Tài liệu

Frequency response

Science and Technology

INTRODUCTION

In sinusoidal circuit analysis, we have learned how to find voltages and currents in a circuit with a constant frequency source. If we left the amplitude of the sinusoidal source remain constant and vary the frequency, we obtain the circuit’s frequency response. The frequency response may be regarded as a complete description of the sinusoidal steady-state behavior of a circuit as a function of frequency.

The frequency response of a circuit is the variation in its behavior with change in signal frequency.

The sinusoidal steady-state frequency responses of circuits are of significance in many applications, especially in communications and control systems. A specific application is in electric filters that block out or eliminate signals with unwanted frequencies and pass signals of the desired frequencies. Filters are used in radio, TV, and telephone systems to separate one broadcast frequency from another.

We begin this chapter by considering the frequency response of simple circuits using their transfer functions. We then consider Bode plots which are the industry-standard way of presenting frequency response. We also consider series and parallel resonant circuits and encounter important concepts such as resonance, quality factor, cutoff frequency and bandwidth. We discuss different kinds of filters and network scaling. In the last section, we consider one practical application of resonant circuits and two applications of filters.

TRANSFER FUNCTION

The transfer function is a useful analytical tool for finding the frequency response of a circuit. In fact, the frequency response of a circuit is the plot of the circuit’s transfer function H( ω size 12{ω} {}) versus ω size 12{ω} {}, with ω size 12{ω} {} varying from ω=0 size 12{ω=0} {} to ω= size 12{ω= infinity } {}.

A transfer function is the frequency-dependent ratio of the forced function to the forcing function (or of an output to an input). The idea of a transfer function was implicit when we used the concepts of impedance and admittance to relate voltage and current. In general, a linear network can be represented by the block diagram shown in [link].

The transfer function H( ω size 12{ω} {}) of a circuit is the frequency-dependent ratio of a phasor output Y( ω size 12{ω} {}) (an element voltage or current) to a phasor input X( ω size 12{ω} {}) (source voltage or current).

Thus

H ( ω ) = Y ( ω ) X ( ω ) size 12{H \( ω \) = { {Y \( ω \) } over {X \( ω \) } } } {}

assuming zero initial conditions. Since the input and output can be either voltage or current at any place in circuit, there are four possible transfer functions:

H ( ω ) = voltage size 12{H \( ω \) = ital "voltage"} {}

gain = V 0 ( ω ) V i ( ω ) size 12{ ital "gain"= { {V rSub { size 8{0} } \( ω \) } over {V rSub { size 8{i} } \( ω \) } } } {}

H ( ω ) = current size 12{H \( ω \) = ital "current"} {}

gain = I 0 ( ω ) I i ( ω ) size 12{ ital "gain"= { {I rSub { size 8{0} } \( ω \) } over {I rSub { size 8{i} } \( ω \) } } } {}

H ( ω ) = transfer size 12{H \( ω \) = ital "transfer"} {}

impedance = V 0 ( ω ) I i ( ω ) size 12{ ital "impedance"= { {V rSub { size 8{0} } \( ω \) } over {I rSub { size 8{i} } \( ω \) } } } {}

H ( ω ) = transfer size 12{H \( ω \) = ital "transfer"} {}

admit tan ce = I 0 ( ω ) V i ( ω ) size 12{ ital "admit""tan" ital "ce"= { {I rSub { size 8{0} } \( ω \) } over {V rSub { size 8{i} } \( ω \) } } } {}

where subscripts i and o denote input and output values. Being a complex quantity, H( ω size 12{ω} {}) has magnitude H( ω size 12{ω} {}) and a phase φ size 12{φ} {}; that is H(ω)=H(ω)φ size 12{H \( ω \) =H \( ω \) ∠φ} {}.

A block diagram reprensentation of a linear network.

To obtain the transfer function using [link], we first obtain the frequency-domain equivalent of the circuit by replacing resistors, inductors, and capacitors with their impedances R, j ω size 12{ω} {}L, and 1/j ω size 12{ω} {}C. we then use any circuit technique to obtain the appropriate quantity in [link]. We can obtain the frequency response of the circuit by plotting the magnitude and phase of the transfer function as the frequency varies. A computer is a real time-saver for plotting the transfer function.

The transfer function H( ω size 12{ω} {}) can be expressed in terms of its numerator polynomial N( ω size 12{ω} {}) and denominator polynomial D( ω size 12{ω} {}) as

H ( ω ) = N ( ω ) D ( ω ) size 12{H \( ω \) = { {N \( ω \) } over {D \( ω \) } } } {}

Where N( ω size 12{ω} {}) and D( ω size 12{ω} {}) are not necessarily the same expressions for the input and output functions, respectively. The representation of H( ω size 12{ω} {}) in [link] assumes that common numerator and denominator factors in H( ω size 12{ω} {}) have canceled, reducing the ratio to lowest terms. The roots of N( ω size 12{ω} {}) = 0 are called the zeros of H( ω size 12{ω} {}) and are usually represented as =z1,z2,... size 12{jω=z rSub { size 8{1} } ,z rSub { size 8{2} } , "." "." "." } {} Similarly, the roots of D( ω size 12{ω} {}) = 0 are the poles of H( ω size 12{ω} {}) and are represented as =p1,p2,... size 12{jω=p rSub { size 8{1} } ,p rSub { size 8{2} } , "." "." "." } {}

A zero as a root of the numerator polynomial, is a value that results in a zero value of the function. A pole, as a root of the denominator polynomial, is a value for which the function is infinite.

To avoid complex algebra, it is expedient to replace j ω size 12{ω} {} temporarily with s when working with H( ω size 12{ω} {}) and replace s with j ω size 12{ω} {} at the end.

THE DECIBEL SCALE

It is not always easy to get a quick plot of the magnitude and phase of the transfer function as we did above. A more systematic way of obtaining the frequency response is to us Bode plots. Before we begin to construct Bode plots, we should take care of two important issues: the use of logarithms and decibels in expressing gain.

Since Bode plots are based on logarithms, it is important that we keep the following properties of logarithms in mind:

  1. log P 1 P 2 = log P 1 + log P 2 size 12{"log"P rSub { size 8{1} } P rSub { size 8{2} } ="log"P rSub { size 8{1} } +"log"P rSub { size 8{2} } } {}
  2. log P 1 / P 2 = log P 1 log P 2 size 12{"log" {P rSub { size 8{1} } } slash {P rSub { size 8{2} } ="log"P rSub { size 8{1} } - "log"P rSub { size 8{2} } } } {}
  3. log P n = n log P size 12{"log"P rSup { size 8{n} } =n"log"P} {}
  4. log 1 = 0 size 12{"log"1=0} {}

In communications systems, gain is measured in bels. Historically, the bel is used to measure the ratio of two levels of power or power gain G; that is,

G = Number of bels = log 10 P 2 P 1 size 12{G= ital "Number" {} cSup {} ital "of" {} cSup {} ital "bels"="log" rSub { size 8{"10"} } { {P rSub { size 8{2} } } over {P rSub { size 8{1} } } } } {}

The decibel (dB) provides us with a unit of less magnitude. It is 1/10th size 12{ {1} slash {"10" rSup { size 8{ ital "th"} } } } {} of a bel and is given by

G dB = 10 log 10 P 2 P 1 size 12{G rSub { size 8{ ital "dB"} } ="10""log" rSub { size 8{"10"} } { {P rSub { size 8{2} } } over {P rSub { size 8{1} } } } } {}

When P1=P2 size 12{P rSub { size 8{1} } =P rSub { size 8{2} } } {}, there is no change in power and the gain is 0 dB. If P2=2P1 size 12{P rSub { size 8{2} } =2P rSub { size 8{1} } } {}, the gain is

G dB = 10 log 10 2 = 3 dB size 12{G rSub { size 8{ ital "dB"} } ="10""log" rSub { size 8{"10"} } 2=3 ital "dB"} {}

And when P2=0.5P1 size 12{P rSub { size 8{2} } =0 "." 5P rSub { size 8{1} } } {}, the gain is

G dB = 10 log 10 0 . 5 = 3 dB size 12{G rSub { size 8{ ital "dB"} } ="10""log" rSub { size 8{"10"} } 0 "." 5= - 3 ital "dB"} {}

[link] and [link] show another reason why logarithms are greatly used: the logarithm of the reciprocal of a quantity is simply negative the logarithm of that quantity.

Alternatively, the gain G can be expressed in terms of voltage or current ratio. To do so, consider the network shown in [link]. If P1 size 12{P rSub { size 8{1} } } {} is the input power, P2 size 12{P rSub { size 8{2} } } {} is the output (load) power, R1 size 12{R rSub { size 8{1} } } {} is input resistance and R2 size 12{R rSub { size 8{2} } } {} is the load resistance, then P1=0.5V12/R1 size 12{P rSub { size 8{1} } =0 "." 5 {V rSub { size 8{1} } rSup { size 8{2} } } slash {R rSub { size 8{1} } } } {} and P2=0.5V22/R2 size 12{P rSub { size 8{2} } =0 "." 5 {V rSub { size 8{2} } rSup { size 8{2} } } slash {R rSub { size 8{2} } } } {}, and [link] becomes

G dB = 10 log 10 P 2 P 1 = 10 log 10 V 2 2 / R 2 V 1 2 / R 1 = 10 log 10 ( V 2 V 1 ) 2 + 10 log 10 R 1 R 2 size 12{G rSub { size 8{ ital "dB"} } ="10""log" rSub { size 8{"10"} } { {P rSub { size 8{2} } } over {P rSub { size 8{1} } } } ="10""log" rSub { size 8{"10"} } { {V rSub { size 8{2} } rSup { size 8{2} } /R rSub { size 8{2} } } over {V rSub { size 8{1} } rSup { size 8{2} } /R rSub { size 8{1} } } } ="10""log" rSub { size 8{"10"} } \( { {V rSub { size 8{2} } } over {V rSub { size 8{1} } } } \) rSup { size 8{2} } +"10""log" rSub { size 8{"10"} } { {R rSub { size 8{1} } } over {R rSub { size 8{2} } } } } {}
G dB = 20 log 10 V 2 V 1 10 log 10 R 2 R 1 size 12{G rSub { size 8{ ital "dB"} } ="20""log" rSub { size 8{"10"} } { {V rSub { size 8{2} } } over {V rSub { size 8{1} } } } - "10""log" rSub { size 8{"10"} } { {R rSub { size 8{2} } } over {R rSub { size 8{1} } } } } {}

For the case when R2=R1 size 12{R rSub { size 8{2} } =R rSub { size 8{1} } } {}, a condition that is often assumed when comparing voltage levels, [link] becomes

G dB = 20 log 10 V 2 V 1 size 12{G rSub { size 8{ ital "dB"} } ="20""log" rSub { size 8{"10"} } { {V rSub { size 8{2} } } over {V rSub { size 8{1} } } } } {}

Instead, if P1=I12R1 size 12{P rSub { size 8{1} } =I rSub { size 8{1} } rSup { size 8{2} } R rSub { size 8{1} } } {} and P2=I22R2 size 12{P rSub { size 8{2} } =I rSub { size 8{2} } rSup { size 8{2} } R rSub { size 8{2} } } {}, for R1=R2 size 12{R rSub { size 8{1} } =R rSub { size 8{2} } } {}, we obtain

G dB = 20 log 10 I 2 I 1 size 12{G rSub { size 8{ ital "dB"} } ="20""log" rSub { size 8{"10"} } { {I rSub { size 8{2} } } over {I rSub { size 8{1} } } } } {}
Voltage-current relationships for a four-terminal network.

Three things are important to note from [link],[link], and [link]:

  1. That 10 log is used for power, while 20 log is used for voltage or current, because of the square relationship between them ( P=V2/R=I2R size 12{P= {V rSup { size 8{2} } } slash {R} =I rSup { size 8{2} } R} {}).
  2. That the dB value is a logarithmic measurement of the ratio of one variable to another of the same type. Therefore, it applies in expressing the transfer function H in [link] and [link], which are dimensionless quantities, but not in expressing H in [link] and [link].
  3. it is important to note that we only use voltage and current magnitude in [link] and [link]. Negative signs and angles will be handled independently as we will see in section 4.

With this in mind, we now apply the concepts of logarithms and decibels to construct Bode plots.

BODE PLOTS

Obtaining the frequency response from the transfer function as we did in section 2 is an uphill task. The frequency range required in frequency response is often so wide that it is inconvenient to use a linear scale for the frequency axis. Also, there is a more systematic way of locating the important features of the magnitude and phase plots of the transfer function. For these reasons, it has become standard practice to use a logarithmic scale for the frequency axis and a linear scale in each of the separate plots of magnitude and phase. Such semilogarithmic plots of the transfer function-known as Bode plots have become the industry standard.

Bode plots are semilog plots of the magnitude (in decibels) and phase (in degrees) of a transfer function versus frequency.

Bode plots contain the same information as the nonlogarithmic plots discussed in the previous section, but they are much easier to construct, as we shall see shortly.

The transfer function can be written as

H = H φ = He size 12{H=H∠φ= ital "He" rSup { size 8{jφ} } } {}

Taking the natural logarithm of both sides,

ln H = ln H + ln e = ln H + size 12{"ln"H="ln"H+"ln"e rSup { size 8{jφ} } ="ln"H+jφ} {}

Thus, the real part of ln H is a function of the magnitude while the imaginary part is the phase. In a Bode magnitude plot, the gain

H dB = 20 log 10 H size 12{H rSub { size 8{ ital "dB"} } ="20""log" rSub { size 8{"10"} } H} {}

is plotted in decibels (dB) versus frequency. [link] provides a few values of H with the corresponding values in decibels. In a Bode phase plot, φ size 12{φ} {} is plotted in degrees versus frequency. Both magnitude and phase plots are made on semilog graph paper.

Specific gains and their decibel values.
Magnitude H 20 log10H size 12{"log" rSub { size 8{"10"} } H} {} (dB)
0.001 -60
0.01 -40
0.1 -20
0.5 -6
1 / 2 size 12{ {1} slash { sqrt {2} } } {} -3
1 0
2 size 12{ sqrt {2} } {} 3
2 6
10 20
20 26
100 40
1000 60

A transfer function in the form of [link] may be written in terms of factors that have real and imaginary parts. One such representation might be

H ( ω ) = K ( ) ± 1 ( 1 + / z 1 ) [ 1 + j2ζ 1 ω / ω k + ( / ω k ) 2 ] ( 1 + / p 1 ) [ 1 + j2ζ 2 ω / ω n + ( / ω n ) 2 ] size 12{H \( ω \) = { {K \( jω \) rSup { size 8{ +- 1} } \( 1+jω/z rSub { size 8{1} } \) \[ 1+j2ζ rSub { size 8{1} } ω/ω rSub { size 8{k} } + \( jω/ω rSub { size 8{k} } \) rSup { size 8{2} } \] } over { \( 1+jω/p rSub { size 8{1} } \) \[ 1+j2ζ rSub { size 8{2} } ω/ω rSub { size 8{n} } + \( jω/ω rSub { size 8{n} } \) rSup { size 8{2} } \] } } } {}

which is obtained by dividing out the poles and zeros in H( ω size 12{ω} {}). The representation of H( ω size 12{ω} {}) as in [link] is called the standard form. In this particular case, H( ω size 12{ω} {}) has seven different factors that can appear in various combination in a transfer function. These are:

  1. A gain K
  2. A pole ()1 size 12{ \( jω \) rSup { size 8{ - 1} } } {} or zero (j ω size 12{ω} {}) at the origin
  3. a simple pole 1/(1+/p1) size 12{ {1} slash { \( 1+ {jω} slash {p rSub { size 8{1} } \) } } } {} or zero (1+/z1) size 12{ \( 1+ {jω} slash {z rSub { size 8{1} } \) } } {}

In constructing a Bode plot, we plot each factor separately and then combine them graphically. The factors can be considered one at time and then combined additively because of the logarithm that makes Bode plots powerful engineering tool.

We will now make straight-line plots of the factors listed above. We shall find that these straight-line plots known as Bode plots approximate the actual plots to a surprising degree of accuracy.

Constant term: for the gain K, the magnitude is 20 log10 size 12{"log" rSub { size 8{"10"} } } {}K and the phase is 00; both are constant with frequency. Thus, the magnitude and phase plots of the gain are shown in [link]. If K is negative, the magnitude remains 20 log10K size 12{"log" rSub { size 8{"10"} } lline K rline } {} but the phase is ±1800 size 12{ +- "180" rSup { size 8{0} } } {}.

Bode plots for gain K: a)magnitude plot, b) phase plot.

Pole/zero at the origin: for the zero (j ω size 12{ω} {}) at the origin, the amplitude is 20 log10 size 12{"log" rSub { size 8{"10"} } } {}ω size 12{ω} {} and the phase is 900. These are plotted in [link], where we notice that the slope of the magnitude plot is 20 dB/decade, while the phase is constant with frequency.

The Bode plots for the pole ()1 size 12{ \( jω \) rSup { size 8{ - 1} } } {} are similar except that the slope of the magnitude plot is -20 dB/decade while the phase is 900 size 12{ - "90" rSup { size 8{0} } } {}. In general, for ()N size 12{ \( jω \) rSup { size 8{N} } } {} is an integer, the magnitude plot will have a slope of 20N dB/decade, while the phase is 90N degrees.

Bode plot for a zero (jw) at the origin: a) magnitude plot, b) phase plot.

Simple pole/zero: for the simple zero (1+/z1) size 12{ \( 1+ {jω} slash {z rSub { size 8{1} } \) } } {}, the magnitude is 20log10/1+/z1/ size 12{"20""log" rSub { size 8{"10"} } lline 1+ {jω} slash {z rSub { size 8{1} } } rline } {} and the phase is tan1ω/z1 size 12{"tan" rSup { size 8{ - 1} } {ω} slash {z rSub { size 8{1} } } } {}. We notice that

H dB = 20 log 10 / 1 + z 1 / -> 20 log 10 1 = 0 as ω 0 size 12{H rSub { size 8{ ital "dB"} } ="20""log" rSub { size 8{"10"} } lline 1+ { {jω} over {z rSub { size 8{1} } } } rline drarrow "20""log" rSub { size 8{"10"} } 1=0 {} cSup {} ital "as" {} cSup {} ω rightarrow 0} {}
H dB = 20 log 10 / 1 + z 1 / -> 20 log 10 ω z 1 as ω alignl { stack { {} # size 12{H rSub { size 8{ ital "dB"} } ="20""log" rSub { size 8{"10"} } lline 1+ { {jω} over {z rSub { size 8{1} } } } rline drarrow "20""log" rSub { size 8{"10"} } { {ω} over {z rSub { size 8{1} } } } {} cSup {} ital "as" {} cSup {} ω rightarrow infinity } {} } } {}

showing that we can approximate the magnitude as zero (a straight line with zero slope) for small values of ω size 12{ω} {} and by a straight line with slope 20 dB/decade for large values of ω size 12{ω} {}. The frequency ω=z1 size 12{ω=z rSub { size 8{1} } } {} where the two asymptotic lines meet is called the corner frequency or break frequency. Thus, the approximate magnitude plot is shown in [link]a, where the actual plot is also shown. Notice that the approximate plot is close to actual plot except at the break frequency, where ω=z1 size 12{ω=z rSub { size 8{1} } } {} and deviation is 20log10/1+j1/=20log102=3dB size 12{"20""log" rSub { size 8{"10"} } lline 1+j rSub { size 8{1} } rline ="20""log" rSub { size 8{"10"} } sqrt {2} =3 ital "dB"} {}.

As a straight-line approximation, we let φ0 size 12{φ approx 0} {} for ωz1/10 size 12{ω <= {z rSub { size 8{1} } } slash {"10"} } {}, φ450 size 12{φ approx "45" rSup { size 8{0} } } {} for ω=z1 size 12{ω=z rSub { size 8{1} } } {}, and φ900 size 12{φ approx "90" rSup { size 8{0} } } {} for ω10z1 size 12{ω >= "10"z rSub { size 8{1} } } {}. As shown in [link]b along with the actual plot, the straight-line plot has a slope of 450 size 12{"45" rSup { size 8{0} } } {} per decade.

Bode plots of zero (1+jw/z1): a) magnitude plot, b) phase plot.

The Bode plots for the pole 1/(1+/p1) size 12{ {1} slash { \( 1+ {jω} slash {p rSub { size 8{1} } \) } } } {} are similar to those in [link] except that the corner frequency is at ω=p1 size 12{ω=p rSub { size 8{1} } } {}, the magnitude has a slope of – 20 dB/decade, and the phase has a slope 450 size 12{ - "45" rSup { size 8{0} } } {} per decade

Quadric pole/zero: the magnitude of the quadric pole ωn)21+j2ς2ω/ωn+(/1/ size 12{ {1} slash { lbrace 1+j2ς rSub { size 8{2} } {ω} slash {ω rSub { size 8{n} } + \( {jω} slash {ω rSub { size 8{n} } \) rSup { size 8{2} } rbrace } } } } {} is ωn)21+j2ς2ω/ωn+(/20log10 size 12{ - "20""log" rSub { size 8{"10"} } lline lbrace 1+j2ς rSub { size 8{2} } {ω} slash {ω rSub { size 8{n} } + \( {jω} slash {ω rSub { size 8{n} } \) rSup { size 8{2} } rbrace } } rline } {} and the phase is tan1(2ω/ωn)/1ω2/ωn)2) size 12{ - "tan" rSup { size 8{ - 1} } { \( 2ς rSub { size 8{2} } {ω} slash {ω rSub { size 8{n} } \) } } slash {1 - {ω rSup { size 8{2} } } slash {ω rSub { size 8{n} } \) rSup { size 8{2} } \) } } } {}. But

H dB = 20 log 10 / 1 + j2ζ 2 ω ω n + ( ω n ) 2 / -> 0 as ω size 12{H rSub { size 8{ ital "dB"} } = - "20""log" rSub { size 8{"10"} } lline 1+ { {j2ζ rSub { size 8{2} } ω} over {ω rSub { size 8{n} } } } + \( { {jω} over {ω rSub { size 8{n} } } } \) rSup { size 8{2} } rline drarrow 0 {} cSup {} ital "as" {} rSup {} ω rightarrow infinity } {}

and

H dB = 20 log 10 / 1 + j2ζ 2 ω ω n + ( ω n ) 2 / -> 40 log 10 ω ω n as ω size 12{H rSub { size 8{ ital "dB"} } = - "20""log" rSub { size 8{"10"} } lline 1+ { {j2ζ rSub { size 8{2} } ω} over {ω rSub { size 8{n} } } } + \( { {jω} over {ω rSub { size 8{n} } } } \) rSup { size 8{2} } rline drarrow - "40""log" rSub { size 8{"10"} } { {ω} over {ω rSub { size 8{n} } } } {} cSup {} ital "as" {} cSup {} ω rightarrow infinity } {}

Thus, the amplitude plot consists of two straight asymptotic lines: one with zero slope for ω<ωn size 12{ω<ω rSub { size 8{n} } } {} and the other with slope -40dB/decade for ω>ωn size 12{ω>ω rSub { size 8{n} } } {}, with ωn size 12{ω>ω rSub { size 8{n} } } {} as the corner frequency. [link]a shows the approximate and actual amplitude plots. Note that the actual plot depends on the damping factor ς2 size 12{ς rSub { size 8{2} } } {} as well as the corner frequency ωn size 12{ω rSub { size 8{n} } } {}. the significant peaking in the neighborhood of the corner frequency should be added to the straight-line approximation if a high level of accuracy is desired. However, we will use the straight-line approximation for the sake of simplicity.

Bode plots of quadratic pole 1 + j2ζ 2 ω ω n + ( ω n ) 2 size 12{H rSub { size 8{ ital "dB"} } = - "20""log" rSub { size 8{"10"} } lline 1+ { {j2ζ rSub { size 8{2} } ω} over {ω rSub { size 8{n} } } } + \( { {jω} over {ω rSub { size 8{n} } } } \) rSup { size 8{2} } rline drarrow - "40""log" rSub { size 8{"10"} } { {ω} over {ω rSub { size 8{n} } } } {} cSup {} ital "as" {} cSup {} ω rightarrow infinity } {} : a) magnitude plot , b) phase plot.

The phase plot is a straight line with a slope of 900 size 12{"90" rSup { size 8{0} } } {} per decade starting at ωn/10 size 12{ {ω rSub { size 8{n} } } slash {"10"} } {} and ending at 10ωn size 12{"10"ω rSub { size 8{n} } } {}, as shown in [link]b. We see again that the difference between the actual plot and the straight-line plot is due to the damping factor. Notice that the straight-line approximations for both magnitude and phase plots for the quadratic pole are the same as those for a double pole, i.e. (1+jω/ωn)2 size 12{ \( 1+j {ω} slash {ω rSub { size 8{n} } \) rSup { size 8{ - 2} } } } {}. We should expect this because the double pole (1+jω/ωn)2 size 12{ \( 1+j {ω} slash {ω rSub { size 8{n} } \) rSup { size 8{ - 2} } } } {} equals the quadrate pole ωn)21+j2ς2ω/ωn+(/1 size 12{ lbrace 1+j2ς rSub { size 8{2} } {ω} slash {ω rSub { size 8{n} } + \( {jω} slash {ω rSub { size 8{n} } \) rSup { size 8{2} } rbrace } } rSup { size 8{ - 1} } } {} when ς2=1 size 12{ς rSub { size 8{2} } =1} {}. Thus, the quadratic pole can be treated as a double pole as fa as straight-line approximation is concerned.

For the quadratic zero ωn)21+j2ς2ω/ωn+(/2 size 12{ lbrace 1+j2ς rSub { size 8{2} } {ω} slash {ω rSub { size 8{n} } + \( {jω} slash {ω rSub { size 8{n} } \) rSup { size 8{2} } rbrace } } rSup { size 8{ - 2} } } {}, the plots in [link] are inverted because the magnitude plot has a slope of 40 dB/decade while the phase plot has a slope of 900 size 12{"90" rSup { size 8{0} } } {} per decade.

[link] presents a summary of Bode plots for the seven factors. To sketch the Bode plots for a function H( ω size 12{ω} {}) in the form of [link], for example, we first record the corner frequencies on the semilog graph paper, sketch the factors one at a time as discussed above, and then combine additively the graphs of the factors. The combined graph is often drawn from left to right, changing slopes appropriately each time a corner frequency is encountered.

Summary of Bode straight-line magnitude and phase plots.

SERIES RESPONSE

The most prominent feature of frequency response of a circuit may be the sharp peak (or resonant peak) exhibited in its amplitude characteristic. The concept of resonance applies in several areas of science and engineering. Resonance occurs in any system that has a complex conjugate pair of poles; it is the cause of oscillations of stored energy from one form to another. It is the phenomenon that allows frequency discrimination in communication networks. Resonance occurs in any circuit that has at least one inductor and one capacitor.

esonance is a condition in an RLC circuit in which the capacitive and inductive reactances are equal in magnitude, thereby resulting in a purely reactive impedance.

Resonant circuits (series or parallel) are useful for constructing filters, as their transfer functions can be highly frequency selective. They are used in any applications such as selecting the desired stations in radio and TV receivers.

Consider the series RLC circuit shown in [link] in the frequency domain. The input impedance is

Z = H ( ω ) = V s I = R + jωL + 1 jωC size 12{Z=H \( ω \) = { {V rSub { size 8{s} } } over {I} } =R+jωL+ { {1} over {jωC} } } {}

or

Z = R + j ( ωL 1 ωC ) size 12{Z=R+j \( ωL - { {1} over {ωC} } \) } {}

Resonance results when the imaginary part of the transfer function is zero, or

Im ( Z ) = ( ωL 1 ωC ) = 0 size 12{"Im" \( Z \) = \( ωL - { {1} over {ωC} } \) =0} {}

The value of ω size 12{ω} {} that satisfies this condition is called the resonant frequency ω0 size 12{ω rSub { size 8{0} } } {}. Thus, the resonance condition is

ω 0 L = 1 ω 0 C size 12{ω rSub { size 8{0} } L= { {1} over {ω rSub { size 8{0} } C} } } {}

Or

ω 0 = 1 LC rad / s size 12{ω rSub { size 8{0} } = { {1} over { sqrt { ital "LC"} } } ital "rad"/s} {}

Since ω0=2πf0 size 12{ω rSub { size 8{0} } =2πf rSub { size 8{0} } } {}.

f 0 = 1 LC Hz size 12{f rSub { size 8{0} } = { {1} over {2π sqrt { ital "LC"} } } ital "Hz"} {}
The series resonant circuit.

Note that at resonance:

1. The impedance is purely resistive, thus, Z = R. in other words, the LC series combination acts like a short circuit, and the entire voltage is across R.

2. The voltage Vs size 12{V rSub { size 8{s} } } {} and the current I are in phase, so that the power factor is unity.

3. The magnitude of the transfer function H( ω size 12{ω} {}) = Z( ω size 12{ω} {}) is minimum.

4. The inductor voltage and capacitor voltage can be much more than the source voltage.

The frequency response of circuit’s current magnitude

I = / I / = V m R 2 + ( ωL 1 ωC ) 2 size 12{I= lline I rline = { {V rSub { size 8{m} } } over { sqrt {R rSup { size 8{2} } + \( ωL - { {1} over {ωC} } \) rSup { size 8{2} } } } } } {}

Is shown in [link]; the plot only shows the symmetry illustrated in this graph when the frequency axis is a logarithm. The average power dissipated by the RLC circuit is

P ( ω ) = 1 2 I 2 R size 12{P \( ω \) = { {1} over {2} } I rSup { size 8{2} } R} {}

The highest power dissipated occurs at resonance, when I=Vrp/R size 12{I= {V rSub { size 8{ ital "rp"} } } slash {R} } {}, so that

P ( ω 0 ) = 1 2 V m 2 R size 12{P \( ω rSub { size 8{0} } \) = { {1} over {2} } { {V rSub { size 8{m} } rSup { size 8{2} } } over {R} } } {}

At certain frequencies ω=ω1=ω2 size 12{ω=ω rSub { size 8{1} } =ω rSub { size 8{2} } } {}, the dissipated power is the half the maximum value; that is,

P ( ω 1 ) = P ( ω 2 ) = ( V m / 2 ) 2 2R = V m 2 4R size 12{P \( ω rSub { size 8{1} } \) =P \( ω rSub { size 8{2} } \) = { { \( V rSub { size 8{m} } / sqrt {2} \) rSup { size 8{2} } } over {2R} } = { {V rSub { size 8{m} } rSup { size 8{2} } } over {4R} } } {}

Hence, ω1 size 12{ω rSub { size 8{1} } } {} and ω2 size 12{ω rSub { size 8{2} } } {} are called the half-power frequencies.

The current amplitude versus frequency for the series resonant circuit of Figure 7.

The half-power frequencies are obtained by setting Z equal to 2R size 12{ sqrt {2} R} {} and writing

R 2 + ( ωL 1 ωC ) 2 = 2 R size 12{ sqrt {R rSup { size 8{2} } + \( ωL - { {1} over {ωC} } \) rSup { size 8{2} } } = sqrt {2} R} {}

Solving for ω size 12{ω} {}, we obtain

ω 1 = R 2L + ( R 2L ) 2 + 1 LC size 12{ω rSub { size 8{1} } = - { {R} over {2L} } + sqrt { \( { {R} over {2L} } \) rSup { size 8{2} } + { {1} over { ital "LC"} } } } {}

ω 2 = R 2L + ( R 2L ) 2 + 1 LC size 12{ω rSub { size 8{2} } = { {R} over {2L} } + sqrt { \( { {R} over {2L} } \) rSup { size 8{2} } + { {1} over { ital "LC"} } } } {}

We can relate the half-power frequencies with the resonant frequency. From [link] and [link],

ω 0 = ω 1 ω 2 size 12{ω rSub { size 8{0} } = sqrt {ω rSub { size 8{1} } ω rSub { size 8{2} } } } {}

Solving that the resonant frequency is the geometric mean of the half-power frequencies. Note that ω1 size 12{ω rSub { size 8{1} } } {} and ω2 size 12{ω rSub { size 8{2} } } {}are in general not symmetrical around resonant frequency ω0 size 12{ω rSub { size 8{0} } } {}, because the frequency response is not generally symmetrical. However, as will be explained shortly, symmetry of the half-power frequencies around the resonant frequency is often a reasonable approximation.

Although the height of the curve in [link] is determined by R, the width of the curve depends on other factors. The width of response curve depends on the bandwidth B, which is defined as the difference between the two half-power frequencies,

B = ω 2 ω 1 size 12{B=ω rSub { size 8{2} } - {}} {}

This definition of bandwidth is just one of several that are commonly used. Strickly speaking, B in[link] is a half-power bandwidth, because it is the width of the frequency band between the half-power frequencies.

The “sharpness” of the resonance in a resonant circuit is measured quantitatively by the quality factor Q. at resonance, the reactive energy in the circuit oscillates between the inductor and the capacitor. The quality factor relates the maximum or peak energy stored to the energy dissipated in the circuit per cycle of oscillation:

Q = 2 π size 12{π} {}__________Peak energy stored in the circuit_____________

Energy dissipated by the circuit in one period at resonance

It is also regarded a measure of the energy storage property of a circuit in relation to its energy dissipation property. In the series RLC circuit, the peak energy stored is 12LI2 size 12{ { {1} over {2} } ital "LI" rSup { size 8{2} } } {}, while the energy dissipated in one period is 12(I2R)(1/f) size 12{ { {1} over {2} } \( I rSup { size 8{2} } R \) \( {1} slash {f \) } } {}. Hence,

Q = 1 2 LI 2 1 2 I 2 R ( 1 / f ) = fL R size 12{Q=2π { { { {1} over {2} } ital "LI" rSup { size 8{2} } } over { { {1} over {2} } I rSup { size 8{2} } R \( 1/f \) } } = { {2π ital "fL"} over {R} } } {}

Or

Q = ω 0 L R = 1 ω 0 CR size 12{Q= { {ω rSub { size 8{0} } L} over {R} } = { {1} over {ω rSub { size 8{0} } ital "CR"} } } {}

Notice that the quality factor is dimentionless. The relationship between the bandwidth B and the quality factor Q is obtained by substituting [link] into [link] and utilizing [link]

B = R L = ω 0 Q size 12{B= { {R} over {L} } = { {ω rSub { size 8{0} } } over {Q} } } {}

or B=ω02CR size 12{B=ω rSub { size 8{0} } rSup { size 8{2} } ital "CR"} {}. Thus

The quality factor of a resonant circuit is the ratio of its resonant frequency to its bandwidth.

Keep in mind that [link], [link], [link], and [link] only apply to a series RLC circuit.

As illustrated in [link], the higher the vale of Q, the more selective the circuit is but the smaller the bandwidth. The selectivity of an RLC circuit is the ability of the circuit to respond to a certain frequency and discriminate against all other frequencies. If the band of frequencies to be selected or rejected is narrow, the quality factor of the resonant circuit must be high. If the band of frequencies is wide, the quality factor must be low.

A resonant circuit is designed to operate at or near its resonant frequency. It is said to be a high-Q circuit when its quality factor is equal to or greater than 10. For high -Q circuits ( Q10 size 12{Q >= "10"} {}), the half-power frequencies are, for all practical purposes, symmetrical around the resonant frequency and can be approximated as

ω 1 ω 0 B 2 , ω 2 ω 0 + B 2 size 12{ω rSub { size 8{1} } approx ω rSub { size 8{0} } - { {B} over {2} } ,ω rSub { size 8{2} } approx ω rSub { size 8{0} } + { {B} over {2} } } {}
The higher circuit Q, the smaller the bandwidth.

High-Q circuits are used often in Communications network.

We see that a resonant circuit is characterized by five related parameters: the two half-power frequencies ω1 size 12{ω rSub { size 8{1} } } {} and ω2 size 12{ω rSub { size 8{2} } } {}, the resonant frequency ω0 size 12{ω rSub { size 8{0} } } {}, the bandwidth B, and the quality factor Q.

PARALLEL RESONANCE

The parallel RLC circuit in [link] is dual of series RLC circuit. So we will avoid needless repetition. The admittance is

Y = H ( ω ) = I V = 1 R + jωC + 1 jωL size 12{Y=H \( ω \) = { {I} over {V} } = { {1} over {R} } +jωC+ { {1} over {jωL} } } {}

Or

Y = 1 R + j ( ωC 1 ωL ) size 12{Y= { {1} over {R} } +j \( ωC - { {1} over {ωL} } \) } {}
The prallel resonant circuit.

Resonance occurs when the imaginary part of Y is zero,

ωC 1 ωL = 0 size 12{ωC - { {1} over {ωL} } =0} {}

or

ω 0 = 1 LC rad / s size 12{ω rSub { size 8{0} } = { {1} over { sqrt { ital "LC"} } } ital "rad"/s} {}

which is the same as [link] for the series resonant circuit. The voltage |V| is sketched in [link] as a function of frequency. Notice that at resonance, the parallel LC combination acts like an open circuit, so that the entire current flows through R. also, the inductor and capacitor current can be much more than the source current at resonance.

We exploit the duality between [link] and [link] by comparing [link] with [link]. by replacing R, L, and C in the expressions for the series circuit with 1/R, 1/C, and 1/L respectively, we obtain for the parallel circuit

The current amplitude versus frequency for the parallel resonant circuit of Figure 10.
ω 1 = 1 2 RC + ( 1 2 RC ) 2 + 1 LC size 12{ω rSub { size 8{1} } = - { {1} over {2 ital "RC"} } + sqrt { \( { {1} over {2 ital "RC"} } \) rSup { size 8{2} } + { {1} over { ital "LC"} } } } {}

ω 2 = 1 2 RC + ( 1 2 RC ) 2 + 1 LC size 12{ω rSub { size 8{2} } = { {1} over {2 ital "RC"} } + sqrt { \( { {1} over {2 ital "RC"} } \) rSup { size 8{2} } + { {1} over { ital "LC"} } } } {}

B = ω 2 ω 1 = 1 RC size 12{B=ω rSub { size 8{2} } - ω rSub { size 8{1} } = { {1} over { ital "RC"} } } {}
Q = ω 0 B = ω 0 RC = R ω 0 L size 12{Q= { {ω rSub { size 8{0} } } over {B} } =ω rSub { size 8{0} } ital "RC"= { {R} over {ω rSub { size 8{0} } L} } } {}

Using [link] and [link], we can express the half-power frequencies in terms of the quality factor. The result is

ω 1 = ω 0 1 + ( 1 2Q ) 2 ω 0 2Q size 12{ω rSub { size 8{1} } =ω rSub { size 8{0} } sqrt {1+ \( { {1} over {2Q} } \) rSup { size 8{2} } } - { {ω rSub { size 8{0} } } over {2Q} } } {}

ω 2 = ω 0 1 + ( 1 2Q ) 2 + ω 0 2Q size 12{ω rSub { size 8{2} } =ω rSub { size 8{0} } sqrt {1+ \( { {1} over {2Q} } \) rSup { size 8{2} } } + { {ω rSub { size 8{0} } } over {2Q} } } {}

Again, for high -Q circuits ( Q10 size 12{Q >= "10"} {})

ω 1 ω 0 B 2 size 12{ω rSub { size 8{1} } approx ω rSub { size 8{0} } - { {B} over {2} } } {}

[link] presents a summary of the characteristics of the series and parallel resonant circuits.

Summary of the characteristics of resonant RLC circuits.
Characteristic Series circuit Parallel circuit
Resonant frequency, ω0 size 12{ω rSub { size 8{0} } } {} 1 LC size 12{ { {1} over { sqrt { ital "LC"} } } } {} 1 LC size 12{ { {1} over { sqrt { ital "LC"} } } } {}
Quality factor, Q ω 0 L R or 1 ω 0 RC size 12{ { {ω rSub { size 8{0} } L} over {R} } ital "or" { {1} over {ω rSub { size 8{0} } ital "RC"} } } {} R ω 0 L or ω 0 RC size 12{ { {R} over {ω rSub { size 8{0} } L} } ital "or"ω rSub { size 8{0} } ital "RC"} {}
Bandwidth, B ω 0 Q size 12{ { {ω rSub { size 8{0} } } over {Q} } } {} ω 0 Q size 12{ { {ω rSub { size 8{0} } } over {Q} } } {}
Half-power frequencies ω 0 1 + ( 1 2Q ) 2 ± ω 0 2Q size 12{ω rSub { size 8{0} } sqrt {1+ \( { {1} over {2Q} } \) rSup { size 8{2} } } +- { {ω rSub { size 8{0} } } over {2Q} } } {} ω 0 1 + ( 1 2Q ) 2 ± ω 0 2Q size 12{ω rSub { size 8{0} } sqrt {1+ \( { {1} over {2Q} } \) rSup { size 8{2} } } +- { {ω rSub { size 8{0} } } over {2Q} } } {}
For Q10 size 12{Q >= "10"} {}, ω1 size 12{ω rSub { size 8{1} } } {}, ω2 size 12{ω rSub { size 8{2} } } {} ω 0 ± B 2 size 12{ω rSub { size 8{0} } +- { {B} over {2} } } {} ω 0 ± B 2 size 12{ω rSub { size 8{0} } +- { {B} over {2} } } {}

PASSIVE FILTERS

The concept of filters has been an integral part of electrical engineering from the beginning. Several technological achievements would not have been possible without electrical filters. Because of this prominent role of filters, much effort has been expended on the theory, design, and construction of filters and many articles and books have been written on them. Our discussion in this chapter should be considered introductory.

Filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate others.

As a frequency-selective device, a filter can be used to limit the frequency spectrum of a signal to some specified band of frequencies. Filters are the circuits used in radio and TV receivers to allow us to select one desired signal out of a multitude of broadcast signals in the environment.

A filter is a passive filter if it consists of only passive elements R, L, and C. it is said to be an active filter if it consists of active elements (such as transistors and op amps) in addition to passive elements R, L, and C. we consider passive filters in this section and active filters in the next section. LC filters have been used in practical applications for more than eight decades. LC filter technology feeds related areas such as equalizers, impedance-matching networks, transformers, shaping networks, power dividers, attenuators, and directional couplers, and is continuously providing practicing engineers with opportunities to innovate and experiment. Besides the LC filters we study in these sections, there are other kinds of filters- such as digital filters, electromechanical filters, and microwave filters-which are beyond the level of the text.

As shown in [link]. There are four types of filters whether passive or active:

  1. A lowpass filter passes low frequencies and stop high frequencies, as shown ideally in [link]a.
  2. A high pass filter passes high frequencies and rejects low frequencies, as shown ideally in [link]b.
  3. A bandpass filter passes frequencies within a frequency band and blocks or attenuates frequencies outside the band, as shown ideally in [link]c.
  4. A bandstop filter passes frequencies outside a frequency band and blocks or attenuates frequencies within the band, as shown ideally in [link]d.
Ideal frequency of four types of filter: a) lowpass filter, b) highpass filter, c) bandpass filter, d) bandstop filter.

[link] presents a summary of the characteristics of these filters. Be aware that the characteristics in [link] are only valid for first or second order filters – but one should not have the impression that only these kinds of filter exist. We now consider typical circuits for realizing the filters shown in [link].

Summary of the characteristics of filters.
Type of filter H(0) H( ) size 12{ infinity \) } {} H( ωc size 12{ω rSub { size 8{c} } } {}) or H( ω0 size 12{ω rSub { size 8{0} } } {})
Lowpass 1 0 1 / 2 size 12{1/ sqrt {2} } {}
Highpass 0 1 1 / 2 size 12{1/ sqrt {2} } {}
Bandpass 0 0 1
Bandstop 1 1 0

Lowpass Filter

A typical lowpass filter is formed when the output of an RC circuit is taken off the capacitor as shown in [link]. The transfer function is

H ( ω ) = V 0 V i = 1 / jωC R + 1 / jωC = 1 1 + RC size 12{H \( ω \) = { {V rSub { size 8{0} } } over {V rSub { size 8{i} } } } = { {1/jωC} over {R+1/jωC} } = { {1} over {1+jω ital "RC"} } } {}
A lowpass filter.

Note that H(0) = 1, H( size 12{ infinity } {}) = 0. [link] shows the plot of H(ω) size 12{ lline H \( ω \) rline } {}, along with the ideal characteristic. The half-power frequency, which is equivalent to the corner frequency on the Bode plots but in the context of filters is usually known as the cutoff frequency ωc size 12{ω rSub { size 8{c} } } {}, is obtained by setting the magnitude of H( ω size 12{ω} {}) equal to 1/ 2 size 12{ sqrt {2} } {}, thus

H ( ω ) = 1 1 + ω c 2 R 2 C 2 = 1 2 size 12{H \( ω \) = { {1} over { sqrt {1+ω rSub { size 8{c} } rSup { size 8{2} } R rSup { size 8{2} } C rSup { size 8{2} } } } } = { {1} over { sqrt {2} } } } {}

or

ω c = 1 RC size 12{ω rSub { size 8{c} } = { {1} over { ital "RC"} } } {}
Ideal and actual frequency response of a lowass filter.

The cutoff frequency is also called the cutoff frequency.

A lowpass filter is designed to pass only frequencies from dc up to the cutoff frequency ωc size 12{ω rSub { size 8{c} } } {}

A low pass filter can also be formed when the output of an RL circuit is taken off the resistor. Of course, there are many other circuits for lowpass filters.

Highpass Filter

A highpass filter is formed when the output of an RC circuit is taken off the resistor as shown in [link]. The transfer function is

H ( ω ) = V 0 V i = R R + 1 / jωC = RC 1 + RC size 12{H \( ω \) = { {V rSub { size 8{0} } } over {V rSub { size 8{i} } } } = { {R} over {R+1/jωC} } = { {jω ital "RC"} over {1+jω ital "RC"} } } {}
A highpass filter.

Note that H(0) = 0, H( size 12{ infinity } {}) = 1. [link] shows the plot of H(ω) size 12{ lline H \( ω \) rline } {}. Again, the corner or cutoff frequency is

ω c = 1 RC size 12{ω rSub { size 8{c} } = { {1} over { ital "RC"} } } {}
Idela and actual frequency response of a highpass filter.

A highpass filter is designed to pass all frequencies above its cutoff frequency ωc size 12{ω rSub { size 8{c} } } {}.

A highpass filter can also be formed when the output of an RL circuit is taken off the inductor.

Bandpass Filter

The RLC series resonant circuit provides a bandpass filter when the output is taken off the resistor as shown in [link]. The transfer function is

H ( ω ) = V 0 V i = R R + j ( ωL 1 / ωC ) size 12{H \( ω \) = { {V rSub { size 8{0} } } over {V rSub { size 8{i} } } } = { {R} over {R+j \( ωL - 1/ωC \) } } } {}

We observe that H(0) = 0, H( size 12{ infinity } {}) = 0. [link] shows the plot of H(ω) size 12{ lline H \( ω \) rline } {}. The bandpass filter passes a band of frequency ( ω1<ω<ω2 size 12{ω rSub { size 8{1} } <ω<ω rSub { size 8{2} } } {}) centered on ω0 size 12{ω rSub { size 8{0} } } {}, the center frequency, which is given by

ω 0 = 1 LC size 12{ω rSub { size 8{0} } = { {1} over { sqrt { ital "LC"} } } } {}
A bandpass filter.

A bandpass filter is designed to pass all frequencies within a band of frequencies, ω1<ω<ω2 size 12{ω rSub { size 8{1} } <ω<ω rSub { size 8{2} } } {}.

Since bandpass filter in [link] is a series resonant circuit, the half-power frequencies, the bandwidth, and the quality factor are determined as in section 5. A bandpass filter can also be formed by cascading the lowpass filter (where ω2=ωc size 12{ω rSub { size 8{2} } =ω rSub { size 8{c} } } {}) in [link] with the highpass filter (where ω1=ωc size 12{ω rSub { size 8{1} } =ω rSub { size 8{c} } } {})

Idela and actual frequency response of a bandpass filter.

Bandstop Filter

A filter that prevents a band of frequencies between two designated values ( ω1 size 12{ω rSub { size 8{1} } } {} and ω2 size 12{ω rSub { size 8{2} } } {}) from passing is variably known as a bandstop, bandreject, or notch filter. A bandstop filter is formed when the output RLC series resonant circuit is taken off the LC series combination as shown in [link]. The transfer function is

H ( ω ) = V 0 V i = j ( ωL 1 / ωC ) R + j ( ωL 1 / ωC ) size 12{H \( ω \) = { {V rSub { size 8{0} } } over {V rSub { size 8{i} } } } = { {j \( ωL - 1/ωC \) } over {R+j \( ωL - 1/ωC \) } } } {}
A bandstop filter.

Notice that H(0) = 1, H( size 12{ infinity } {}) = 1. [link] shows the plot of H(ω) size 12{ lline H \( ω \) rline } {}. Again, the center frequency is given by

ω 0 = 1 LC size 12{ω rSub { size 8{0} } = { {1} over { sqrt { ital "LC"} } } } {}

While the half-power frequencies, the bandwidth, and quality factor are calculated using the formulas in section 5 for a series resonant circuit. Here, ω0 size 12{B=ω rSub { size 8{2} } - ω rSub { size 8{1} } } {} is called the frequency of rejection, while the corresponding bandwidth ( B=ω2ω1 size 12{B=ω rSub { size 8{2} } - ω rSub { size 8{1} } } {}) is known as the bandwidth of rejection. Thus,

A bandstop filter is designed to stop or eliminate all frequencies within a band of frequencies, ω1<ω<ω2 size 12{ω rSub { size 8{1} } <ω<ω rSub { size 8{2} } } {}.

Idela and actual frequency response of a bandstop filter.

Notice that adding the transfer function of the bandpass anf the bandstop gives unity at any frequency for the same values of R, L, and C. Of course, this is not true in general but true for the circuits treated here. This is due to the fact that the characteristic of one is the inverse of the other.

In concluding this section, we should note that:

  1. From [link], [link], [link], the maximum gain of a passive filter is unity. To generate a gain greater than unity, one should use an active filter as the next section shows.
  2. There are other ways to get the types of filters treated in this section.
  3. The filter treated here are the simple types. Many other filters have sharper and complex frequency response.

ACTIVE FILTERS

There are three major limitations to the passive filter in the previous section. First, they cannot generate gain greater than 1; passive elements cannot add energy to the network. Second, they may require bulky and expensive inductor. Third, they perform poorly at frequencies below the audio frequency range (300 Hz < f < 3000 Hz). Nevertheless, passive filters are useful at high frequencies.

Active filters consist of combination of resistors, capacitors, and op amps. They offer some advantage over passive RLC filters. First, they are often smaller and less expensive, because they do not require inductors. This makes feasible the integrated circuit realizations of filters. Second, they can provide amplifier gain in addition to providing the same frequency response as RLC filters. Third, active filter can be combined with buffer amplifiers (voltage followers) to isolate each stage of the filter from source and load impedance effects. This isolation allows designing the stages independently and then cascading them to realize the desired transfer function. (Bode plots, being logarithmic, may be added when transfer function are cascaded.) However, active filters are less reliable and less stable. The practical limit of most active filters is about 100 kHz – most active filters operate well below that frequency.

Filters are often classified according to their order (or number of poles) or their specific design type.

First-Order Lowpass Filter

One type of first-order filter is shown in [link]. The components selected for Zi size 12{Z rSub { size 8{i} } } {} and Zf size 12{Z rSub { size 8{f} } } {} determine whether the filter is lowpass or highpass, but one the components must be reactive.

A general first-order active filter.

[link] shows a typical active lowpass filter. For this filter the transfer function is

H ( ω ) = V 0 V i = Z f Z i size 12{H \( ω \) = { {V rSub { size 8{0} } } over {V rSub { size 8{i} } } } = - { {Z rSub { size 8{f} } } over {Z rSub { size 8{i} } } } } {}

Where Zi=Zf size 12{Z rSub { size 8{i} } =Z rSub { size 8{f} } } {} and

Z f = R f 1 jωC f = R f / jωC f R f + 1 / jωC f = R f 1 + jωC f R fi size 12{Z rSub { size 8{f} } =R rSub { size 8{f} } lline rline { {1} over {jωC rSub { size 8{f} } } } = { {R rSub { size 8{f} } /jωC rSub { size 8{f} } } over {R rSub { size 8{f} } +1/jωC rSub { size 8{f} } } } = { {R rSub { size 8{f} } } over {1+jωC rSub { size 8{f} } R rSub { size 8{ ital "fi"} } } } } {}

Therefore,

H ( ω ) = R f R i 1 1 + jωC f R f size 12{H \( ω \) = - { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } { {1} over {1+jωC rSub { size 8{f} } R rSub { size 8{f} } } } } {}

We notice that [link] is similar to [link], except that there is a low frequency ( ω0 size 12{ω rightarrow 0} {}) gain or dc gain of Rf/Ri size 12{ - R rSub { size 8{f} } /R rSub { size 8{i} } } {}. Also, the corner frequency is

ω c = 1 R f C f size 12{ω rSub { size 8{c} } = { {1} over {R"" lSub { size 8{f} } C rSub { size 8{f} } } } } {}

Which does not depend on Ri size 12{R rSub { size 8{i} } } {}. This means that several inputs with different Ri size 12{R rSub { size 8{i} } } {}could be summer if required, and the corner frequency would remain the same for each input.

Active first-order lowpass filter.

First-Order Highpass Filter

[link] shows a typical highpass filter. As before,

H ( ω ) = V 0 V i = Z f Z i size 12{H \( ω \) = { {V rSub { size 8{0} } } over {V rSub { size 8{i} } } } = - { {Z rSub { size 8{f} } } over {Z rSub { size 8{i} } } } } {}

Where Zi=Ri+1/jωCi size 12{Z rSub { size 8{i} } =R rSub { size 8{i} } + {1} slash {jωC rSub { size 8{i} } } } {} and Zf=Rf size 12{Z rSub { size 8{f} } =R rSub { size 8{f} } } {} so that

H ( ω ) = R f R i + 1 / jωC i = jωC i R f 1 + jωC i R i size 12{H \( ω \) = { {R rSub { size 8{f} } } over {R rSub { size 8{i} } +1/jωC rSub { size 8{i} } } } = - { {jωC rSub { size 8{i} } R rSub { size 8{f} } } over {1+jωC rSub { size 8{i} } R rSub { size 8{i} } } } } {}

This is similar to [link], except that at very high frequencies ( ω size 12{ω rightarrow infinity } {}), the gain tends to Rf/Ri size 12{ { - R rSub { size 8{f} } } slash {R rSub { size 8{i} } } } {}. The corner frequency is

ω c = 1 R i C i size 12{ω rSub { size 8{c} } = { {1} over {R rSub { size 8{i} } C rSub { size 8{i} } } } } {}
Active first-order highpass filter.

Bandpass Filter

The circuit in [link] may be combined with that in [link] to form a bandpass filter that will have a gain K over the required range of frequencies. By cascading a unit gain lowpass filter, a unit gain highpass filter, and an inverter with gain Rf/Ri size 12{ { - R rSub { size 8{f} } } slash {R rSub { size 8{i} } } } {} as shown in the block diagram of [link]a, we can construct a bandpass filter whose frequency response is that in [link]b. The actual construction of the bandpass filter is shown in [link].

The analysis of the bandpass filter is relatively simple. Its transfer function is obtained by multiplying [link] and [link] with the gain of the inverter. That is

H ( ω ) = V 0 V i = ( 1 1 + jωC 1 R ) ( jωC 2 R 1 + jωC 2 R ) ( R f R 1 ) = R f R 1 1 1 + jωC 1 R jωC 2 R 1 + jωC 2 R size 12{H \( ω \) = { {V rSub { size 8{0} } } over {V rSub { size 8{i} } } } = \( - { {1} over {1+jωC rSub { size 8{1} } R} } \) \( - { {jωC rSub { size 8{2} } R} over {1+jωC rSub { size 8{2} } R} } \) \( - { {R rSub { size 8{f} } } over {R rSub { size 8{1} } } } \) = - { {R rSub { size 8{f} } } over {R rSub { size 8{1} } } } { {1} over {1+jωC"" lSub { size 8{1} } R} } { {jωC rSub { size 8{2} } R} over {1+jωC rSub { size 8{2} } R} } } {}
Active bandpass filter: a) block diagram, b) frequency response.
Active bandpass filter.

The lowpass section sets upper corner frequency as

ω 2 = 1 RC 1 size 12{ω rSub { size 8{2} } = { {1} over { ital "RC""" lSub { size 8{1} } } } } {}

While the highpass sets the lower corner frequency as

ω 1 = 1 RC 2 size 12{ω rSub { size 8{1} } = { {1} over { ital "RC" rSub { size 8{2} } } } } {}

With these values of ω1 size 12{ω rSub { size 8{1} } } {} and ω2 size 12{ω rSub { size 8{2} } } {}, the center frequency, bandwidth, and quality factor are found as follows:

ω 0 = ω 1 ω 2 size 12{ω rSub { size 8{0} } = sqrt {ω rSub { size 8{1} } ω rSub { size 8{2} } } } {}
B = ω 2 ω 1 size 12{B=ω rSub { size 8{2} } - ω rSub { size 8{1} } } {}
Q = ω 0 B size 12{Q= { {ω rSub { size 8{0} } } over {B} } } {}

To find the passband gain K, we write [link] in the standard form of [link],

H ( ω ) = R f R i / ω 1 ( 1 + / ω 1 ) ( 1 + / ω 2 ) = R f R i j ωω 2 ( ω 1 + ) ( ω 2 + ) size 12{H \( ω \) = { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } { {jω/ω rSub { size 8{1} } } over { \( 1+jω/ω rSub { size 8{1} } \) \( 1+jω/ω rSub { size 8{2} } \) } } = { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } { {j ital "ωω" rSub { size 8{2} } } over { \( ω rSub { size 8{1} } +jω \) \( ω rSub { size 8{2} } +jω \) } } } {}

At the center frequency ω0=ω1ω2 size 12{ω rSub { size 8{0} } = sqrt {ω rSub { size 8{1} } ω rSub { size 8{2} } } } {}, the magnitude of the transfer function is

H ( ω ) = R f R i / ω 1 ( ω 1 + 0 ) ω 2 + 0 = R f R i ω 2 ( ω 1 + ω 2 ) size 12{ lline H \( ω \) rline = \lline { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } { {jω/ω rSub { size 8{1} } } over { \( ω"" lSub { size 8{1} } +jω rSub { size 8{0} } \) left (ω rSub { size 8{2} } +jω rSub { size 8{0} } right )} } \rline = { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } { {ω rSub { size 8{2} } } over { \( ω rSub { size 8{1} } +ω rSub { size 8{2} } \) } } } {}

Thus the passband gain is

K = R f R i ω 2 ( ω 1 + ω 2 ) size 12{K= { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } { {ω rSub { size 8{2} } } over { \( ω rSub { size 8{1} } +ω rSub { size 8{2} } \) } } } {}

Bandreject (or Notch) Filter

A bandreject filter may be constructed by parallel combination of a low pass filter and a highpass filter and summing amplifier, as shown in the block diagram of [link]a. The circuit is designed such that the lower cutoff frequency ω1 size 12{ω rSub { size 8{1} } } {} is set by the lowpass filter while the upper cutoff frequency ω2 size 12{ω rSub { size 8{2} } } {} is set by the highpass filter. The gap between ω1 size 12{ω rSub { size 8{1} } } {} and ω2 size 12{ω rSub { size 8{2} } } {} is the bandwidth of the filter. As shown in [link]b, the filter passes frequencies below ω1 size 12{ω rSub { size 8{1} } } {} and above ω2 size 12{ω rSub { size 8{2} } } {}. The block diagram in [link]a is actually constructed as shown in [link]. The transfer function is

H ( ω ) = V 0 V i = R f R i ( 1 1 + jωC 1 R jωC 2 R 1 + jωC 2 R ) size 12{H \( ω \) = { {V rSub { size 8{0} } } over {V rSub { size 8{i} } } } = { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } \( - { {1} over {1+jωC rSub { size 8{1} } R} } - { {jωC"" lSub { size 8{2} } R} over {1+jωC"" lSub { size 8{2} } R} } \) } {}

The formulas for calculating the values of ω1 size 12{ω rSub { size 8{1} } } {}, ω2 size 12{ω rSub { size 8{2} } } {}, the center frequency, bandwidth, and quality factor are the same as in [link] to [link].

Active bandreject filter.

To determine the passband gain K of the filter, we can write [link] in terms of the upper and lower corner frequencies as

H ( ω ) = R f R i ( 1 1 + / ω 2 + / ω 1 1 + / ω 1 ) = R f R i ( 1 + j2ω / ω 1 + ( ) 2 / ω 1 ω 2 ) ( 1 + / ω 2 ) ( 1 + / ω 1 ) size 12{H \( ω \) = { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } \( { {1} over {1+jω/ω rSub { size 8{2} } } } + { {jω/ω"" lSub { size 8{1} } } over {1+jω/ω rSub { size 8{1} } } } \) = { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } { { \( 1+j2ω/ω rSub { size 8{1} } + \( jω \) rSup { size 8{2} } /ω rSub { size 8{1} } ω rSub { size 8{2} } \) } over { \( 1+jω/ω rSub { size 8{2} } \) \( 1+jω/ω rSub { size 8{1} } \) } } } {}

Comparing this with the standard form in [link] indicates that in the two passbands ( ω0 size 12{ω rightarrow 0} {}) and ( ω size 12{ω rightarrow infinity } {}) the gain is

K = R f R i size 12{K= { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } } {}

We can also find the gain at the center frequency by finding the magnitude of the transfer function at ω0=ω1ω2 size 12{ω rSub { size 8{0} } = sqrt {ω rSub { size 8{1} } ω rSub { size 8{2} } } } {}, writing

H ( ω 0 ) = / R f R i ( 1 + j2ω / ω 1 + ( ) 2 / ω 1 ω 2 ) ( 1 + / ω 2 ) ( 1 + / ω 1 ) / = R f R i 1 ω 1 + ω 2 size 12{H \( ω rSub { size 8{0} } \) = \lline { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } { { \( 1+j2ω/ω rSub { size 8{1} } + \( jω \) rSup { size 8{2} } /ω rSub { size 8{1} } ω rSub { size 8{2} } \) } over { \( 1+jω/ω rSub { size 8{2} } \) \( 1+jω/ω rSub { size 8{1} } \) } } \rline = { {R rSub { size 8{f} } } over {R rSub { size 8{i} } } } { {2ω rSub { size 8{1} } } over {ω rSub { size 8{1} } +ω rSub { size 8{2} } } } } {}

Again, the filters treated in this section are only typical. There are many other active filters that are more complex.

SUMMARY

  1. The transfer function H( ω size 12{ω} {}) is the ratio of the output response Y( ω size 12{ω} {}) to the input excitation X( ω size 12{ω} {}); that is , H( ω size 12{ω} {}) = Y( ω size 12{ω} {})/X( ω size 12{ω} {}).
  2. The frequency response is the variation of the transfer function with frequency.
  3. Zeros of a transfer function H(s) are the values of s = j ω size 12{ω} {} that make H(s) = 0, while poses are the values of s that make H(s) size 12{H \( s \) rightarrow infinity } {}.
  4. The decibel is the unit of logarithmic gain. For a gain G, its decibel equivalent is GdB=20log10G size 12{G rSub { size 8{ ital "dB"} } ="20""log" rSub { size 8{"10"} } G} {}.
  5. Bode plots are semilog plots of the magnitude and phase of the transfer function as it varies with frequency. The straight line approximations of H (in dB) and φ size 12{φ} {} (in degrees) are constructed using the corner frequencies defined by the poles and zeros of H( ω size 12{ω} {}).
  6. The resonant frequency is that frequency at which the imaginary part of a transfer function vanishes. For series and parallel RLC circuits,

ω 0 = 1 LC size 12{ω rSub { size 8{0} } = { {1} over { sqrt { ital "LC"} } } } {}

  1. The half-power frequencies ( ω1 size 12{ω rSub { size 8{1} } } {}, ω2 size 12{ω rSub { size 8{2} } } {}) are those frequencies at which the power dissipated is one-half of that dissipated at the resonant frequency. The geometric mean between the half power frequencies is the resonant frequency, or

ω 0 = ω 1 ω 2 size 12{ω rSub { size 8{0} } = sqrt {ω rSub { size 8{1} } ω rSub { size 8{2} } } } {}

  1. the bandwidth is the frequency band between half-power frequencies:

B = ω 2 ω 1 size 12{B=ω rSub { size 8{2} } - ω rSub { size 8{1} } } {}

  1. The quality factor is a measure of the sharpness of the resonance peak. It is the ratio of the resonant (angular) frequency to the band width,

Q = ω 0 B size 12{Q= { {ω rSub { size 8{0} } } over {B} } } {}

  1. A filter is a circuit designed to pass a band of frequencies and reject others. Passive filters are constructed with resistors, capacitors, and inductors. Active filters are constructed with resistors, capacitors, and active devices, usually an op amp.
  2. Four common types of filters are lowpass, highpass, bandpass, and bandstop. A lowpass filter passes only signals whose frequencies are below the cutoff frequencies ωc size 12{ω rSub { size 8{c} } } {}. A highpass filter passes only signals whose frequencies are above the cutoff frequency ωc size 12{ω rSub { size 8{c} } } {}. A bandpass filter passes only signals whose frequecies are within a prescribed range ( ω1<ω<ω2 size 12{ω rSub { size 8{1} } <ω<ω rSub { size 8{2} } } {}). A bandstop filter passes only signals whose frequencies are outside a prescribed range ( ω1>ω>ω2 size 12{ω rSub { size 8{1} } >ω>ω rSub { size 8{2} } } {}).
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