Regular Paper
Electronically controllable bandpass filters with high quality factor
and reduced capacitor value: An additional approach
Abdullah Yesil
a, Firat Kacar
a, Shahram Minaei
b,⇑ aDept. of Electrical and Electronics Engineering, Istanbul University, 34340 Istanbul, Turkey bDept. of Electronics and Communications Engineering, Dogus University, 34722 Istanbul, Turkey
a r t i c l e i n f o
Article history: Received 5 December 2015 Accepted 10 April 2016 Keywords: Active filter Bandpass filter Amplifier Analog circuit High quality factora b s t r a c t
In this paper, a new method is presented to increase the quality factor of bandpass filters employing active elements. Using the proposed method the values of the externally capacitors in filter structure are reduced. In this way chip area which is an important parameter in the integrated circuits can be minimized. Moreover, bandpass filters are investigated from the point of view of stability conditions and effects of the parasitic and non-ideal elements. Considering these effects, operating conditions and boundaries of the bandpass filter are calculated. To validate the feasibility of the method, an application example of bandpass filter used in intermediate frequency (IF) stage of AM receivers is given and the performance of the circuit is demonstrated by comparing the theory and simulation.
Ó 2016 Elsevier GmbH. All rights reserved.
1. Introduction
Bandpass filters are mainly used in filtering signals as well as many applications in electronic circuits such as mixers, oscillators, etc. The most important parameter of the bandpass filters is the quality factor (Q) which determines the frequency selectivity of the filter. Generally, bandpass filters with high-Q value require high ratio of capacitances. However high-valued capacitors occupy large silicon area in integrated circuits (ICs).
In order to increase quality factor of active filters many researches have been presented in the literature[1–4]. However these filters are employed using several active and/or passive ele-ments. Nevertheless, Lakys and Fabre[5]presented new method to increase pole frequency and quality factor at the same time which is the technique of shadow filters. The application areas of this method are such as cognitive and encrypted communications[6], frequency hopping circuit [7] and frequency agile filters [8]. Another method is presented to increase quality factor of the second order bandpass filter by Biolkova and Biolek[9]. The second order filter has lowpass, bandpass and highpass outputs. Both of lowpass and highpass outputs are added and applied to the input of the feedback circuit. The output of the feedback circuit is added to input signal. Therefore, quality factor is increased by factor of feedback gain without changing the pole frequency of the filter.
As mentioned, the capacitor being major used element in active filters occupies large area in IC fabrication. For this reason, the capacitance multiplier circuits[10–14]are also proposed in the lit-erature to obtain high capacitance values. However, these pre-sented circuits are employed more active and/or passive elements. In this paper, we present a new method to increase the quality factor of the active bandpass filters. The high quality factor value can be obtained and tuned electronically by changing the gain of the feedback circuit. In fact, the proposed method decreases the values of the capacitor up to approximately square root of the feedback gain. The new method employs a feedback circuit and a second order filter which has two outputs, bandpass and highpass responses. The gain of the bandpass filter remains constant while the center frequency can be adjusted by electronically changing of the feedback gain. The active and passive sensitivities of the filter elements are no more than unity. Moreover, taking into account a single pole model for the two stages amplifier used in the feedback circuit, stability conditions are investigated. In addi-tion, the effects of the parasitic elements in whole structure are investigated.
2. The proposed method
The proposed method depicted inFig. 1b is built using second order filter structure and feedback circuit. InFig. 1a second order filter structure called Class-0 has two filter outputs, bandpass and highpass responses. Highpass response of Class-0 is applied to the input of the feedback circuit. Then, output of the feedback
http://dx.doi.org/10.1016/j.aeue.2016.04.009
1434-8411/Ó 2016 Elsevier GmbH. All rights reserved.
⇑Corresponding author.
E-mail addresses:abdullah.yesil@istanbul.edu.tr(A. Yesil),fkacar@istanbul.edu. tr(F. Kacar),sminaei@dogus.edu.tr(S. Minaei).
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International Journal of Electronics and
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circuit is added into the input signal of the system. The whole structure is called Class-1. This is an interesting idea for the bandpass filter output of the Class-1.
Here we assumed that Class-0 and Class-1 structure operate in voltage mode. Nevertheless, the same method can be applied to all different filter modes. The general transfer functions of the Class-0 can be defined the following as
TBP¼ VBP VIN¼ sa1 s2þ sb 1þ b0 ð1Þ THP¼ VHP VIN¼ a2s2 s2þ sb 1þ b0 ð2Þ
where, b1and b0are real positive constants to ensure stability of the second order filter. a1can be real positive or negative constants while a2 is real positive constant. The routine analysis of the Class-1 filter yields the following voltage-mode bandpass transfer function, TBPA¼ VBPA VE ¼ sa1 ð1þAa2Þ s2þ sb1 ð1þAa2Þþ b0 ð1þAa2Þ ð3Þ Here, A is gain of the feedback circuit and is positive constant. From(3)the expressions for pole frequency and quality factor of the Class-1 are given by
x
0A¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b0 ð1 þ Aa2Þ s ð4Þ QA¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ Aa2Þb0 p b1 ð5ÞIt can be clearly seen from Eqs.(4) and (5)that quality factor increases by factor of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Aa2 while pole frequency decreases by factor ofpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Aa2in Class-1 topology with respect to Class-0. This result is the main aim of the proposed method. Sensitivity analyses of the proposed method with respect to gain of the feedback circuit yields
Sx0A A ¼ 1 2þ 1 2ð1 þ Aa2Þ ð6Þ SQA A ¼ 1 2 1 2ð1 þ Aa2Þ ð7Þ
As it is evident from the above analysis, the sensitivity of the pole frequency and quality factor of the proposed method in regard to gain of the feedback circuit do not exceed unity. Note that, to avoid the stability problem, in Eq. (3) the term (1 + Aa2) should ensure to be positive by virtue of Routh Hurwitz criterion [15]. Consequently, both feedback gain and highpass filter gain of the Class-0 must be chosen simultaneously positive or negative.
A summary of the specific parameters of the Class-0 and Class-1 are shown inTable 1. The gain of the bandpass filter remains the same in Class-0 and Class-1. It is obvious fromTable 1that pole fre-quency of the Class-1 diminishes amount of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Aa2as quality factor of the Class-1 rises by factor pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Aa2 with respect to Class-0.
3. Application example
In order to show the feasibility of the proposed method, the voltage mode filter, proposed by Chen[16], is selected. However, this filter is modified input terminal and some connections. The modified filter (Class-0) is shown inFig. 2. Thanks to the modifica-tion in filter, it is not necessary to use adder circuit. The filter in
Fig. 2can simultaneously realize voltage mode bandpass and high-pass filters employing two DDCCs and four grounded high-passive elements.
The port relations of the DDCC can be characterized by VX= VY1 VY2+ VY3and IZ= IX. Routine analysis of the filter shown inFig. 2gives the following filter transfer functions:
VHP VIN ¼ s2 s2þ s 1 C2R2þ 1 C1C2R1R2 ð8Þ VBP VIN¼ s 1 C1R1 s2þ s 1 C2R2þ 1 C1C2R1R2 ð9Þ Gain of the bandpass filter, the natural frequency and quality factor are obtained as:
GBP¼ C2R2 C1R1 ð10Þ
x
0¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 C1C2R1R2 s ð11Þ Q¼ ffiffiffiffiffiffiffiffiffiffiffi C2R2 C1R1 s ð12Þ It can be seen that the natural frequency and quality factor can-not be tuned electronically. Moreover, buffer circuit is required for bandpass output due to possessing high and frequency-dependent output impedance. The Class-1 circuit is shown inFig. 3which is the combination of the Class-0 and a feedback circuit (amplifier). The input terminal of the feedback circuit is connected to highpass output of the modified filter. Then, the output of the feedbackVBP VIN VHP 2nd order filter Class-0
(a)
VBPA VE -AVHP -A 2nd order filter Class-1 Feedback circuit VHP(b)
Fig. 1. (a) Class-0 and (b) Class-1 structure.
Table 1
The specific parameters of the Class-0 and Class-1 inFig. 1. Class-0 Class-1 Gain BP filter at pole frequency a1
b1 a1 b1 Pole frequency x0¼ ffiffiffiffiffib0 p x0A¼ ffiffiffiffi b0 p ffiffiffiffiffiffiffiffiffiffiffi 1þAa2 p Quality factor Q¼ ffiffiffiffib0 p b1 QA¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ Aa2 p ffiffiffiffib0 p b1
circuit is connected to Y3 terminal of the DDCC1. Thus, Class-1 structure is completed with the modified filter and feedback circuit.
It is well-known that the relation of the amplifier can be defined by VO= A(V+ V). A is frequency dependent open-loop gain of the amplifier. It can be assumed that open loop gain of the amplifier is frequency independent in our operating frequency range of interest. Routine analysis of the Class-1 structure shown inFig. 3
yields VBPA VIN ¼ s 1 ð1þAÞC1R1 s2þ s 1 ð1þAÞC2R2þ 1 ð1þAÞC1C2R1R2 ð13Þ Gain of the bandpass filter, the natural frequency and quality factor of the Class-1 are calculated as:
GBPA¼ GBP¼ C2R2 C1R1 ð14Þ
x
0A¼ ffiffiffiffiffiffiffiffiffiffiffiffix
0 1þ A p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ð1 þ AÞC1C2R1R2 s ð15Þ QA¼ Q ffiffiffiffiffiffiffiffiffiffiffiffi 1þ A p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ AÞC2R2 C1R1 s ð16Þ It is clearly seen that gain of bandpass in the Class-1 is the same as the Class-0. In contrast to the Class-0, natural frequency of the Class-1 circuit decreases by a factor ofpffiffiffiffiffiffiffiffiffiffiffiffi1þ Awhile the quality factor of the Class-1 increases by a factor ofpffiffiffiffiffiffiffiffiffiffiffiffi1þ A. Consequently, the results in Eqs.(14)–(16)are confirmed by theoretical results inTable 1. In contrast to the Class-0, buffer circuit is similarly required for bandpass output in Class-1. Also, from(15) and (16)
it is easy to show that the active and passive sensitivities of the parameters
x
0Aand QAareSx0A R1 ¼ S x0A R2 ¼ S x0A C1 ¼ S x0A C2 ¼ 1 2; S x0A A ¼ 1 2þ 1 2ð1 þ AÞ ð17Þ SQA R1 ¼ S QA R2 ¼ S QA C1 ¼ S QA C2 ¼ 1 2; S QA A ¼ 1 2 1 2ð1 þ AÞ ð18Þ
which, are no more than unity in magnitudes. 4. Stability and effects of parasitic elements
Feedback circuit is composed of a two stages amplifier. It is well known that two stages amplifier consists of differential amplifier circuit and common source amplifier. The advantages of two stages amplifier are simple structure and high open loop gain and high bandwidth. The open loop gain of the two stages amplifier can be represented by the well-known first pole roll-off characteristic
[17]
AðsÞ ffi A0
1þ s
xA
ð19Þ where, A0is open loop DC gain,
x
Ais the first pole frequency of the two stages amplifier. For the Class-1 structure inFig. 3, if single pole model of two stages amplifier is taken into account, bandpass trans-fer function of the Class-1 is found as,VBPA VIN ¼ s2 C2R2 xA þ sC2R2 s3 C1C2R1R2 xA þ s 2 ð1 þ A 0ÞC1C2R1R2þCx1RA1 þ s C1R1þx1A þ 1 ð20Þ Here, the effects of the parasitic impedances at the DDCC and at two stages amplifiers terminals are ignored to investigate the effects of feedback circuit pole frequency. From(20)it can be seen that due to parasitic pole of the two stages amplifier, a third order filter response is obtained with undesirable terms in the bandpass transfer function. The Class-1 structure can operate approximately as a second order filter by satisfying the following conditions,
x
A>>ð1þA1 0ÞC2R2x
>> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1C1R1xA C1C2R1R2q ð21Þ
where,
x
is the operating frequency of the Class-1. To investigate the stability of the circuit problem, Routh–Hurwitz criterion is applied to Eq.(20), the conditions obtained by Routh–Hurwitz cri-terion can be obtained as follows,C1C2R1R2> 0 C1R1ð1 þ ð1 þ A0ÞC2R2
x
AÞ > 0 1þA0C2R2xAþC1R1xAð1þð1þA0ÞC2R2xAÞ 1þð1þA0ÞC2R2xA > 0x
A> 0 ð22ÞIt is observed from Eq.(22)that there is no stability problem due to fact that all terms of Routh–Hurwitz criterion is positive. Also, it is important to note that single pole model of the two stages amplifier can give a rough idea about the stability of the fil-ter, because the DDCC and two stages amplifier have other poles and possibly zeroes, due to parasitic impedances of active ele-ments, at high frequencies that may affect the stability. In deepen analysis, taking into account parasitic inductance and resistance in series to the terminal X of DDCCs and single pole model of gain in the two stages amplifier, bandpass transfer function in Class-1 can be obtained as VBPA VIN ¼ C2sðRT2þ LXsÞðs þ
x
AÞ s5C 1C2L2Xþ s4C1C2LXðRT1þ RT2þ LXx
AÞ þs3C 1C2ðLXRT1x
Aþ RT1RT2þ ðRT2þ R1A0ÞLXx
AÞ þs2C 1ðC2RT2x
AðRT1þ A0R1Þ þ R1Þ þsð1 þ C1R1x
AÞ þx
A ð23Þ DDCC1 Z X Y2 Y1 Y3 DDCC2 Z X Y2 Y3 Y1 VBP VHP VIN R1 R2 C2 C1Fig. 2. Modified DDCC based filter structure.
DDCC1 Z X Y2 Y1 Y3 DDCC2 Z X Y2 Y3 Y1 VBPA VIN R1 R2 C2 C1 + -Amplifier
where, RT1= R1+ RSand RT2= R2+ RS. LXand RSare parasitic induc-tance and resisinduc-tance in series to the X terminal of DDCC, respec-tively. It is seen in Eq. (23) that fifth order filter responses is found as the result of the deepen analysis. Due to fifth order filter responses, understandable results cannot be obtained applying Root-Hurwitz criterion. Because of this reason, the results are ana-lyzed by the means of MATLABÒprogramming. The passive element is selected as C1= C2= 15 pF and parasitic inductance and resistance value in series to the terminal X of DDCC are calculated as LX= 2.4
l
H and RS= 113O, respectively. The first pole frequencyx
A and A0of the two stage amplifier depend on its bias current. The first pole and A0are obtained using SPICE when bias current of the two stages amplifier is changed from 10l
A to 100l
A. Thex
A can be represented with an empirical formula using SPICE results, as a function of A0as follows,x
A¼ 2p
ð722:06 A20 1:104 10 6A0þ 3:9404 108Þ ð24Þ
The resistor value in Class-1 with respect to A0restricted by the stability condition is depicted inFig. 4.
The stability analysis result of the Class-1 inFig. 3is given in
Fig. 5with respect to variation of the external resistors, capacitors and feedback gain.
Moreover, the Class-1 structure inFig. 3with parasitic impe-dances of the active elements is taken into consideration. Effects on natural frequency, quality factor and the gain of bandpass filter are investigated. The main parasitic impedances of the DDCC appear parallel parasitic resistance and capacitance RP//(1/(sCP)) at Z terminal. Also, the main parasitic impedance of the two stages amplifier exhibits series parasitic resistance ROat output terminal. Considering the Class-1 structure, the output parasitic resistance of the two stages amplifier is ignored due to fact that Y terminals of the DDCC have theoretically infinite input resistance (connected to gates of MOS transistor). Taking into account the parasitic
impedances, gain of bandpass filter, the natural frequency and quality factor of the Class-1 structure recalculated as
G0BPA¼ CT2R22RP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ AÞ2 C2T1R 2 1R 2 Pþ ðR2 PþR1ðð1þAÞR2þRPÞÞ 2 R2 2 r R1ðð1 þ AÞðCT1þ CT2ÞR2þ CT1RPÞðR2Pþ R1ðð1 þ AÞR2þ RPÞÞ ð25Þ
x
0 0A¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2Pþ R1RPþ ð1 þ AÞR1R2 q RP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ AÞR1R2CT1CT2 p ð26Þ Q0A¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ AÞCT1CT2R2ðð1 þ AÞR1R2þ R1RPþ R2PÞ q ffiffiffiffiffi R1 p ðR2ð1 þ AÞðCT1þ CT2Þ þ CT1RPÞ ð27Þ where CT1= C1+ CZP, CT2= C2+ CZP. It is important to note that Y-terminal capacitances (CY) are ignored due to their small values. It is seen inFig. 3that parasitic capacitance of the DDCCs at the Z ter-minal appears parallel the external capacitor. The natural fre-quency, quality factor and gain of bandpass may be altered slightly due to fact that these parasitic elements appear as additive values. Furthermore, for various feedback gain (A), in Eq. (25) is tested with respect to feedback gain and output impedance of the Z terminal of the DDCC for R1= R2= 1 k and C1= C2= 15 pF.Fig. 6illustrates how to choose the feedback gain and output impedance of the Z terminal of the DDCC. It is observed fromFig. 6that it is required that to increase output impedance of Z terminal of the DDCC, in order to avoid reduced voltage gain when feedback is applied to Class-1 structure. For this reason, cascade current stages are used at the output stage of the DDCC[18,19]so impedance of the output stage is increased significantly.
5. Simulation results
In order to demonstrate the frequency domain performance of the proposed method, the circuit ofFig. 3is simulated by LTSPICE using TSMC 0.18
l
m CMOS process technology model parameters[20]. The modified internal structure of DDCC inFig. 7is derived from simple DDCC structure [21]by employing cascode current stage to obtain the required high output resistance at Z terminal. The internal CMOS structure of the modified DDCC with DC supply voltages of ±1.25 V, bias voltage VB=0.45 V and bias currents IB1= IB2= 10
l
A is depicted inFig. 7. The aspect ratios of the CMOS transistors of the DDCC are given inTable 2. The aspect ratios of thedifferential difference amplifier stage of the modified DDCC are given in [22]. The values of some parasitic impedances of the modified DDCC in Fig. 7 are given as follows: RS= 113.3O, RP= 11.77 MO, CP= 52 fF and LX= 2.4
l
H. Power consumption of the modified DDCC is found to be 0.84 mW.Two stages amplifier is simulated using the schematic imple-mentation shown inFig. 8with DC supply voltages ±1.25 V and bias voltage VB=0.656 V to obtained IB= 50
l
A. The aspect ratios of the two stages amplifier are given inTable 3. The open-loop gain, first pole frequency and the power dissipation of the two stages amplifier are found to be 457, 42 MHz and 0.2 mW, respec-tively. Gain of two stages amplifier is obtained from 500 to 321 when bias current IBchanges between 20l
A and 200l
A.Fig. 9depicts gain-frequency responses of the 0 and Class-1 filters. The passive element values are selected as C1= C2= 15 pF, R1= R2= 1 kO. From the simulation results, center frequency of f0= 10.5 MHz and quality factor of 1.65 are obtained for Class-0 fil-ter. For the same passive values, center frequency of f0= 453 kHz and quality factor of 18.68 are obtained for Class-1 filter. Center frequency and quality factor of the Class-0 filter are theoretically calculated as 10.6 MHz and 1, respectively. In a similar way, Class-1’s center frequency and quality factor are theoretically cal-culated as 495 kHz and 21.35, respectively. For Class-1, the errors of center frequency and quality factor between theoretically and simulation results are found to be 8.5% and 12.5%, respectively. The difference between theoretical and simulation results stems from non-ideal gain and parasitic impedances that are especially Z terminal of the DDCCs. It is clearly seen that quality factor of the Class-1 is significantly provided to increase by means of two stages amplifier circuit.
As a second test, the Class-0’s passive element values are selected as R1= R2= 1 kO, C1= C2= 312 pF to obtained the center
Fig. 5. The changing of stability analysis with respect to resistor and capacitor values as well as feedback gain in Class-1 (Unstable region is defined by star).
frequency of Class-1 filter (f0= 453 kHz) while the passive element values of the Class-1 structure are the same as mentioned above. The gain-frequency responses of the Class-1 and Class-0 filters are shown inFig. 10. Note that while both center frequency and resistor values are the same for both the Class-0 and Class-1 filters, the capacitor values are just altered. It is clearly observed that the external capacitor values in the Class-1 filter are 20.8 times smaller than ones in the Class-0 structure. Furthermore, quality factor of the Class-1 filter has already been 16.5 times bigger than ones in the Class-0 structure.
Variation of feedback gain of the two stages amplifier and cen-ter frequency of the bandpass filcen-ter in Class-1 is shown inFig. 11.
VSS VDD M10 M13 X Z M4 M3 M8 M7 M9 M12 M6 M2 M1 M5 VB Y2 Y3 Y1 M11 VSS VDD M14 M15 M16 IB1 IB2
Fig. 7. The realization of the modified DDCC.
Table 2
Aspect ratios of the modified DDCC.
W/L M1–M6, M12,M13 4.5lm/0.9lm M7–M10 9lm/0.9lm M14–M15 10lm/1.2lm M11–M16 20lm/1.2lm VSS VDD Out M2 M1 M4 M3 M5 M7 M6 + VB -IB
Fig. 8. The realization of two stages amplifier.
Table 3
Aspect ratios of two stages amplifier.
W/L M1–M5 3lm/0.18lm
M6 10lm/0.36lm
M7 5lm/0.36lm
Fig. 9. The gain frequency responses of the Class-0 and Class-1.
Fig. 10. The gain frequency responses of the Class-0 and Class-1 filters for different passive element values.
Note that gain of feedback circuit is adjusted by electronically the biasing current IBof feedback circuit.Fig. 12depicts variation of between quality factor and feedback gain. It is observed from
Fig. 12 that quality factor increases with feedback gain towards 400 however quality factor value decreases after feedback gain of 400. The reason of drop in quality factor stems from parasitic impedances of X and Z terminal in DDCC and parasitic pole of two stages amplifier. Also, it is clearly seen fromFig. 6that gain of bandpass filter in Class-1 decreases due to the non-infinite par-asitic parallel impedance of Z terminal while feedback gain increases. It should be mentioned that output resistance at Z termi-nals of the DDCC can be enhanced by bias current of cascode cur-rent mirror in DDCC or by using improved active feedback cascode current mirror[23]instead of cascode current mirror. By this way, the drop in gain of bandpass filter can be prevented.
The time domain responses of the Class-1 structure are investi-gated by applying a sinusoidal input voltage signal with an ampli-tude of 50 mV peak to peak at f0= 450 kHz. Input and output signals of the bandpass filter of the Class-1 are depicted as
Fig. 13. The total harmonic distortion of the output signal for Class-1 filter is given inFig. 14.
To verify accuracy of the stability analyses, phase responses of the bandpass output in the Class-1 are given inFig. 15for different resistor values. Capacitors and feedback gain are selected as 15 pF and 457, respectively. It is observed from the phase responses that the filter is stable for R1= R2= 1000X, 750X, 500X, 400Xand
Fig. 11. Variation of between the center frequency and feedback gain for Class-1 filter.
Fig. 12. Dependence of the quality factor on feedback gain for Class-1 filter.
Fig. 13. Time domain response of the Class-1 bandpass filter.
Fig. 14. Dependence of the output harmonic distortion of the Class-1 filter on input signal amplitude.
Fig. 15. Phase responses of the Class-1 bandpass response for different resistor values.
300X(the phase changes from 90 degree to90 degree) while it is unstable for 100Xand 200X(the phase changes from 90 degree to 270 degree). Comparing these values with the theoretical results shown inFig. 4, it is clearly seen that simulations results agree quite with theoretical analyses.
6. Conclusion
In this paper a new method is presented to increase the quality factor value in active bandpass filters and in order to decrease capacitance values. The structure of the method consists of second order filter and simple and flexible feedback circuit. It is shown that the quality factor value increases approximately by factor of square root of feedback gain for the modified bandpass filter. Also, center frequency and the quality factor of bandpass filter can be tuned electronically by feedback gain. In addition, considering sin-gle pole model for feedback circuit, stability conditions are deter-mined. Also effects of the main parasitic elements in whole structure are investigated. A bandpass filter example is presented as an application example which is center frequency of 450 kHz and quality factor of 18.7. Simulation results are found in close agreement with the theoretical results.
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