Frequency response

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 bandawidth. 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.

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

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:

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

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

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

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

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 jω=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 jω=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.

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,

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

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

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

[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

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

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

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.

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

Taking the natural logarithm of both sides,

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

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

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 (jω)−1 size 12{ ( jω ) rSup { size 8{ - 1} } } {} or zero (j ω size 12{ω} {}) at the origin
3. a simple pole 1/(1+jω/p1) size 12{ {1} slash { ( 1+ {jω} slash {p rSub { size 8{1} } ) } } } {} or zero (1+jω/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 log10∣K∣ 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 (jω)−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 (jω)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+jω/z1) size 12{ ( 1+ {jω} slash {z rSub { size 8{1} } ) } } {}, the magnitude is 20log10/1+jω/z1/ size 12{"20""log" rSub { size 8{"10"} } lline 1+ {jω} slash {z rSub { size 8{1} } } rline } {} and the phase is tan−1ω/z1 size 12{"tan" rSup { size 8{ - 1} } {ω} slash {z rSub { size 8{1} } } } {}. We notice that

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 φ≈0 size 12{φ approx 0} {} for ω≤z1/10 size 12{ω <= {z rSub { size 8{1} } } slash {"10"} } {}, φ≈450 size 12{φ approx "45" rSup { size 8{0} } } {} for ω=z1 size 12{ω=z rSub { size 8{1} } } {}, and φ≈900 size 12{φ approx "90" rSup { size 8{0} } } {} for ω≥10z1 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+jω/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+(jω/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+(jω/−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 −tan−1(2ς2ω/ωn)/1−ω2/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

and

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 + ( jω 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+(jω/−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+(jω/−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.

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

or

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

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

Or

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

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

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

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

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,

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

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