Frequency Response & Resonant Circuits

FREQUENCY RESPONSE & RESONANT CIRCUITSFilters, frequency response, time domain connection, bode plots, resonant circuits.

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

FILTERS

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.

FILTERS In general, there are two classifications of

filters: Passive filters-gain always<1 Active filters-gain can be >1

Circuit or system

T(jω)=T(s)Vi Vo

Gain works for DC alwaysFor AC, only for amplitude/phase of sine/cosine

Gain=Vo/ViCan also be representedBy a differential eqn.

FILTERS

FIG. 21.7 Defining the four broad categories of filters.

Couple of definitions:Impedance of a circuit element is the AC analog of resistance. Only works for sines and cosines!

Z-impedanceR-real part of impedance-resistanceX-imag part of impedance-REACTANCE

Resistance is always positiveReactance can be positive or negative

Positive reactance-inductiveNegative reactance-capacitive

FILTERS

FIG. 21.7 Defining the four broad categories of filters.

Impedance of a circuit element is the AC analog of resistance. Only works for sines and cosines! WHY?

if input and output are sine or cosine, we can represent them as

Then, the “resistance-like” ratio of the amplitudes of current and voltage

Why?

R-C LOW-PASS FILTER

FIG. 21.8 Low-pass filter.

FIG. 21.9 R-C low-pass filter at low frequencies.

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.

Applying voltage divider

Zc-impedanceXc-reactance

We must take the magnitudeTo get the size of the gain. This magnitude gives rise to the square root we see in the textbook.

We take the “argument” or angleTo get the phase of the gain.

R-C LOW-PASS FILTER

FIG. 21.12 Normalized plot of Fig. 21.11.

Links to help with sketching sines and cosines

http://www.youtube.com/watch?v=ijTIr-aykUk&feature=relmfu

http://www.youtube.com/watch?feature=endscreen&v=RzZyyIu9IvA&NR=1

θ-phase change

At a given frequency, gain is a complex number that can be drawn as

Real

Imaginary/”reactive”

Length is amplitude gain








R-C LOW-PASS FILTER-PHASE

FIG. 21.13 Angle by which Vo leads Vi.

R-C LOW-PASS FILTER

FIG. 21.14 Angle by which Vo lags Vi.

R-C LOW-PASS FILTER

FIG. 21.15 Low-pass R-L filter.

FIG. 21.16 Example 21.5.

R-C LOW-PASS FILTER

FIG. 21.17 Frequency response for the low-pass R-C network in Fig. 21.16.

Remember,

R-C LOW-PASS FILTER


R-C HIGH-PASS FILTER

FIG. 21.19 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.

As w=0, |gain|=0As winf, |gain|1


FIG. 21.22 Vo versus frequency for a high-pass R-C filter.




FIG. 21.24 Phase-angle response for the high-pass R-C filter.

Same as Boylestad result

R-L HIGH-PASS FILTER

FIG. 21.25 High-pass R-L filter.

What is L/R?

w=0, |gain|=0winf., |gain|1

What kind of filter is this?


FIG. 21.26 Normalized plots for a low-pass and a high-pass filter using the same elements.


FIG. 21.27 Phase plots for a low-pass and a high-pass filter using the same elements.

PASS-BAND FILTERS

FIG. 21.28 Series resonant pass-band filter.

CR gives you high pass, LR gives you low pass.

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.

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).

SERIES RESONANT CIRCUITAt resonance, ZC=-ZL, or jXC=j(-XL)Therefore, reactances cancel! Only resistive components remain

FIG. 21.28 Series resonant pass-band filter.

ω

reactance

fr, resonance

LC

PASS-BAND FILTERS

FIG. 21.29 Parallel resonant pass-band filter.

PASS-BAND FILTERS

FIG. 21.30 Series resonant pass-band filter for Example 21.7.

PASS-BAND FILTERS

FIG. 21.31 Pass-band response for the network.

PASS-BAND FILTERS

FIG. 21.32 Normalized plots for the pass-band filter in Fig. 21.30.

SELECTIVITY

FIG. 20.15 Effect of R, L, and C on the selectivity curve for the series resonant circuit.

Q is the “sharpness” or selectivity of the resonance

We define it as:

Q –Quality factorQ high, narrow BW, sharp, High quality resonance.

Q small, wide BWPoor quality resonance

SELECTIVITY

FIG. 20.16 Approximate series resonance curve for Qs ≥ 10.

PASS-BAND FILTERS

FIG. 21.33 Pass-band filter.

This is a common shorthand notation

T1(s) T2(s)

T(s) is the ‘transfer’ function, or the ‘gain’ functionDescribes how voltage is ‘transferred’ from in to out

Total Gain=T(s)=T1(s)xT2(s)

PASS-BAND FILTERS

FIG. 21.34 Pass-band characteristics.

PASS-BAND FILTERS

FIG. 21.35 Pass-band filter.

FIG. 21.36 Pass-band characteristics for the filter in Fig. 21.35.

PASS-BAND FILTERS

FIG. 21.37 Network of Fig. 21.35 at f = 994.72 kHz.

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.

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.

BAND-REJECT FILTERS

FIG. 21.39 Band-reject filter using a series resonant circuit.

BAND-REJECT FILTERS

FIG. 21.40 Band-reject filter using a parallel resonant network.

BAND-REJECT FILTERS

FIG. 21.41 Band-reject filter.

BAND-REJECT FILTERS

FIG. 21.42 Band-reject characteristics.

OPERATIONAL AMPLIFIERSActive filters

AMPLIFIERS GIVE GAIN

Simple amp-1 input and 1 outputGain, A=Vout/Vin

EXAMPLE

If the amplifier above gives an output voltage of 1000V with an input voltage of 50V, what is the gain?

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

http://www.youtube.com/watch?v=TQB1VlLBgJE

NEGATIVE FEEDBACK

Negative feedback (NF) tries to reduce the differencewith NF, Vplus=Vminus ALWAYS

summing point constraintsvirtual ground.

INVERTING AMPLIFIER

Input goes into Vminus input-INVERTING inputGain, Ainv=-R2/R1, gain is negative because inverting

INVERTING AMPLIFIER

Vplus=VminusInputs draw no current

NON-INVERTING AMPLIFIER

Input goes into Vplus input-NON-INVERTING inputGain, Ainv=1+R2/R1, gain is positive

UNITY GAIN BUFFER

Gain is 1 i.e. Vin=VoutUsed to isolate one side from the other

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

http://www.national.com/mpf/LM/LM324.html%23Overview

BANDPASS FILTER AMPLIFIER

f1=0.3Hz, f2=10HzHigh freq., cap is short, low freq., cap is open

FREQUENCY<F1

all caps are open.What is the gain?

F1<FREQUENCY<F2

C1 is short. C2 is open.What is the gain?-midband gain.

FREQUENCY>F2

All caps are shortsWhat is the gain?

FILTER OP-AMP

What is T(s)?

FILTER OP-AMP

zero at s=0poles at 1/R1C1 and 1/R2C2What happens at the zero? At the poles?

DECIBELS & BODE PLOTSThe key to amplifiers and control systems.

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.

Disambiguate now:

Vo/Vi-refers to |T(s)| i.e. ratio of amplitudesT(s) is the full “gain” equation before taking the absolute value

INTRODUCTIONLOGARITHMSBasic Relationships

Let us first examine the relationship between the variables of the logarithmic function.

The mathematical expression:

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.

INTRODUCTIONLOGARITHMS

FIG. 21.1 Semilog graph paper.


FIG. 21.2 Frequency log scale.


FIG. 21.3 Finding a value on a log plot.


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

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.

DECIBELS Power Gain Voltage Gain Human Auditory Response

DECIBELS

TABLE 21.1

DECIBELS

TABLE 21.2 Typical sound levels and their decibel levels.

DECIBELS

FIG. 21.5 LRAD (Long Range Acoustic Device) 1000X. (Courtesy of the American Technology Corporation.)

DECIBELSINSTRUMENTATION

FIG. 21.6 Defining the relationship between a dB scale referenced to 1 mW, 600Ω and a 3 V rms voltage scale.

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.

t

ttt

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

BODE PLOTS There is a technique for sketching the

frequency response of such factors as filters, amplifiers, and systems on a decibel scale that can save a great deal of time and effort and provide an excellent way to compare decibel levels at different frequencies.

The curves obtained for the magnitude and/or phase angle versus frequency are called Bode plots (Fig. 21.44). Through the use of straight-line segments called idealized Bode plots, the frequency response of a system can be found efficiently and accurately.

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.

LINEAR SYSTEMS

T(s) has zeros when T(s)=0T(s) has poles when T(s)=infinity

POLES & ZEROS Write T(s) in the form A(s)/B(s), where A and B

do not have any fractions in them. They should look like factored polynomials.

All transfer functions have poles and zeros. Zeros are when T(s)=0 i.e. A(s)=0 Poles are when 1/T(s)=0 or T(s)=∞ i.e. B(s)=0 These contribute very distinct behaviors to

the frequency response of a system. They tell us that these are the critical frequencies in the system.

TRANSFER FUNCTION FOR LOW-PASS Again we go to our good friend, the low-pass

filter.


Now, we will redo this in the language of “transfer function”

0.16uF

LOW PASS FILTER TRANSFER FUNCTION


Zeros: NONEPoles: s=-1/RC6.28x103 rad/s or 1kHz(We can ignore the minus signs)Why?

T(s)1 as w0, so gain is 1 (0dB) and phase is 0T(s)-j/wC as winf., so gain0 and phase-90deg

0.16uF

BODE PLOTS There is a technique for sketching the

frequency response of such factors as filters, amplifiers, and systems on a decibel scale that can save a great deal of time and effort and provide an excellent way to compare decibel levels at different frequencies.

The curves obtained for the magnitude and/or phase angle versus frequency are called Bode plots (Fig. 21.44). Through the use of straight-line segments called idealized Bode plots, the frequency response of a system can be found efficiently and accurately.

WHAT DOES A BODE PLOT TELL YOU? Only works for sine/cosine If input is , cos(wt) output becomes

|T(s)|cos(wt+phase(T(s)))

Circuit or system

T(jω)=T(s)

Vi=Cos(wt) Vo=|T(s)|cos(wt+phase(T(s)))

BODE PLOTS Once you have the transfer function, there is a

clear recipe that lets you sketch the frequency response by hand very accurately.

There is a surprisingly easy way to do this. List the poles and zeros from low to high. Go to w=0 and w=infinity to get start/end points. At each pole/zero, apply the 20dB/dec

add/subtract At each pole/zero apply the 90deg phase shift These contributions are additive.

Because of the simplicity, there are some errors, but only small ones near the critical cutoff frequencies.

LOW-PASS RESPONSE-MAGNITUDE |T(S)|

FIG. 21.52 Bode plot for the high-frequency region of a low-pass R-C filter.

LOW PASS RESPONSE ANGLE(T(S))

MORE RULES Phase starts at 0deg UNLESS

-sign, starts at +/-180 deg. Pole/zero at 0 (see next slide).

Gain starts with slope of 0dB/dec UNLESS Pole/zero at 0 (see next slide) (Remember, the starting value of gain could be

any value and must be determined from the transfer function)

SPECIAL CASE POLES AND ZEROS AT W=0

We will now work some examples that have been posted

‘REVERSE’ BODE PLOTS If I give you the Bode plot, and tell you what

the input is, can you sketch the output?

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Frequency Response & Resonant Circuits