Frequency Response & Resonant Circuits
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Transcript of Frequency Response & Resonant Circuits
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FREQUENCY RESPONSE & RESONANT CIRCUITSFilters, frequency response, time domain connection, bode plots, resonant circuits.
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OUTLINE AND TOPICS Low-pass filters High-pass filters Other filters Resonance (Ch 20) Ideal op-amps and active filters Decibels & log scales Linear systems and transfer functions Bode plots
Reading1. Boylestad Ch 21.1-
21.112. Boylestad Ch 20.1-20.8
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FILTERS
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FILTERS Any combination of passive (R, L, and
C) and/or active (transistors or operational amplifiers) elements designed to select or reject a band of frequencies is called a filter.
In communication systems, filters are used to pass those frequencies containing the desired information and to reject the remaining frequencies.
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FILTERS In general, there are two classifications of
filters: Passive filters-gain always<1 Active filters-gain can be >1
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FILTERS
FIG. 21.7 Defining the four broad categories of filters.
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R-C LOW-PASS FILTER
FIG. 21.8 Low-pass filter.
FIG. 21.9 R-C low-pass filter at low frequencies.
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R-C LOW-PASS FILTER
FIG. 21.10 R-C low-pass filter at high frequencies.
FIG. 21.11 Vo versus frequency for a low-pass R-C filter.
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R-C LOW-PASS FILTER
FIG. 21.12 Normalized plot of Fig. 21.11.
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R-C LOW-PASS FILTER
FIG. 21.13 Angle by which Vo leads Vi.
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R-C LOW-PASS FILTER
FIG. 21.14 Angle by which Vo lags Vi.
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R-C LOW-PASS FILTER
FIG. 21.15 Low-pass R-L filter.
FIG. 21.16 Example 21.5.
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R-C LOW-PASS FILTER
FIG. 21.17 Frequency response for the low-pass R-C network in Fig. 21.16.
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R-C LOW-PASS FILTER
FIG. 21.18 Normalized plot of Fig. 21.17.
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R-C HIGH-PASS FILTER
FIG. 21.19 High-pass filter.
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R-C HIGH-PASS FILTER
FIG. 21.20 R-C high-pass filter at very high frequencies.
FIG. 21.21 R-C high-pass filter at f = 0 Hz.
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R-C HIGH-PASS FILTER
FIG. 21.22 Vo versus frequency for a high-pass R-C filter.
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R-C HIGH-PASS FILTER
FIG. 21.23 Normalized plot of Fig. 21.22.
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R-C HIGH-PASS FILTER
FIG. 21.24 Phase-angle response for the high-pass R-C filter.
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R-C HIGH-PASS FILTER
FIG. 21.25 High-pass R-L filter.
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R-C HIGH-PASS FILTER
FIG. 21.26 Normalized plots for a low-pass and a high-pass filter using the same elements.
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R-C HIGH-PASS FILTER
FIG. 21.27 Phase plots for a low-pass and a high-pass filter using the same elements.
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PASS-BAND FILTERS
FIG. 21.28 Series resonant pass-band filter.
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RLC CIRCUITS-RESONANCE! The resonant electrical circuit must
have both inductance and capacitance.
In addition, resistance will always be present due either to the lack of ideal elements or to the control offered on the shape of the resonance curve.
When resonance occurs due to the application of the proper frequency ( fr), the energy absorbed by one reactive element is the same as that released by another reactive element within the system.
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SERIES RESONANT CIRCUIT A resonant circuit (series or parallel) must
have an inductive and a capacitive element. A resistive element is always present due to
the internal resistance of the source (Rs), the internal resistance of the inductor (Rl), and any added resistance to control the shape of the response curve (Rdesign).
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SERIES RESONANT CIRCUIT
FIG. 20.2 Series resonant circuit.
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PASS-BAND FILTERS
FIG. 21.29 Parallel resonant pass-band filter.
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PASS-BAND FILTERS
FIG. 21.30 Series resonant pass-band filter for Example 21.7.
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PASS-BAND FILTERS
FIG. 21.31 Pass-band response for the network.
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PASS-BAND FILTERS
FIG. 21.32 Normalized plots for the pass-band filter in Fig. 21.30.
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PASS-BAND FILTERS
FIG. 21.33 Pass-band filter.
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PASS-BAND FILTERS
FIG. 21.34 Pass-band characteristics.
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PASS-BAND FILTERS
FIG. 21.35 Pass-band filter.
FIG. 21.36 Pass-band characteristics for the filter in Fig. 21.35.
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PASS-BAND FILTERS
FIG. 21.37 Network of Fig. 21.35 at f = 994.72 kHz.
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BAND-REJECT FILTERS Since the characteristics of a band-reject
filter (also called stop-band or notch filter) are the inverse of the pattern obtained for the band-pass filter, a band-reject filter can be designed by simply applying Kirchhoff’s voltage law to each circuit.
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BAND-REJECT FILTERS
FIG. 21.38 Demonstrating how an applied signal of fixed magnitude can be broken down into a pass-band and band-reject response curve.
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BAND-REJECT FILTERS
FIG. 21.39 Band-reject filter using a series resonant circuit.
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BAND-REJECT FILTERS
FIG. 21.40 Band-reject filter using a parallel resonant network.
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BAND-REJECT FILTERS
FIG. 21.41 Band-reject filter.
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BAND-REJECT FILTERS
FIG. 21.42 Band-reject characteristics.
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OPERATIONAL AMPLIFIERSActive filters
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AMPLIFIERS GIVE GAIN
Simple amp-1 input and 1 outputGain, A=Vout/Vin
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EXAMPLE
If the amplifier above gives an output voltage of 1000V with an input voltage of 50V, what is the gain?
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IDEAL OPERATIONAL-AMPLIFIER(OP-AMP)
Inputs draw no current-infinite input impedaceVout=A(Vplus-Vminus) A-open loop gain.
A is ideally infinity-How is this useful?Output can provide as much voltage/current as needed-zero output impedance
http://www.youtube.com/watch?v=TQB1VlLBgJE
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NEGATIVE FEEDBACK
Negative feedback (NF) tries to reduce the differencewith NF, Vplus=Vminus ALWAYS
summing point constraintsvirtual ground.
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INVERTING AMPLIFIER
Input goes into Vminus input-INVERTING inputGain, Ainv=-R2/R1, gain is negative because inverting
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INVERTING AMPLIFIER
Vplus=VminusInputs draw no current
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NON-INVERTING AMPLIFIER
Input goes into Vplus input-NON-INVERTING inputGain, Ainv=1+R2/R1, gain is positive
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UNITY GAIN BUFFER
Gain is 1 i.e. Vin=VoutUsed to isolate one side from the other
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REAL OP-AMPS
Output voltage determined by rails (power supply)Open loop gain not infinityInputs draw small amount of current-nA’s or less
Quad LM324Single LM741
http://www.national.com/mpf/LM/LM324.html#Overview
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BANDPASS FILTER AMPLIFIER
f1=0.3Hz, f2=10HzHigh freq., cap is short, low freq., cap is open
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FREQUENCY<F1
all caps are open.What is the gain?
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F1<FREQUENCY<F2
C1 is short. C2 is open.What is the gain?-midband gain.
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FREQUENCY>F2
All caps are shortsWhat is the gain?
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LINEAR SYSTEMS
T(s) has zeros when T(s)=0T(s) has poles when T(s)=infinity
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LINEAR SYSTEMSAny voltage signal can be represented by a sum of sinusoidal voltage signals-Fourier/Laplace theoremsIf s=jω, the input voltage is represented by:
V0exp(jωt)= V0exp(st) Allows us to use algebra instead of differential eqns.
RLC circuit, for example.
t
ttt
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FILTER OP-AMP
What is T(s)?
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FILTER OP-AMP
zero at s=0poles at 1/R1C1 and 1/R2C2What happens at the zero? At the poles?
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DECIBELS & BODE PLOTSThe key to amplifiers and control systems.
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INTRODUCTION The unit decibel (dB), defined by a
logarithmic expression, is used throughout the industry to define levels of audio, voltage gain, energy, field strength, and so on.
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INTRODUCTIONLOGARITHMSBasic Relationships
Let us first examine the relationship between the variables of the logarithmic function.
The mathematical expression:
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INTRODUCTIONLOGARITHMSSome Areas of Application
The following are some of the most common applications of the logarithmic function: 1. The response of a system can be plotted for a
range of values that may otherwise be impossible or unwieldy with a linear scale.
2. Levels of power, voltage, and the like can be compared without dealing with very large or very small numbers that often cloud the true impact of the difference in magnitudes.
3. A number of systems respond to outside stimuli in a nonlinear logarithmic manner.
4. The response of a cascaded or compound system can be rapidly determined using logarithms if the gain of each stage is known on a logarithmic basis.
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INTRODUCTIONLOGARITHMS
FIG. 21.1 Semilog graph paper.
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INTRODUCTIONLOGARITHMS
FIG. 21.2 Frequency log scale.
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INTRODUCTIONLOGARITHMS
FIG. 21.3 Finding a value on a log plot.
FIG. 21.4 Example 21.1.
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PROPERTIES OF LOGARITHMS There are a few characteristics of logarithms
that should be emphasized: The common or natural logarithm of the
number 1 is 0 The log of any number less than 1 is a
negative number The log of the product of two numbers is the
sum of the logs of the numbers The log of the quotient of two numbers is the
log of the numerator minus the log of the denominator
The log of a number taken to a power is equal to the product of the power and the log of the number
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PROPERTIES OF LOGARITHMSCALCULATOR FUNCTIONS Using the TI-89 calculator, the
common logarithm of a number is determined by first selecting the CATALOG key and then scrolling to find the common logarithm function.
The time involved in scrolling through the options can be reduced by first selecting the key with the first letter of the desired function—in this case, L, as shown below, to find the common logarithm of the number 80.
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DECIBELS Power Gain Voltage Gain Human Auditory Response
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DECIBELS
TABLE 21.1
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DECIBELS
TABLE 21.2 Typical sound levels and their decibel levels.
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DECIBELS
FIG. 21.5 LRAD (Long Range Acoustic Device) 1000X. (Courtesy of the American Technology Corporation.)
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DECIBELSINSTRUMENTATION
FIG. 21.6 Defining the relationship between a dB scale referenced to 1 mW, 600Ω and a 3 V rms voltage scale.
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LINEAR SYSTEMS RLC circuits, op-amps are linear circuit
elements i.e. a differential equation can describe them.
You can add solutions at a given ω i.e. if exp(jωt) and exp(-jωt) are solutions, exp(jωt)+exp(-jωt)=2cos(ωt) is a solution.
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TRANSFER FUNCTION Transfer function T(s), or H(s) describes how
the output is affected by the input. i.e. T(s)=Vo/Vi s=jω, so ZC=1/sC and ZL=sL The ‘s’ notation is convenient shorthand, but
is also important in the context of Laplace Transforms, which you will see later in the class.
Transfer because it describes how voltage is “transferred” from the input to output.
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TRANSFER FUNCTION FOR LOW-PASS Again we go to our good friend, the low-pass
filter.
FIG. 21.16 Example 21.5.
Now, we will redo this in the language of “transfer function”
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POLES & ZEROS All transfer functions have poles and zeros. Zeros are when T(s)=0 Poles are when 1/T(s)=0 or T(s)=∞ These contribute very distinct behaviors to
the frequency response of a system. For our friend the RC lowpass circuit,
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BODE PLOTS