔𝜔  Conversion from dB to a linear scale: 𝐻𝐻 𝜔𝜔 = 10 𝐻𝐻 𝜔𝜔 𝑑𝑑𝑑𝑑 20
  • 19. K. Webb ENGR 202 19 Decibels – dB  Multiplying two gain values corresponds to adding their values in dB  E.g., the overall gain of cascaded systems 𝐻𝐻1 𝜔𝜔 ⋅ 𝐻𝐻2 𝜔𝜔 𝑑𝑑𝑑𝑑 = 𝐻𝐻1 𝜔𝜔 𝑑𝑑𝑑𝑑 + 𝐻𝐻2 𝜔𝜔 𝑑𝑑𝑑𝑑  Negative dB values corresponds to sub-unity gain  Positive dB values are gains greater than one dB Linear 60 1000 40 100 20 10 0 1 dB Linear 6 2 -3 1/√2 = 0.707 -6 0.5 -20 0.1
  • 20. K. Webb ENGR 202 20 Value of Logarithmic Axes - dB  Gain axis is linear in dB  A logarithmic scale  Allows for displaying detail at very large and very small levels on the same plot  Gain plotted in dB  Two resonant peaks clearly visible  Linear gain scale  Smaller peak has disappeared
  • 21. K. Webb ENGR 202 21 Value of Logarithmic Axes - Frequency  Frequency axis is logarithmic  Allows for displaying detail at very low and very high frequencies on the same plot  Log frequency axis  Can resolve frequency of both resonant peaks  Linear frequency axis  Lower resonant frequency is unclear
  • 22. K. Webb ENGR 202 22 Interpreting Bode Plots Bode plots tell you the gain and phase shift at all frequencies: choose a frequency, read gain and phase values from the plot For a 10KHz sinusoidal input, the gain is 0dB (1) and the phase shift is 0°. For a 10MHz sinusoidal input, the gain is -32dB (0.025), and the phase shift is -176°.
  • 23. K. Webb ENGR 202 23 Interpreting Bode Plots
  • 24. K. Webb ENGR 202 Example Problems 24
  • 25. K. Webb ENGR 202 A measured signal has the frequency spectrum shown here. Assuming the larger signal component has an amplitude of 500 mV, and that both signal components are in phase, write a time- domain expression for the measured signal.
  • 26. K. Webb ENGR 202 Determine the frequency response function, 𝐻𝐻 𝜔𝜔 , for the following circuit. What are the circuit’s gain and phase at 200 kHz?
  • 28. K. Webb ENGR 202 The input to a circuit with the following Bode plot is 𝑣𝑣𝑖𝑖 𝑡𝑡 = 1.2𝑉𝑉 ⋅ cos 2𝜋𝜋 ⋅ 10𝑘𝑘𝑘𝑘𝑘𝑘 ⋅ 𝑡𝑡 What is the output, 𝑣𝑣𝑜𝑜 𝑡𝑡 ?
  • 29. K. Webb ENGR 202 Filters are classified by the ranges of frequencies they pass and those they filter out Types of Filters 29
  • 30. K. Webb ENGR 202 30 Filter Operation  Frequency spectrum describes frequency content of electrical signals  Frequency response describes system (circuit) gain and phase at different frequencies  Can design circuits (i.e. filters) to have high gain at desirable frequencies and low gain at undesirable frequencies  Want to filter out high frequencies?  Design a filter with low gain at high frequencies and high gain at low frequencies.  Want to filter out all signals between 1 MHz and 10 MHz?  Design a filter with low gain in this range and high gain everywhere else.
  • 31. K. Webb ENGR 202 31 Types of Filters  Filters are classified according to the ranges of frequencies they pass and those they filter out  Low pass filters: pass low frequencies, filter out high frequencies  High pass filters: pass high frequencies, filter out low frequencies  Band pass filters: pass only a range of frequencies, filter out everything else  Band stop (notch) filters: filter out only a certain range of frequencies, pass all others
  • 32. K. Webb ENGR 202 32 Ideal Filters  Ideal filter gain characteristics:  Unity gain in the pass band  Range of frequencies to be passed  Zero gain in the stop band  Range of frequencies to be filtered out  Abrupt transition between pass band and stop band  Signals with frequency components in the pass band pass through the filter unaltered  Signals with frequency components in the stop band are completely filtered out – removed from the signal
  • 33. K. Webb ENGR 202 33 Ideal Filters – Magnitude Response Ideal Low Pass Filter Ideal High Pass Filter Ideal Band Pass Filter Ideal Band Stop Filter pass band stop band stop band pass band pass band stop band stop band stop band pass band pass band  Note the use of a linear gain scale here  Stop band gain of zero translates to −∞ dB  Ideal filters often referred to as brick wall filters
  • 34. K. Webb ENGR 202 34 Real Filters – Magnitude Response Magnitude response for a real low pass filter: pass band stop band Pass band edge is freq. at which gain is down by 3 dB – the -3 dB frequency. This is the filter’s bandwidth. Roll-off rate between pass band and stop band depends on the type of filter – particularly, the order of the filter.
  • 35. K. Webb ENGR 202 First-order – only one energy-storage element Passive – contain only resistors and capacitors or inductors – no opamps or transistors First-Order Passive Filters 35
  • 36. K. Webb ENGR 202 36 Filters as Voltage Dividers  Already familiar with resistive voltage dividers: 𝑣𝑣𝑜𝑜 = 𝑣𝑣𝑠𝑠 𝑅𝑅2 𝑅𝑅1 + 𝑅𝑅2  Frequency response function: 𝐻𝐻 𝜔𝜔 = 𝐕𝐕𝑜𝑜 𝐕𝐕𝑠𝑠 = 𝑅𝑅2 𝑅𝑅1 + 𝑅𝑅2  A real constant – independent of frequency  Same gain at all frequencies  No phase shift at any frequency  Now consider a circuit whose resistances have been replaced with impedances : 𝐻𝐻 𝜔𝜔 = 𝐕𝐕𝑜𝑜 𝐕𝐕𝑠𝑠 = 𝑍𝑍2 𝑍𝑍1 + 𝑍𝑍2  Frequency response is now a complex function of frequency  Gain and phase vary as a function of frequency  Basis for the design of first-order filters
  • 37. K. Webb ENGR 202 RC Low Pass Filter 37
  • 38. K. Webb ENGR 202 38 RC Low Pass Filter  Now, let 𝑍𝑍1 be resistive and 𝑍𝑍2 be capacitive  Frequency response: 𝐻𝐻 𝜔𝜔 = 𝑉𝑉 𝑜𝑜 𝑉𝑉 𝑠𝑠 = 𝑍𝑍2 𝑍𝑍1 + 𝑍𝑍2 = � 1 𝑗𝑗𝑗𝑗𝑗𝑗 𝑅𝑅 + � 1 𝑗𝑗𝑗𝑗𝑗𝑗 𝐻𝐻 𝜔𝜔 = 1 1 + 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗  Recall from ENGR 201 that the transient response of this same circuit is characterized by its time constant, 𝜏𝜏 = 𝑅𝑅𝑅𝑅  In the frequency domain, this is the corner frequency or break frequency 𝜔𝜔𝑐𝑐 = 1 𝜏𝜏 = 1 𝑅𝑅𝑅𝑅 and 𝑓𝑓𝑐𝑐 = 1 2𝜋𝜋𝜋𝜋𝜋𝜋  The frequency at which gain is down by 3 dB  The -3 dB frequency  Frequency at which the magnitude of R and C impedances are equal
  • 39. K. Webb ENGR 202 39 RC Low Pass Filter  To gain insight into the behavior of this filter circuit, consider two limiting cases  As 𝑓𝑓 → 0,  Capacitor→ open circuit  𝑖𝑖 𝑡𝑡 → 0  𝑣𝑣𝑜𝑜 → 𝑣𝑣𝑠𝑠  Gain → unity  As 𝑓𝑓 → ∞  Capacitor → short circuit  𝑣𝑣𝑜𝑜 shorted to ground  Gain → zero
  • 40. K. Webb ENGR 202 40 RC Low Pass Filter – Bode Plot
  • 41. K. Webb ENGR 202 Gain is -3 dB at 𝑓𝑓𝑐𝑐, the bandwidth of the filter 41 RC Low Pass Filter – Magnitude Response Response rolls off at -20dB/decade, or -6dB/octave above 𝑓𝑓𝑐𝑐 Low-frequency asymptote at 0dB 𝑓𝑓 𝑐𝑐 = 200 𝑘𝑘𝑘𝑘𝑘𝑘
  • 42. K. Webb ENGR 202 42 RC Low Pass Filter – Phase Response 𝑓𝑓 𝑐𝑐 = 200 𝑘𝑘𝑘𝑘𝑘𝑘 Phase approximation rolls off at -45°/decade around 𝑓𝑓𝑐𝑐 Phase is -45° at 𝑓𝑓𝑐𝑐 Low-frequency asymptote at 0° High-frequency asymptote at -90° One decade above 𝑓𝑓𝑐𝑐 One decade below 𝑓𝑓𝑐𝑐
  • 43. K. Webb ENGR 202 43 RC Low Pass Filter – Magnitude Response  Known slope can be used to relate gains at different frequencies 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐻𝐻 𝑓𝑓2 𝑑𝑑𝑑𝑑 − 𝐻𝐻 𝑓𝑓1 𝑑𝑑𝑑𝑑 log10 𝑓𝑓2 − log10 𝑓𝑓1  For example: −20 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐻𝐻 7𝑀𝑀𝑀𝑀𝑀𝑀 − 𝐻𝐻 1𝑀𝑀𝑀𝑀𝑀𝑀 𝑑𝑑𝑑𝑑 log10 7𝑀𝑀𝑀𝑀𝑀𝑀 − log10 1𝑀𝑀𝑀𝑀𝑀𝑀 −20 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐻𝐻 7𝑀𝑀𝑀𝑀𝑀𝑀 − −14𝑑𝑑𝑑𝑑 6.845 − 6 𝐻𝐻 7𝑀𝑀𝑀𝑀𝑀𝑀 = −30.9𝑑𝑑𝑑𝑑 -20dB/dec 𝐻𝐻 1𝑀𝑀𝑀𝑀𝑀𝑀 = −14𝑑𝑑𝑑𝑑 𝐻𝐻 7𝑀𝑀𝑀𝑀𝑀𝑀 = ??
  • 44. K. Webb ENGR 202 44 RC LP Filter – Application Example  Simple first-order RC low pass filters provide a quick and easy way to remove noise from electrical signals  Consider for example a dual-tone multi-frequency (DTMF) signal  Touch-tone telephone signal (key 5 in this example)  Tone is the sum of two sinusoids (key 5 = 1336Hz and 770Hz)  Pressing the “5” key generates the DTMF signal  Noise on the DTMF signal makes decoding impossible  Filter noise to enabling decoding
  • 45. K. Webb ENGR 202 45 RC LP Filter – Application Example  Key number 5 is pressed  DTMF signal generated  Sum of 770 Hz and 1336 Hz sinusoids  Decoder at the receiving end decodes the DTMF signal and determines that a 5 was pressed
  • 46. K. Webb ENGR 202 46 RC LP Filter – Application Example  Consider a more realistic scenario  DTMF signal corrupted by a significant amount of noise  The decoder is no longer able to determine that a 5 was pressed
  • 47. K. Webb ENGR 202 47 RC LP Filter – Application Example  The goal is to filter the received signal so that the decoder can accurately interpret the DTMF signal  Designing the low pass filter  White noise  Flat frequency spectrum  Equal power at all frequencies  DTMF frequency range: 697 Hz – 1633 Hz  Want to attenuate as much noise as possible  Want to attenuate DTMF signals as little as possible  RC LPF with corner frequency at 10 kHz will limit DTMF-band attenuation to < 0.2 dB Filter
  • 48. K. Webb ENGR 202 48 RC LP Filter – Application Example  RC LPF design  Need to select R and C to set the corner frequency 𝑓𝑓𝑐𝑐 = 1 2𝜋𝜋𝜋𝜋𝜋𝜋 = 10 𝑘𝑘𝑘𝑘𝑘𝑘  Say we have a 0.1 𝜇𝜇F capacitor available  Solve for R 𝑅𝑅 = 1 2𝜋𝜋𝑓𝑓𝐶𝐶𝐶𝐶 𝑅𝑅 = 1 2𝜋𝜋 ⋅ 10 𝑘𝑘𝑘𝑘𝑘𝑘 ⋅ 0.1𝜇𝜇𝜇𝜇 = 159 Ω 𝑅𝑅 = 159 Ω, 𝐶𝐶 = 0.1 𝜇𝜇𝜇𝜇
  • 49. K. Webb ENGR 202 49 RC LP Filter – Application Example Bode plot of the resulting filter: noise attenuated DTMF signals lie in this range – passed through the filter with little attenuation
  • 50. K. Webb ENGR 202 50 RC LP Filter – Application Example  Filter allows DTMF signal to pass mostly unaltered  Noise below 10 kHz is mostly passed through  Noise above 10 kHz is attenuated, but not removed  Received signal is not noiseless, but clean enough to be decoded
  • 51. K. Webb ENGR 202 Example Problems 51
  • 52. K. Webb ENGR 202 Design a filter to pass the desired 500 Hz signal and to attenuate the unwanted 100 kHz by 40 dB. What is the signal-to-noise ratio (SNR) at the output of the filter?
  • 55. K. Webb ENGR 202 RC High Pass Filter 55
  • 56. K. Webb ENGR 202 56 RC High Pass Filter  Now, swap the locations of the resistor and capacitor  Frequency response: 𝐻𝐻 𝜔𝜔 = 𝑉𝑉 𝑜𝑜 𝑉𝑉 𝑠𝑠 = 𝑍𝑍2 𝑍𝑍1 + 𝑍𝑍2 = 𝑅𝑅 𝑅𝑅 + � 1 𝑗𝑗𝑗𝑗𝑗𝑗 𝐻𝐻 𝜔𝜔 = 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 1 + 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗  Corner frequency is the same as for the low pass filter 𝜔𝜔𝑐𝑐 = 1 𝜏𝜏 = 1 𝑅𝑅𝑅𝑅 and 𝑓𝑓𝑐𝑐 = 1 2𝜋𝜋𝜋𝜋𝜋𝜋  The frequency at which gain is down by 3 dB  Frequency at which the capacitor impedance magnitude is equal to the resistor impedance magnitude  Now, gain is constant above 𝑓𝑓𝑐𝑐 and rolls off below 𝑓𝑓𝑐𝑐
  • 57. K. Webb ENGR 202 57 RC High Pass Filter  To gain insight into the behavior of this filter circuit, consider two limiting cases  As 𝑓𝑓 → 0,  Capacitor→ open circuit  𝑖𝑖 𝑡𝑡 → 0  𝑣𝑣𝑜𝑜 → 0  Gain → zero  As 𝑓𝑓 → ∞  Capacitor → short circuit  𝑣𝑣𝑜𝑜 shorted to 𝑣𝑣𝑠𝑠  Gain → unity
  • 58. K. Webb ENGR 202 58 RC High Pass Filter – Bode Plot
  • 59. K. Webb ENGR 202 59 RC High Pass Filter – Magnitude Response 𝑓𝑓 𝑐𝑐 = 1 𝑘𝑘𝑘𝑘𝑘𝑘 Response rolls off at 20 dB/decade, or 6 dB/octave below 𝑓𝑓𝑐𝑐 Gain is -3dB at 𝑓𝑓𝑐𝑐 High-frequency asymptote at 0 dB
  • 60. K. Webb ENGR 202 60 RC High Pass Filter – Phase Response 𝑓𝑓 𝑐𝑐 = 1 𝑘𝑘𝑘𝑘𝑘𝑘 Phase approximation rolls off at -45°/decade around 𝑓𝑓𝑐𝑐 Phase is +45° at 𝑓𝑓𝑐𝑐 Low-frequency asymptote at +90° High-frequency asymptote at 0° One decade above 𝑓𝑓𝑐𝑐 One decade below 𝑓𝑓𝑐𝑐
  • 61. K. Webb ENGR 202 61 RC HP Filter – Application Example  High pass filters are useful for removing low-frequency content, including DC, from electrical signals.  For example, consider the following scenario:  Instrumented a flow loop in the lab  Pumps, temperature sensors, pressure sensors, and flow meters  Flow meter output seems to be erroneous every ~1 msec  Suspected cause: coupled through the +12V power supply from one of the pumps  Want to measure the flow meter’s +12V power supply with a channel on our data acquisition system (DAQ)  Dynamic range of DAQ input: ±5 V  Use a high-pass filter to remove the +12V DC component from the power supply voltage
  • 62. K. Webb ENGR 202 62 RC HP Filter – Application Example  Want to a +12 V supply with a ±5 V DAQ input  High pass filter will remove the DC component of the supply voltage  High pass filter used to remove DC signal components  Couples only AC signal components to the DAQ input  AC coupling  Similar to the AC coupling setting on the scopes in the lab
  • 63. K. Webb ENGR 202 63 RC HP Filter – Application Example  High pass filter design  Want to remove DC  Low corner frequency  High RC time constant  Large R and C  Arbitrarily set 𝑓𝑓𝑐𝑐 = 10 𝐻𝐻𝐻𝐻  DAQ system  Datasheet says 𝑅𝑅𝑖𝑖𝑖𝑖 = 10 𝑀𝑀Ω  Let 𝑅𝑅𝑖𝑖𝑖𝑖 be the filter resistance  Calculate C to get desired 𝑓𝑓𝑐𝑐 𝐶𝐶 = 1 2𝜋𝜋𝑓𝑓𝑐𝑐𝑅𝑅 = 1 2𝜋𝜋 ⋅ 10𝐻𝐻𝐻𝐻 ⋅ 10𝑀𝑀Ω 𝐶𝐶 = 15.9 𝑛𝑛𝑛𝑛  Or anything in that neighborhood  Not critical – just want to block DC
  • 64. K. Webb ENGR 202 64 RC HP Filter – Application Example RC high pass filter: High pass filter Bode plot: The +12V DC component of the power supply voltage is completely removed.
  • 65. K. Webb ENGR 202 65 RC HP Filter – Application Example The noisy +12V power supply at the malfunctioning flow meter: High pass filter output – AC coupled power supply voltage:  DC value of signal is +12 V  Outside ±5 V DAQ input dynamic range  DC value of signal is now 0 V  Within ±5 V DAQ input range  Glitches clearly measured with the DAQ
  • 66. K. Webb ENGR 202 66 Oscilloscopes – AC Coupling  Scope inputs allow you to select between DC and AC coupling  Usually under the channel menu  DC coupling: input signal is terminated in 1MΩ and connected directly to the preamp and ADC in the scope  AC coupling: input signal is switched through a capacitor that forms a high pass filter with the 1MΩ input resistor  𝑓𝑓𝑐𝑐 ≈ 3.5 𝐻𝐻𝐻𝐻 – removes DC  Useful for looking at power supply ripple, etc.
  • 67. K. Webb ENGR 202 67 Oscilloscopes – AC Coupling High-impedance scope front-end: Configured for DC coupling: Configured for AC coupling:
  • 68. K. Webb ENGR 202 RL Filters 68
  • 69. K. Webb ENGR 202 69 First-order RL filters  Can also use inductors to make RL low pass and high pass filters  Capacitors are usually preferable for simple first-order filters  Smaller  Cheaper  Draw no DC current RL low pass filter: RL high pass filter: Corner frequency: 𝑓𝑓𝑐𝑐 = 𝑅𝑅 2𝜋𝜋𝜋𝜋 Corner frequency: 𝑓𝑓𝑐𝑐 = 𝑅𝑅 2𝜋𝜋𝜋𝜋
  • 70. K. Webb ENGR 202 70 RL Low Pass Filter  RL low pass filter  Frequency response: 𝐻𝐻 𝜔𝜔 = 𝑉𝑉 𝑜𝑜 𝑉𝑉 𝑠𝑠 = 𝑍𝑍2 𝑍𝑍1 + 𝑍𝑍2 = 𝑅𝑅 𝑅𝑅 + 𝑗𝑗𝑗𝑗𝑗𝑗 𝐻𝐻 𝜔𝜔 = 𝑅𝑅 𝑅𝑅 + 𝑗𝑗𝑗𝑗𝐿𝐿  Corner frequency is one over the time constant 𝜔𝜔𝑐𝑐 = 1 𝜏𝜏 = 𝑅𝑅 𝐿𝐿 and 𝑓𝑓𝑐𝑐 = 𝑅𝑅 2𝜋𝜋𝜋𝜋  The frequency at which gain is down by 3 dB  Frequency at which the inductor impedance magnitude is equal to the resistor impedance magnitude  Bode plot identical to that of the RC low pass filter  As it is for all first-order low pass systems
  • 71. K. Webb ENGR 202 71 RL Low Pass Filter  Again consider the filter’s behavior for two limiting cases  As 𝑓𝑓 → 0,  Inductor → short circuit  𝑣𝑣𝑜𝑜 shorted to 𝑣𝑣𝑠𝑠  Gain → unity  As 𝑓𝑓 → ∞  Inductor → open circuit  𝑖𝑖 𝑡𝑡 → 0  𝑣𝑣𝑜𝑜 → 0  Gain → zero
  • 72. K. Webb ENGR 202 72 RL High Pass Filter  Now, swap the locations of the resistor and inductor  Frequency response: 𝐻𝐻 𝜔𝜔 = 𝑉𝑉 𝑜𝑜 𝑉𝑉 𝑠𝑠 = 𝑍𝑍2 𝑍𝑍1 + 𝑍𝑍2 = 𝑗𝑗𝑗𝑗𝑗𝑗 𝑅𝑅 + 𝑗𝑗𝑗𝑗𝑗𝑗 𝐻𝐻 𝜔𝜔 = 𝑗𝑗𝑗𝑗𝑗𝑗 𝑅𝑅 + 𝑗𝑗𝑗𝑗𝐿𝐿  Corner frequency is the same as for the low pass filter 𝜔𝜔𝑐𝑐 = 1 𝜏𝜏 = 𝑅𝑅 𝐿𝐿 and 𝑓𝑓𝑐𝑐 = 𝑅𝑅 2𝜋𝜋𝜋𝜋  Bode plot is identical to that of the RC high pass filter  Gain is constant above 𝑓𝑓𝑐𝑐 and rolls off below 𝑓𝑓𝑐𝑐
  • 73. K. Webb ENGR 202 73 RL High Pass Filter  Again, consider the two limiting frequency cases  As 𝑓𝑓 → 0,  Inductor → short circuit  𝑣𝑣𝑜𝑜 shorted to ground  Gain → zero  As 𝑓𝑓 → ∞  Inductor → open circuit  𝑖𝑖 𝑡𝑡 → 0  𝑣𝑣𝑜𝑜 → 𝑣𝑣𝑠𝑠  Gain → unity
  • 74. K. Webb ENGR 202 Audio Filter Demo 74
  • 75. K. Webb ENGR 202 75 Analog Discovery Instrument  2-chan. Scope  14-bit, 100MSa/s  5MHz bandwidth  2-chan. function generator  14-bit, 100MSa/s  5MHz bandwidth  2-chan. spectrum analyzer  Network analyzer  Voltmeter  ±5V power supplies  16-chan. logic analyzer  16-chan. digital pattern generator  USB connectivity
  • 76. K. Webb ENGR 202 76 Analog Discovery – Audio Demo  Demo board plugs in to Analog Discovery module  Summation of multiple tones  Optional filtering of audio signal  3.5 mm audio output jack
  • 77. K. Webb ENGR 202 77 Analog Discovery – Audio Demo