Fundamentals of Microelectronics

Analog Filters

Our treatment of microelectronics thus far has mostly concentrated on the problem of amplification. Another important function vastly used in electronic systems is “filtering.” For example, a cellphone incorporates filters to suppress “interferers” that are received in addition to the desired signal. Similarly, a high-fidelity audio system must employ filters to eliminate the 60-Hz (50-Hz) ac line interference. This chapter provides an introduction to analog filters. The outline is shown below.

14.1 General Considerations

In order to define the performance parameters of filters, we first take a brief look at some applications. Suppose a cellphone receives a desired signal, X(f), with a bandwidth of 200 kHz at a center frequency of 900 MHz [Fig. 14.1(a)]. As mentioned in Chapter 1, the receiver may

Figure 14.1 (a) Desired channel in a receiver, (b) large interferer, (c) use of filter to suppress the interferer.

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translate this spectrum to zero frequency and subsequently “detect” the signal.

Now, let us assume that, in addition to X(f), the cellphone receives a large interferer centered at 900 MHz + 200 kHz [Fig. 14.1(b)].1 After translation to zero center frequency, the desired signal is still accompanied by the large interferer and cannot be detected properly. We must therefore “reject” the interferer by means of a filter [Fig. 14.1(c)].

14.1.1 Filter Characteristics

Which characteristics of the above filter are important here? First, the filter must not affect the desired signal; i.e., it must provide a “flat” frequency response across the bandwidth of X(f). Second, the filter must sufficiently attenuate the interferer; i.e., it must exhibit a “sharp” transition [Fig. 14.2(a)]. More formally, we divide the frequency response of filters into three regions:

Figure 14.2 (a) Generic and (b) ideal filter characteristics.

the “passband,” the “transition band,” and the “stopband.” Depicted in Fig. 14.2(b), the characteristics of the filter in each band play a critical role in the performance. The “flatness” in the passband is quantified by the amount of “ripple” that the magnitude response exhibits. If excessively large, the ripple substantially (and undesirably) alters the frequency contents of the signal. In Fig. 14.2(b), for example, the signal frequencies between f2 and f3 are attenuated whereas those between f3 and f4 are amplified.

The width of the transition band determines how much of the interferer remains alongside the signal, i.e., the inevitable corruption inflicted upon the signal by the interferer. For this reason, the transition band must be sufficiently narrow, i.e., the filter must provide sufficient “selectivity.”

The stopband “attenuation” and ripple also impact the performance. The attenuation must be large enough to suppress the interferer to well below the signal level. The ripple in this case proves less critical than that in the passband, but it simply subtracts from the stopband attenuation. In Fig. 14.2(b), for example, the stopband attenuation is degraded between f5 and f6 as a result of the ripple.

Example 14.1

In a wireless application, the interferer in the adjacent channel may be 25 dB higher than the desired signal. Determine the required stopband attenuation of the filter in Fig. 14.2(b) if the signal power must exceed the interferer power by 15 dB for proper detection.

Solution

As illustrated in Fig. 14.3, the filter must suppress the interferer by 40 dB, requiring the same

1This is called the “adjacent channel.”

25 dB

15 dB

Filter

Figure 14.3

amount of stopband attenuation.

In addition to the above characteristics, other parameters of analog filters such as linearity, noise, power dissipation, and complexity must also be taken into account. These issues are described in [1].

14.1.2 Classification of Filters

Filters can be categorized according to their various properties. We study a few classifications of filters in this section.

One classification of filters relates to the frequency band that they “pass” or “reject.” The example illustrated in Fig. 14.2(b) is called a “low-pass” filter as it passes low-frequency signals and rejects high-frequency components. Conversely, one can envision a “high-pass” filter, wherein low-frequency signals are rejected (Fig. 14.4).

Stopband

Figure 14.4 High-pass filter frequency response.

Example 14.2

We wish to amplify a signal in the vicinity of 1 kHz but the circuit board and wires pick up a strong 60-Hz component from the line electricity. If this component is 40 dB higher than the desired signal, what filter stopband attenuation is necessary to ensure the signal level remains 20 dB above the interferer level?

Solution

As shown in Fig. 14.5, the high-pass filter must provide a stopband attenuation of 60 dB at 60 Hz.

Figure 14.5

Exercise

A signal in the audio frequency range is accompanied by an interferer at 100 kHz. If the interferer is 30 dB above the signal level, what stopband attenuation is necessary if the signal must be 20 dB above the interferer.

Some applications call for a “bandpass” filter, i.e., one that rejects both lowand highfrequency signals and passes a band in between (Fig. 14.6). The example below illustrates the

Figure 14.6 .

need for such filters.

Example 14.3

Receivers designed for the Global Positioning System (GPS) operate at a frequency of approximately 1.5 GHz. Determine the interferers that may corrupt a GPS signal and the type of filters necessary to suppress them.

Solution

The principal sources of interference in this case are cellphones operating in the 900-MHz and 1.9-GHz bands.2 A bandpass filter is therefore required to reject these interferers (Fig. 14.7).

Exercise

Bluetooth transceivers operate at 2.4 GHz. What type of filter is required to avoid corrupting Bluetooth signals by PCS signals?

2The former is called the “cellular band” and the latter, the “PCS band,” where PCS stands for Personal Communication System.

Figure 14.7

Figure 14.8 summarizes four types of filters, including a “band-reject” response, which sup-

f f f f 1 f 2 f

Figure 14.8 Summary of filter responses.

presses components between f1 and f2.

Another classification of analog filters concerns their circuit implementation and includes “continuous-time” and “discrete-time” realizations. The former type is exemplified by the familiar RC circuit depicted in Fig. 14.9(a), where C1 exhibits a lower impedance as the frequency

Figure 14.9 (a) Continuous-time and (b) discrete-time realizations of a low-pass filter.

increases, thus attenuating high frequencies. The realization in Fig. 14.9(b) replaces R1 with a “switched-capacitor” network. Here, C2 is periodically switched between two nodes having voltages V1 and V2. We prove that this network acts as a resistor tied between the two nodes - an observation first made by James Maxwell in the 19th century.

In each cycle, C2 stores a charge of Q1 = C2V1 while connected to V1 and Q2 = C2V2 while tied to V2. For example, if V1 > V2, C2 absorbs charge from V1 and delivers it to V2, thus approximating a resistor. We also observe that the equivalent value of this resistor decreases as the switching is performed at a higher rate because the amount of charge delivered from V1 to V2 per unit time increases. Of course, practical switched-capacitor filters employ more sophisticated topologies.

The third classification of filters distinguishes between “passive” and “active” implementations. The former incorporates only passive devices such as resistors, capacitors, and inductors, whereas the latter also employs amplifying components such as transistors and op amps.

The concepts studied in Chapter 8 readily provide examples of passive and active filters. A low-pass filter can be realized as the passive circuit in Fig. 14.10(a) or the active topology

Figure 14.10 (a) Passive and (b) active realizations of a low-pass filter.

(integrator) in Fig. 14.10(b). Active filters provide much more flexibility in the design and find wide application in many electronic systems. Table 14.1 summarizes these classifications.

Frequency

Response

Continuous−Time

and Discrete−Time

Passive and

Active

Table 14.1 Classifications of filters.

14.1.3 Filter Transfer Function

The foregoing examples of filter applications point to the need for a sharp transition (a high selectivity) in many cases. This is because (1) the interferer frequency is close to the desired signal band and/or (2) the interferer level is quite higher than the desired signal level.

How do we achieve a high selectivity? The simple low-pass filter of Fig. 14.11(a) exhibits

Figure 14.11 (a) First-order filter along with its frequency response, (b) addition of another RC section to sharpen the selectivity.

Sec. 14.1 General Considerations 727

a slope of only ,20 dB/dec beyond the passband, thus providing only a tenfold suppression as the frequency increases by a factor of ten. We therefore postulate that cascading two such stages may sharpen the slope to ,40 dB/dec, providing a suppression of 100 times for a tenfold increase in frequency [Fig. 14.11(b)]. In other words, increasing the “order” of the transfer function can improve the selectivity of the filter.

The selectivity, ripple, and other attributes of a filter are reflected in its transfer function, H(s):

Example 14.4

Construct the pole-zero diagram for the circuits shown in Fig. 14.12.

obtaining a real pole at ,1=(R1C1). For the topology in Fig. 14.12(b),

The circuit therefore contains a zero at ,1=(R1C2) and a pole at ,1=[R1(C1 + C2)]. Note that the zero arises from C2. The arrangement in Fig. 14.12(c) provides the following transfer function:

Figure 14.13

The circuit exhibits a zero at zero frequency and two poles that may be real or complex depending

on whether L2 , 4R2L1C1 is positive or negative. Figure 14.13 summarizes our findings for the

1

three circuits, where we have assumed Hc(s) contains complex poles.

Exercise

Repeat the above example if the capacitor and the inductor in Fig. 14.12(c) are swapped.

Example 14.5

Explain why the poles of the circuits in Fig. 14.12 must lie in the right half plane.

Solution

Figure 14.14

14.14(b)]. Thus, we require k < 0 for the system to remain stable [Fig. 14.14(c)].

Exercise

Redraw the above waveforms if !k is doubled.

It is instructive to make several observations in regards to Eq. (14.1). (1) The order of the numerator, m, cannot exceed that of the denominator; otherwise, H(s) ! 1 as s ! 1, an unrealistic situation. (2) For a physically-realizable transfer function, complex zeros or poles must occur in conjugate pairs, e.g., z1 = 1 + j!1 and z2 = 1 , j!1. (3) If a zero is located on the j! axis, z1;2 = j!1, then H(s) drops to zero at a sinusoidal input frequency of !1 (Fig.

14.15). This is because the numerator contains a product such as (s , j!1)(s + j!1) = s2 + !2,

1

H (ω)

Figure 14.15 Effect of imaginary zero on the frequency response.

which vanishes at s = j!1. In other words, imaginary zeros force jHj to zero, thereby providing significant attenuation in their vicinity. For this reason, imaginary zeros are placed only in the stopband.

14.1.4 Problem of Sensitivity

The frequency response of analog filters naturally depends on the values of their constituent components. In the simple filter of Fig. 14.10(a), for example, the ,3-dB corner frequency is given by 1=(R1C1). Such dependencies lead to errors in the cut-off frequency and other parameters in two situations: (a) the value of components varies with process and temperature (in integrated circuits), or (b) the available values of components deviate from those required by the design (in discrete implementations).3

We must therefore determine the change in each filter parameter in terms of a given change (tolerance) in each component value.

Example 14.6

In the low-pass filter of Fig. 14.10(a), resistor R1 experiences a (small) change of R1. Determine the error in the corner frequency, !0 = 1=(R1C1).

Solution

3For example, a particular design requires a 1.15-k resistor but the closest available value is 1.2 k .

Since we are usually interested in the relative (percentage) error in !0 in terms of the relative change in R1, we write (14.7) as

Exercise

Repeat the above example if C1 experiences a small change of C.

The above example leads to the concept of “sensitivity,” i.e., how sensitive each filter parameter is with respect to the value of each component. Since in the first-order circuit, jd!0=!0j = jdR1=R1j, we say the sensitivity of !0 with respect to R1 is unity in this example. More formally, the sensitivity of parameter P with respect to the component value C is defined as

dP

C

Sensitivities substantially higher than unity are undesirable as they make it difficult to obtain a reasonable approximation of the required transfer function in the presence of component variations.

Example 14.7

Calculate the sensitivity of !0 with respect to C1 for the low-pass filter of Fig. 14.10(a).

Solution