ADVANCED POWER ELECTRONICS

Homework #2

Student: Dao Ngoc Dat / ID: 21650109

Subject: Filter study

Low-pass, high-pass, band-pass, band-stop filters: first, second, third order Analog circuits for these filters, and s-domain transfer function, Bode plot Digital implementation of these filters, z-domain transfer function. Comparison of output performance

Report:

1. INTRODUCTION

A filter is a device that passes electric signals at certain frequencies or frequency ranges while preventing the passage of others. — Webster.

Filter circuits are used in a wide variety of applications. In the field of telecommunication, band-pass filters are used in the audio frequency range (0 kHz to 20 kHz) for modems and speech processing. High-frequency band-pass filters (several hundred MHz) are used for channel selection in telephone central offices. Data acquisition systems usually require anti-aliasing low-pass filters as well as low-pass noise filters in their preceding signal conditioning stages. System power supplies often use band-rejection filters to suppress the 60-Hz line frequency and high frequency transients.

1.1 Filter Terminology

The range of signal frequencies that are allowed to pass through a filter, with little or no change to the signal level, is called the passband. The passband cutoff frequency (or cutoff point) is the passband edge where there is a 3 dB reduction in signal amplitude (the half-power point). The range of signal frequencies that are reduced in amplitude by an amount specified in the design, and effectively prevented from passing, is called the stopband. In between the passband and the stopband is a range of frequencies called the transition band, where the reduction in signal amplitude (also known as the attenuation) changes rapidly. These features are illustrated in Figure 1, which gives the frequency response of a lowpass filter.

Fig.1: Low pass filter Fig. 2: Frequency responses of basic filters

There are four possible frequency domain responses: lowpass, highpass, bandpass, and bandstop. Simplistic graphical representations are given above in Figure 2

1.2 Frequency Domain Responses

(a) Lowpass filters pass low frequencies. That is, they allow frequencies from DC up to what is known as the cutoff frequency with minimal loss of amplitude.

(b) Highpass filters pass high frequencies. They have the opposite function to that of lowpass filters, in that they allow frequencies above the cutoff to pass with minimal loss. They do not pass DC.

(c) Bandpass filters pass a band of frequencies between the lower and upper cutoff points. The upper cutoff determines the maximum frequency passed (with minimal loss). The lower cutoff decides the minimum frequency to be passed; DC is blocked.

(d) Bandstop filters stop a band of frequencies between the lower and upper cutoff points. They are the opposite of bandpass filters and allow two frequency bands to pass. One band that is passed goes from DC to the lower cutoff frequency. The other band passed covers all frequencies above the upper cutoff point.

1.3 Active Filters

At high frequencies (> 1 MHz), all of these filters usually consist of passive components such as inductors (L), resistors (R), and capacitors (C). They are then called LRC filters.

In the lower frequency range (1 Hz to 1 MHz), however, the inductor value becomes very large and the inductor itself gets quite bulky, making economical production difficult.

In these cases, active filters become important. Active filters are circuits that use an operational amplifier (op amp) as the active device in combination with some resistors and capacitors to provide an LRC-like filter performance at low frequencies (Figure 3).

This report focuses only on active filter design using operational amplifiers.

Fig. 3: Second-Order Passive Low-Pass and Second-Order Active Low-Pass

2. LOW-PASS FILTER

2.1 First-Order Low-Pass Filters

The transfer function of a first-order low-pass filter has the general form: �(�) =���

����

Fig. 4: First-order low-pass filter: (a) Low-pass characteristic with K =1, (b) Filter design

A first-order filter that uses an RC network for filtering is shown in Fig. 4(b). The op-amp operates as a noninverting amplifier, which has the characteristics of a very high input impedance and a very low output impedance.

The voltage (�� in Laplace’s domain of s) at the noninverting terminal of the op-amp can be found by the voltage divider rule:

The output voltage of the noninverting amplifier is:

which gives the voltage transfer function H(s) as:

where the DC gain is:

Substituting � = �� into the transfer function, we get:

which gives the cutoff frequency �� at 3-dB gain as:

2.2 Second-Order Low-Pass Filters

The roll-off of a first-order filter is only 20 dB/decade in the stop band. A second-order filter exhibits a stop-band roll-off of 40 dB/decade and thus is preferable to a first-order filter.

Fig. 5: Second-order low-pass filter: (a) Low-pass characteristic with K = 1, (b) Filter design

where K is the DC gain, Q is the quality factor and ��is the undamped natural (or resonant) frequency. A typical frequency characteristic is shown in Fig. 5(a); for high values of Q, overshoots will be exhibited at the resonant frequency��. For frequencies above��, the gain rolls off at the rate of 40 dB/decade.

A first-order filter can be converted to a second-order filter by adding an additional RC network, known as the Sallen–Key circuit, as shown in Fig. 5(b). The transfer function of the filter network is:

where � = (1 + �� ��)⁄ is the DC gain

Setting the denominator equal to zero gives the characteristic equation:

which will have two real parts and two equal roots. Setting � = �� in the above equation and then equating the real parts to zero, we get:

which gives the cutoff frequency as:

To simplify the design of second-order filters, equal resistances and capacitances are normally used—that is, R1 = R2 = R3 = R, C2 = C3 = C. Then the transfer function can be simplified to:

3. HIGH-PASS FILTERS

High-pass filters can be classified broadly into two types: first-order and second-order. Higher-order filters can be synthesized from these two basic types. Since the frequency scale of a low-pass filter is 0 to �� and that of a high-pass filter is �� to ∞, their frequency scales have a reciprocal relationship. Therefore, if we can design a low-pass filter, we can convert it to a high-pass filter by applying an RC-CR transformation. This transformation can be accomplished by replacing �� by �� and �� by��. The op-amp, which is modeled as a voltage-controlled voltage source, is not affected by this transformation. The resistors that are used to set the DC gain of the op-amp circuit are not affected either.

3.1 First-Order High-Pass Filters

The transfer function of a first-order high-pass filter has the general form

A typical high-pass frequency characteristic is shown in Fig. 6(a). A first-order high-pass filter can be formed by interchanging the frequency-dependent resistor and capacitor of the low-pass filter of Fig. 5(b). This arrangement is shown in Fig. 6(b).

Fig. 6: First-order high-pass filter: (a) High-pass characteristic, (b) Filter design

The voltage at the noninverting terminal of the op-amp can be found by the voltage divider rule. That is,

The output voltage of the noninverting amplifier is

which gives the voltage gain as

the cutoff frequency �� at 3-dB gain as

The magnitude and phase angle of the filter gain can be found from:

and:

This filter passes all signals with frequencies higher than��. However, the high-frequency limit is determined by the bandwidth of the op-amp itself. The gain–bandwidth product of a practical µ741-type op-amp is 1 MHz.

3.2 Second-Order High-Pass Filters

A second-order high-pass filter has a stop-band characteristic of 40 dB/decade rise. The general form of a second-order high-pass filter is:

where K is the DC gain, Q is the quality factor and ��is the undamped natural (or resonant) frequency. Figure 7(a) shows a typical frequency response. As in the case of the first-order filter, a second-order high-pass filter can be formed from a second-order low-pass filter by interchanging the frequency-dominant resistors and capacitors. Figure 7(b) shows a second-order high-pass filter derived from the Sallen–Key circuit of Fig. 6(b). The transfer function can be derived by applying the RC-to-CR transformation and substituting 1/s for s in the transfer function of a second-order low-pass filter. For R1 = R2 = R3 = R and C2 = C3 = C, the transfer function becomes:

The cutoff frequency is deduced as:

Fig. 7: Second-order high-pass filter: (a) High-pass characteristic with K = 1, (b) Filter design

4. BAND-PASS FILTERS

A band-pass filter has a passband between two cutoff frequencies �� and �� such that�� > ��. Any frequency outside this range is attenuated. The transfer function of a BP filter has the general form

where KPB is the pass-band gain and �� is the center frequency in radians per second. There are two types of band-pass filters: wide band pass and narrow band pass. A filter may be classified as wide band pass if Q ≤ 10 and narrow band pass if Q > 10.

The relationship of Q to 3-dB bandwidth and center frequency �� is given by

For a wide-band-pass filter, the center frequency �� can be defined as:

where �� is the low cutoff frequency, in hertz, and �� is the high cutoff frequency, in hertz. In a narrow-bandpass filter, the output peaks at the center frequency��.

4.1 Wide-Band-Pass Filters

The frequency characteristic of a wide-band-pass filter is shown in Fig. 8(a), where�� > ��.

As shown in Fig. 8(b), we use two filters: one low-pass filter and one high-pass filter. The output is obtained by multiplying the low-frequency response by the high-frequency response; this solution can be implemented simply by cascading the first-

order (or second-order) high-pass and low-pass sections. The order of the band-pass filter depends on the order of the high-pass and low-pass sections. This arrangement has the advantage that the falloff, rise, and midband gain can be set independently. However, it requires more op-amps and components.

Figure 8(c) shows a 20 dB/decade wide-band-pass filter implemented with first-order highpass and first-order low-pass filters. In this case, the magnitude of the voltage gain is equal to the product of the voltage gain magnitudes of the high-pass and low-pass filters. In this case, the transfer function of the wide-midband filter for first-order implementation becomes

And the transfer function for second-order implementation is:

where KPB overall pass-band gain high-pass gain KH low-pass gain KL.

Fig. 8: Wide-band-pass filter

4.2 Narrow-Band-Pass Filters

A typical frequency response of a narrow-band-pass filter is shown in Fig. 9(a). This characteristic can be derived by setting a high Q-value for the band-pass filter shown in Fig. 9(b). This filter uses only one op-amp in the inverting mode. Because it has two feedback paths, it is also known as a multiple feedback filter. For a low Q-value, it can also exhibit the characteristic of a wide-band-pass filter.

A narrow-band-pass filter is generally designed for specific values of fC and Q or fC and BW. The op-amp, along with C2 and R2, can be regarded as an inverting differentiator such that

The equivalent filter circuit is shown in Fig. 9(c). The transfer function of the filter network is:

Fig. 9: Narrow-band-pass filter: (a) Narrow-band characteristic, (b) Filter design, (c) Equivalent circuit

5. BAND-REJECT FILTERS

A band-reject filter attenuates signals in the stop band and passes those outside this band. It is also called a band-stop or band-elimination filter. The transfer function of a second-order band-reject filter has the general form where KPB is the pass-band gain. Band-reject filters can be classified as wide band reject or narrow band reject. A narrow-band-reject filter is commonly called a notch filter. Because of its higher Q (>10), the bandwidth of a narrow-band-reject filter is much smaller than that of a wide-band-reject filter.

5.1 Wide-Band-Reject Filters

The frequency characteristic of a wide-band-reject filter is shown in Fig. 10(a). This characteristic can be obtained by adding a low-pass response to a high-pass response, as shown in Fig. 10(b); the solution can be implemented by summing the responses of a first-order (or second-order) high-pass section and low-pass section through a summing amplifier. This arrangement is shown in Fig. 10(c).

The order of the band-reject filter depends on the order of the high-pass and low-pass sections. For a band-reject response to be realized, the cutoff frequency fL of the high-pass filter must be larger than the cutoff frequency fH of the low-pass filter. In addition, the pass-band gains of the high-pass and low-pass sections must be equal. With an inverting summer (A3), the output will be inverted.

5.2 Narrow-Band-Reject Filters

A typical frequency response of a narrow-band-reject filter is shown in Fig. 11(a). This filter, often called a notch filter, is commonly used in communication and biomedical instruments to eliminate undesired frequencies such as the 60-Hz power line frequency hum. A twin-T network, which is composed of two T-shaped networks, as shown in Fig. 11(b), is commonly used for a notch filter. One network is made up of two resistors and a capacitor; the other uses two capacitors and a resistor. To increase the Q of a twin-T network, it is used with a voltage follower. The transfer function of a twin-T network is given by

where:

Fig. 10: Wide-band-reject filter

Fig. 11: Narrow-band-reject filter: (a) Narrow-band-reject characteristic, (b) Filter design

6. SUMMARY

The table below summarizes the characteristics of second-order filters

with K is the DC gain, Q is the quality factor and ��is the undamped natural (or resonant) frequency and: , then transfer function in frequency domain of second order filters can be deduced as:

Case 1: Second order Low-pass filter:

Case 2: Second order High-pass filter:

Case 3: Second order Band-pass filter:

Case 4: Second order Band-stop filter:

7. COMPARISON

This part will give example designs of those filters described above. The Bode graphs (with gain(dB) and phase delay) are showed in order to easily compare the output performance between these designs.

7.1 Low-pass filter design

(a) First order Low-pass filter

(b) Second order Low-pass filter

(c) Third order Low-pass filter

Fig. 12: Low-pass filter design example, with cutoff frequency is 1kHz

7.2 High-pass filter design

(a) First order High-pass filter

(b) Second order High-pass filter

(c) Third order High-pass filter

Fig. 13: High-pass filter design example, with cutoff frequency is 10 kHz

7.3 Band-pass filter design

(a) Second order Band-pass filter

(b) Fourth order Band-pass filter

Fig. 14: Band-pass filter design example, with center frequency is 1 kHz, bandwidth is 200 Hz

7.4 Band-stop filter design

(a) Second order Band-stop filter

(b) Fourth order Band-stop filter

Fig. 15: Band-pass filter design example, with center frequency is 1 kHz, bandwidth is 200 Hz

8. DIGITAL IMPLEMENT

After finding the transfer functions of the filters, we use z-transform to obtain a real-time discrete equivalent of a filter in the continuous system.

The bilinear transform can be used to convert continuous-time filters (represented in the Laplace domain) into discrete-time filters (represented in the Z-domain), and vice versa. The following substitution is used:

� =�

�

���

���=

�

�

�����

����� (T: sampling time)

8.1 Low-pass filter

8.1.a First order Low-pass filter

1

1

1

1

22

1

22

1

22

1

112

1

1

12)(

:)(

zTT

zT

zTT

z

Tz

T

z

zzT

z

z

z

T

zH

frequencycutoffs

sH

cc

c

cc

c

cc

c

c

c

c

c

c

c

cLPF

with (�) = �(�) �(�)⁄ , �����(�) = �(� − 1)

����(�) = �(� − 1) , then from the equation above, we have:

12

12

2

1122

kukuT

Tky

T

Tky

kukuTkyTkyT

c

c

c

c

ccc

8.1.b Second order Low-pass filter

)(2442844)(

)(2442844)(

)(11121214)(

)(1

122

1

12)(

:2

)(

2221222222212222

22222222222222

2222222

22

2

22

2

zuzTzTTzTTzTTTzy

zuTzTzTTTzTzTTzy

zuTzTzzTzzzy

zuz

z

Tz

z

Tzy

frequencycutoffss

sH

cccccccc

cccccccc

ccc

ccc

c

cc

cLPF

244

144

2

44

244

441

44

28

221

44228144

22

22

22

22

22

22

22

22

22

22

222222

222222

kuTT

T

kuTT

Tku

TT

T

kyTT

TTky

TT

Tky

TkuTkuTku

TTkyTkyTTky

cc

c

cc

c

cc

c

cc

cc

cc

c

ccc

ccccc

8.2 High-pass filter

8.2.a First order High-pass filter

1

1

22

12

22

12

112

12

1

121

12

)(

:)(

zTT

z

TzT

z

zTz

z

z

z

T

z

z

TzH

frequencycutoffs

ssH

cccccc

c

c

HPF

12

21

2

2

12122

kukuT

kyT

Tky

kukukyTkyT

cc

c

cc

8.2.b Second order High-pass filter

)(484442844)(

)(484442844)(

)(141121214)(

)(1

12

1

122

1

12)(

:2

)(

2122212222

22222222

22222

2

2

2

22

2

zuzzzTTzTTTzy

zuzzTTzTzTTzy

zuzTzzTzzzy

zuz

z

Tz

z

Tz

z

Tzy

frequencycutoffss

ssH

ccccc

ccccc

cc

cc

c

cc

HPF

244

4

144

8

44

4

244

441

44

28

24184

44228144

22

2222

22

22

22

22

222222

kuTT

kuTT

kuTT

kyTT

TTky

TT

Tky

kukuku

TTkyTkyTTky

cc

cccc

cc

cc

cc

c

ccccc

8.3 Band-pass filter

)(12242824)(

)(.211112114)(

)(1

12

1

12

1

12)(

:

:

)(

22222222

2222

2

2

22

zuzTQ

TTQ

zTzTTQ

zy

zuTzzQ

TzzTzQ

zzy

zuz

z

TQz

z

TQz

z

Tzy

BQ

frequencybandB

frequencycenterf

sQ

s

sQ

sH

bb

bbb

b

bb

b

bb

b

b

b

bb

b

BPF

2

24

2

2

24

24

1

24

28

)(12242824)(

2222

22

22

22

222212222

kuku

TTQ

TQ

ky

TTQ

TTQ

ky

TTQ

Tky

zuzTQ

zTTQ

zTTTQ

zy

bb

b

bb

bb

bb

b

bb

bbb

b

8.4 Band-stop filter

)(4284

242824)(

)(1

12

1

12

1

12)(

:

:

)(

2222222

2222222

2

2

2

2

22

22

zuTzTzT

TTQ

zTzTTQ

zy

zuz

z

Tz

z

TQz

z

Tzy

BQ

frequencybandB

frequencycenterf

sQ

s

ssH

bbb

bb

bbb

bbb

b

b

bb

bBSF

2

24

41

24

28

24

4

2

24

24

1

24

28

22

22

22

22

22

22

22

22

22

22

ku

TTQ

Tku

TTQ

Tku

TTQ

T

ky

TTQ

TTQ

ky

TTQ

Tky

bb

b

bb

b

bb

b

bb

bb

bb

b