Noise transfer functions for second-order LP filter (*e
1. Introduction
The application of Matlab, combining its symbolic and numeric calculation capabilities, to calculate noise and sensitivity properties of allpole active-RC filters is shown. Transfer function coefficients calculations, as well as plotting of amplitude-frequency and phase-frequency characteristics (Bode plots) have been performed using Matlab. Thus, using Matlab a comparison of different design strategies of active-RC filters is done. It is shown that active-RC filters can be designed to have low sensitivity to passive components and at the same time possess low output thermal noise. The classical methods were used to determine output noise of the filters. It was found that low-sensitivity filters with minimum noise have reduced resistance levels, low Q-factors, low-noise operational amplifiers (opamps) and use impedance tapering design. The design procedure of low-noise and low-sensitivity, positive- and negative-feedback, second- and third-order low-pass (LP), high-pass (HP) and band-pass (BP) allpole filters, using impedance tapering, is presented. The optimum designs, regarding both performances of most useful filter sections are summarized (as a cookbook programmed in Matlab) and demonstrated on examples. The relationship between the low sensitivity and low output noise, that are the most important performance of active-RC filters, is investigated, and optimum designs that reduce both performances are presented.
A considerable improvement in sensitivity of single-amplifier active-RC allpole filters to passive circuit components is achieved using the design technique called 'impedance tapering' (Moschytz, 1999), and as shown in (Jurisic et al., 2010 a)at the same time they will have low output thermal noise. The improvement in noise and sensitivity comes free of charge, in that it requires simply the selection of appropriate component values. Preliminary results of the investigation of the relation between low sensitivity and low thermal noise performances using impedance tapering on the numeric basis using Matlab have been presented in (Jurisic & Moschytz, 2000; Jurisic, 2002).
For LP filters of second- and third-order the complete analytical proofs for noise properties of the desensitized filters are given in (Jurisic et al., 2010 a). By means of classical methods as in (Jurisic et al., 2010 a) closed-form expressions are derived in (Jurisic et al., 2010 c), providing insight into noise characteristics of the LP, HP and BP active-RC filters using different designs. LP, HP and BP, low-sensitivity and low-noise filter sections using positive and negative feedback, that have been considered in (Jurisic et al., 2010 c) are presented here. These filters are of low power because they use only one opamp per circuit. The design of optimal second- and third-order sections referred to as 'Biquads' and 'Bitriplets', regarding low passive and active sensitivities has been summarized in the table form as a cookbook in (Jurisic et al., 2010 b). For common filter types, such as Butterworth and Chebyshev, design tables with normalized component values for designing single-amplifier LP filters up to the sixth-order with low passive sensitivity to component tolerances have been presented in (Jurisic et al., 2008). The filter sections considered in (Jurisic et al., 2010 c) and repeated here have been recommended in (Moschytz & Horn, 1981; Jurisic et al., 2010 b) as high-quality filter sections. It was shown in (Jurisic & Moschytz, 2000; Jurisic, 2002; Jurisic et al., 2008, 2010 a, 2010b , 2010c), that both noise and sensitivity are directly proportional to the pole Q’s and, therefore, to the pass band ripple specified by the filter requirements. The smaller the required ripple, the lower the pole Q’s. Besides, it is wise to keep the filter order
2. A brief review of noise and sensitivity of active-RC filters
2.1. Output noise and dynamic range
Thermal (or Johnson) noise is a result of random fluctuations of voltages or currents that seriously limit the processing of signals by analog circuits. Because this noise is caused by random motion of free charges and is proportional to temperature, it is referred to as thermal noise (Jurisic et al., 2010 a).The most important sources of noise in active-RC filters are resistors and opamps. For the purpose of noise analysis, appropriate noise models for resistors and opamps must be used. Resistors are represented by the well-known Nyquist voltage or current noise models shown in Figure 1(a) and (b), consisting of noiseless resistors and noise sources whose values are defined by the squared noise voltage density within the narrow frequency band
or the squared noise current density given by
where
The noise defined by (Eq. 1) and (Eq. 2) has a constant spectrum over the frequency band, and is referred to as 'white noise'. The squared noise spectral density in (Eq. 1) has the dimension [V2/Hz], unless written
where
The dynamic range is defined by:
where
2.2. Sensitivity to passive component variations
Sensitivity analysis provides information on network changes caused by small deviations of passive component values. Given the network function F(s, x1,, x
where
If several components deviate from the nominal value, a criterion for assessing the deviation of the function F due to the change of several parameters must be used. With x
3. Application to second-order LP filter
3.1. Calculating transfer function coefficients and parameters using 'symbolic toolbox' in Matlab
Consider the second-order low-pass active-RC allpole filter circuit (Biquad) shown in Figure 2(a). This circuit belongs to the positive feedback or class-4 (Sallen and Key) filters (Moschytz & Horn, 1981). In Figure 2(b)there is a simplified version of the same circuit with the voltage-controlled voltage source (VCVS) having voltage gain . For an ideal opamp in the non-inverting mode it is given by
Note that the voltage gain of the class-4 circuit is positive and larger than or equal to unity. Voltage transfer function for the filters in Figure 2 expressed in terms of the coefficients a
and in terms of the pole frequency
where
To calculate the voltage transfer function T(s)=Vout(s)/Vg(s) of the Biquad in Figure 2(a), consider the following system of nodal equations (note that the last equation represents the opamp):
The system of Equation 12 can be solved using 'Symbolic toolbox' in Matlab. The following Matlab code solves the system of equations:Matlab command syms defines symbolic variables in Matlab's workspace:
syms A R1 R2 C1 C2 RF RG s Vg V1 V2 V3 V4 V5;
Matlab command solve is used to solve analytically above system of five Equation 12 for the five voltages V1 to V5 as unknowns. The unknowns are defined in the last row of command solve. Note that all variables used in solve are defined as symbolic.
CircuitEquations=solve(...
'V1=Vg',...
'-V1*1/R1 + V2*(1/R1+1/R2+s*C1)-V3*1/R2 - V5*s*C1=0',...
'-V2*1/R2 + V3*(1/R2+s*C2)=0',...
'V4*(1/RG+1/RF)-V5/RF=0',...
'(V3-V4)*A =V5',...
'V1','V2','V3','V4','V5');
Once all variables are known simple symbolic division of V5/V1 yields the desired transfer function (limit value for A∞ has to be applied, as well):
Tofs=CircuitEquations.V5/CircuitEquations.V1;
Tofsa=limit(Tofs,A,Inf);
Another way of presentation polynomials is by collecting all coefficients that multiply 's':
Tofsc=collect(Tofsa,s);
Transfer function coefficients and parameters readily follow.To obtain coefficients, it is useful to separate numerator and denominator using the following command:
[numTa,denTa]=numden(Tofsa);
syms a2 a1 a0 wp qp k;
denLP2=coeffs(denTa,s)/RG;
numLP2=coeffs(numTa,s)/RG;
Now coefficients follow
a0=denLP2(1)/denLP2(3);
a1=denLP2(2)/denLP2(3);
a2=denLP2(3)/denLP2(3);
And parameters
k=numLP2;
wp=sqrt(a0);
qp=wp/a1;
Typing command whos we obtain the following answer about variables in Matlab workspace:
>> whos
Name Size Bytes Class
A 1x1 126 sym object
C1 1x1 128 sym object
C2 1x1 128 sym object
CircuitEquations 1x1 2828 struct array
Tofs 1x1 496 sym object
Tofsa 1x1 252 sym object
Tofsc 1x1 248 sym object
R1 1x1 128 sym object
R2 1x1 128 sym object
RF 1x1 128 sym object
RG 1x1 128 sym object
V1 1x1 128 sym object
V2 1x1 128 sym object
V3 1x1 128 sym object
V4 1x1 128 sym object
V5 1x1 128 sym object
Vg 1x1 128 sym object
a0 1x1 150 sym object
a1 1x1 210 sym object
a2 1x1 126 sym object
denTa 1x1 232 sym object
denLP2 1x3 330 sym object
k 1x1 144 sym object
numTa 1x1 134 sym object
numLP2 1x1 144 sym object
qp 1x1 254 sym object
s 1x1 126 sym object
wp 1x1 166 sym object
Grand total is 1436 elements using 7502 bytes
It can be seen that all variables that are defined and calculated so far are of symbolic type.We can now check the values of the variables. For example we are interested in voltage transfer function Tofsa. Matlab gives the following answer, when we invoke the variable:
>> Tofsa
Tofsa =
(RF+RG)/(s*C2*R2*RG+R2*s^2*C1*R1*C2*RG-s*C1*R1*RF+RG+R1*s*C2*RG)
The command pretty presents the results in a more beautiful way.
>> pretty(Tofsa)
RF + RG
-------------------------------------------------------------
2
s C2 R2 RG + R2 s C1 R1 C2 RG - s C1 R1 RF + RG + R1 s C2 RG
Or we could invoke variable Tofsc (see above that Tofsc is the same as Tofsa, but with collected coefficients that multiply powers of 's').
>> pretty(Tofsc)
RF + RG
-----------------------------------------------------------
2
R2 s C1 R1 C2 RG + (C2 R2 RG - C1 R1 RF + R1 C2 RG) s + RG
Other variables follow using pretty command.
>> pretty(a0)
1
-----------
R2 C1 R1 C2
>> pretty(a1)
C2 R2 RG - C1 R1 RF + R1 C2 RG
------------------------------
RG R2 C1 R1 C2
>> pretty(a2)
1
>> pretty(wp)
/ 1 \1/2
|-----------|
\R2 C1 R1 C2/
>> pretty(qp)
/ 1 \1/2
|-----------| RG R2 C1 R1 C2
\R2 C1 R1 C2/
-------------------------------
C2 R2 RG - C1 R1 RF + R1 C2 RG
>> pretty(k)
RF + RG
-------
RGNext, according to simplified circuit in Figure 2(b) having the replacement of the gain element by defined in (Eq. 10), we can substitute values for R
>> syms beta
>> a1=subs(a1,RF,'(beta-1)*RG');
>> pretty(a1)
C2 R2 RG - C1 R1 (beta - 1) RG + R1 C2 RG
-----------------------------------------
RG R2 C1 R1 C2
Note that we have obtained R
>> pretty(simple(a1))
1 beta 1 1
----- - ----- + ----- + -----
C1 R1 R2 C2 R2 C2 R2 C1
>> pretty(simplify(a1))
-C2 R2 + C1 R1 beta - C1 R1 - R1 C2
- -----------------------------------
R2 C1 R1 C2
The final form of the coefficient a1 is the simplest one, and is the same as in (11c) above. Using the same Matlab procedures as presented above, we have calculated all coefficients and parameters of the different filters' transfer functions in this Chapter.If we want to calculate the numerical values of coefficients a
>> R1=1;R2=1;C1=0.5;C2=2;
>> a0val=subs(a0)
a0val =
1>> whos a0 a0val
Name Size Bytes Class
a0 1x1 150 sym object
a0val 1x1 8 double array
Grand total is 15 elements using 158 bytes
Note that the new variable a0val is of the double type and has numerical value equal to 1, whereas the symbolic variable a0 did not change its type. Numerical variables are of type double.
3.2. Drawing amplitude- and phase-frequency characteristics of transfer function using symbolic and numeric calculations in Matlab
Suppose we now want to plot Bode diagram of the transfer function, e.g. of the Tofsa, using the symbolic solutions already available (see above). We present the usage of the Matlab in numeric way, as well. Suppose we already have symbolic values in the Workspace such as:
>> pretty(Tofsa)
RF + RG
-------------------------------------------------------------
2
RG - C1 R1 RF s + C2 R1 RG s + C2 R2 RG s + C1 C2 R1 R2 RG s
Define set of element values (normalized):
>> R1=1;R2=1;C1=1;C2=1;RG=1;RF=1.8;
Now the variables representing elements R1, R2, C1, C2, R
>> Tofsa1=subs(Tofsa);
>> pretty(Tofsa1)
14
----------------
/ 2 s \
5 | s + - + 1 |
\ 5 /
Note that in new transfer function Tofsa1 an independent variable is symbolic variable s. To calculate the amplitude-frequency characteristic, i.e., the magnitude of the filter's voltage transfer function we first have to define frequency range of , as a vector of discrete values in wd, make substitution s=j into T(s) (in Matlab represented by Tofsa1) to obtain T(j), and finally calculate absolute value of the magnitude in dB by α(ω)=20 log |T(jω)|. The phase-frequency characteristic is φ(ω)=arg T(jω) and is calculated using atan2(). This can be performed in following sequence of commands:
wd = logspace(-1,1,200);
ad1 = subs(Tofsa1,s,i*wd);
Alphad=20*log10(abs(ad1));
semilogx(wd, Alphad, 'g-');
axis([wd(1) wd(end) -40 30]);
title('Amplitude Characteristic');
legend('Circuit 1 (normalized)');
xlabel('Frequency /rad/s');ylabel('Magnitude / dB');
grid;
Phid=180/pi*atan2(imag(ad1),real(ad1));
semilogx(wd, Phid, 'g-');
axis([wd(1) wd(end) -180 0]);
title('Phase Characteristic');
legend('Circuit 1 (normalized)');
xlabel('Frequency /rad/s');ylabel('Phase / deg');
grid;
Commands are self-explanatory. The amplitude- and phase-frequency characteristics thus obtained are shown in Figure 3. Note that we have generated vectors of values wd, Alphad and Phid to be plotted in logarithmic scale by the command semilogx (instead, we could have used command plot to generate linear axis).The next example defines new set of second-order LP filter element values (those are obtained when above normalized elements are denormalized to the frequency 0=286103 rad/s and impedance R0=37k; see in (Jurisic et al., 2008) how):
>> R1=37e3;R2=37e3;C1=50e-12;C2=50e-12;RG=1e4;RF=1.8e4;
Those element values were calculated starting from transfer function parameters
>> Tofsa2=subs(Tofsa);
>> pretty(Tofsa2)
28000
--------------------------------------------------------------------------------------
2
800318296602402496323046008438980478515625 s 4473025532574128109375 s
-------------------------------------------------- + ------------------------- + 10000
23384026197294446691258957323460528314494920687616 1208925819614629174706176
It is seen that the denormalized-transfer-function presentation in symbolic way is not very useful. It is possible rather to use numeric and vector presentation of the Tofsa2. First we have to separate numerator and denominator of Tofsa2 by typing:
>> [num2, den2]=numden(Tofsa2);
then we have to convert obtained symbolic data of num2 and den2 into vectors n2 and d2:
>> n2=sym2poly(num2)
n2 =
6.5475e+053
>> d2=sym2poly(den2)
d2 =
1.0e+053 *
0.0000 0.0000 2.3384
and finally use command tf to write transfer function which uses vectors with numeric values:
>> tf(n2,d2)
Transfer function:
6.548e053
---------------------------------------
8.003e041 s^2 + 8.652e046 s + 2.338e053
If we divide numerator and denominator by the coefficient of s2 in the denominator, i.e., d2(1), we have a more appropriate form:
>> tf(n2/d2(1),d2/d2(1))
Transfer function:
8.181e011
-----------------------------
s^2 + 1.081e005 s + 2.922e011
Obviously, the use of Matlab (numeric) vectors provides a more compact and useful representation of the denormalized transfer function.Finally, note that when several (N) filter sections are connected in a cascade, the overall transfer function of that cascade can be very simply calculated by symbolic multiplication of sections' transfer functions T
3.3. Calculating noise transfer function using symbolic calculations in Matlab
Using the noise models for the resistors and opamps from Figure 1, we obtain noise spot sources shown in Figure 4(a).
The noise transfer functions as in (Eq. 3) T
The system of Equation 13 can be solved using Matlab Symbolic toolbox in the same way as the system of Equation 12 presented above. The following Matlab code solves the system of Equation 13:
CircuitEquations=solve(...
'V1=0',...
'-V1*1/R1 + V2*(1/R1+1/R2+s*C1)-V3*1/R2 - V5*s*C1=InR1',...
'-V2*1/R2 + V3*(1/R2+s*C2)=0',...
'V4*(1/RG+1/RF)-V5/RF=0',...
'(V3-V4)*A =V5',...
'V1','V2','V3','V4','V5');
IR1ofs=CircuitEquations.V5/InR1;
IR1ofsa=limit(IR1ofs,A,Inf);
[numIR1a,denIR1a]=numden(IR1ofsa);
syms a2 a1 a0 b0
denIR1=coeffs(denIR1a,s)/RG;
numIR1=coeffs(numIR1a,s)/RG;
%Coefficients of the transfer function
a0=denIR1(1)/denIR1(3);
a1=denIR1(2)/denIR1(3);
a2=denIR1(3)/denIR1(3);
b0=numIR1/denIR1(3);
In Matlab workspace we can check the value of each coefficient calculated by above program, simply, by typing the corresponding variable. For example, we present the value of the coefficient b0 in the numerator by typing:
>> pretty(b0)
-- RF + RG --
| ----------- |
-- C1 C2 R2 RG --
The coefficients a0, a1 and a2 are the same as those of the voltage transfer function calculated in Section 3.1 above, which means that two transfer functions have the same denominator, i.e., D(s). Thus, the only useful data is the coefficient b0. The transfer resistance T
3.4. Drawing output noise spectral density of active-RC filters using numeric calculations in Matlab
Noise transfer functions for second-order LP filter, generated using Matlab in Section 3.3, are shown in Table 1. We can retype them and use Matlab in only numerical mode to calculate noise spectral density curves at the output, that are defined as a square root of (Eq. 3). Define set of element values (Circuit 1)
>> R1=37e3;R2=37e3;C1=50e-12;C2=50e-12;RG=1e4;RF=1.8e4;
We draw the curve:
% FREQUENCY RANGE
Nfreq=200;
Fstart=1e4; %Hz
Fstop=1e6; %Hz
fd =logspace(log10(Fstart),log10(Fstop),Nfreq);
% NOISE SOURCES at temperature T=295K (22 deg C)
IR1=sqrt(4*1.38e-23*295/R1);
IR2=sqrt(4*1.38e-23*295/R2);
IRF=sqrt(4*1.38e-23*295/RF);
IRG=sqrt(4*1.38e-23*295/RG);
EP=17e-9;
IP=0.01e-12;
IM=0.01E-12;
% TRANSFER FUNCTIONS OF EVERY NOISE SOURCE
D=1/(R1*R2*C1*C2) - (fd*2*pi).^2 +...
i*(fd*2*pi)*(1/(R1*C1)+1/(R2*C1)-RF/(R2*C2*RG));
H=(1/(R1*R2*C1*C2)*(1+RF/RG))./D;
numerator=(1/(R1*R2*C1*C2)*(1+RF/RG))*conj(D);
phase=atan(imag(numerator)./real(numerator));
TR1=(1/(R2*C1*C2)*(1+RF/RG))./D;
TR2=((1+RF/RG)*(1/(R1*C1*C2)+i*(fd*2*pi)*1/C2))./D;
TIP=((1+RF/RG)*(1/(R1*C1*C2)+1/(R2*C1*C2)+i*(fd*2*pi)*1/C2))./D;
TIM=-RF*(1/(R1*R2*C1*C2)-(fd*2*pi).^2 +...
i*(fd*2*pi)*(1/(R1*C1)+1/(R2*C1)+1/(R2*C2)))./D;
TRG=TIM;
TRF=TIM;
TEP=(1+RF/RG)*(1/(R1*R2*C1*C2)-...
(fd*2*pi).^2+i*(fd*2*pi)*(1/(R1*C1)+1/(R2*C1)+1/(R2*C2)))./D;
% SQUARES OF TRANS. FUNCTIONS
TR1A =(abs(TR1)).^2;
TR2A =(abs(TR2)).^2;
TIPA =(abs(TIP)).^2;
TIMA =(abs(TIM)).^2;
TRGA =TIMA;
TRFA =TIMA;
TEPA =(abs(TEP)).^2;
% SPECTRAL DENSITY OF EVERY NOISE SOURCE
UR1 =TR1A*IR1^2;
UR2 =TR2A*IR2^2;
UIP =TIPA*IP^2;
UIM =TIMA*IM^2;
UEP =TEPA*EP^2;
URG =TRGA*IRG^2;
URF =TRFA*IRF^2;
% OVERALL SPECTRAL DENSITY PLOT
U2=sqrt(UR1+UR2+UIP+UIM+URF+UEP+URG);
semilogx(fd,U2,'k-');
titletext=sprintf('Output Noise');title(titletext);
xlabel('Frequency / kHz');
ylabel('Noise Spectral Density / \muV/\surdHz');
axis ([fd(1) fd(Nfreq) 0 3e-6]); grid;
% Numerical integration of Total Noise Power at the Output (RMS)
Eno = sqrt(sum(U22(1:Nfreq))/(Nfreq-1)*(fd(Nfreq)-fd(1)));
To draw the second curve, apply the following method. Define the second set of element values, that are represented as example 4) ideally tapered filter (=4, and r=4), (see Equation 18and Table 3 in Section 4 below). We refer to those values as 'Circuit 2'.
>> R1=23.1e3;R2=92.4e3;C1=80e-12;C2=20e-12;RG=1e4;RF=1.05e4;
>> hold on;
>> redo all above equations; use 'r--' for the second curve shape
>> hold off;
>> legend('Circuit 1', 'Circuit 2');
Output noise spectral density is shown in Figure 5.Furthermore, two values of rms voltages E
3.5. Sensitivity characteristic of active-RC filter using both symbolic and numeric calculations in Matlab
To efficiently calculate multi-parametric sensitivity in (Eq. 9), we use a mixture of symbolic and numeric capabilities of Matlab.Suppose F in (Eq. 7)–( Eq. 9) is our transfer function T(s)=N(s)/D(s) defined by (Eq. 11), where x
To construct (Eq. 14), we proceed as follows. The following code reveal numerator and denominator as function of components. (Division of both numerator and denominator by R
>> den=simplify(denTa/RG);
>> pretty(den)
2 C1 R1 RF s
C2 R1 s + C2 R2 s + C1 C2 R1 R2 s - ---------- + 1
RG
>> denofw = subs(den,s,i*wd)
denofw =
C2*R2*wd*i - C1*C2*R1*R2*wd^2 + 1 - (C1*R1*RF*wd*i)/RG + C2*R1*wd*i
(To calculate all components and frequency values as real variables we have to retype real and imaginary parts of denofw.)
>> syms wd;
>> redenofw= - C1*C2*R1*R2*wd^2 + 1;
>> imdenofw= C2*R2*wd - (C1*R1*RF*wd)/RG + C2*R1*wd;
>> absden=sqrt(redenofw^2+imdenofw^2);
>> pretty(absden)
/ / C1 R1 RF wd \2 2 2 \1/2
| | C2 R1 wd + C2 R2 wd - ----------- | + (C1 C2 R1 R2 wd - 1) |
\ \ RG / /
>> SDR1=diff(absden,R1)*R1/absden;
>> pretty(SDR1)
/ / C1 RF wd \ / C1 R1 RF wd \ 2 2 \
R1 | 2 | C2 wd - -------- | | C2 R1 wd + C2 R2 wd - ----------- | + 2 C1 C2 R2 wd (C1 C2 R1 R2 wd - 1) |
\ \ RG / \ RG / /
----------------------------------------------------------------------------------------------------------
/ / C1 R1 RF wd \2 2 2 \
2 | | C2 R1 wd + C2 R2 wd - ----------- | + (C1 C2 R1 R2 wd - 1) |
\ \ RG / /
The same calculus (with simpler results) can be done for the numerator:
>> num=simplify(numTa/RG);
>> pretty(num)
RF
-- + 1
RG
>> numofw = subs(num,s,i*wd)
numofw =
RF/RG + 1
>> renumofw= RF/RG + 1;
>> imnumofw= 0;
>> absnum=sqrt(renumofw^2+imnumofw^2);
>> pretty(absnum)
/ / RF \2 \1/2
| | -- + 1 | |
\ \ RG / /
>> SNR1=diff(absnum,R1)*R1/absnum;
>> pretty(SNR1)
0
Sensitivity of the numerator to R1 is zero. We have obviously obtained too long result to be analyzed by observation. We continue to form sensitivities to all remaining components in symbolic form.
>> SDR2=diff(absden,R2)*R2/absden;
>> SDC1=diff(absden,C1)*C1/absden;
>> SDC2=diff(absden,C2)*C2/absden;
>> SDRF=diff(absden,RF)*RF/absden;
>> SDRG=diff(absden,RG)*RG/absden;
>> SNR2=diff(absnum,R2)*R2/absnum;
>> SNC1=diff(absnum,C1)*C1/absnum;
>> SNC2=diff(absnum,C2)*C2/absnum;
>> SNRF=diff(absnum,RF)*RF/absnum;
>> SNRG=diff(absnum,RG)*RG/absnum;
By application of rule (Eq. 14), we form sensitivities to each component, whose squares we finally have to sum, and form (Eq. 9).
>>SCH=(SNR1-SDR1)^2+(SNR2-SDR2)^2+(SNC1-SDC1)^2+(SNC2-SDC2)^2+...
(SNRF-SDRF)^2+(SNRG-SDRG)^2;
The resulting analytical form of multi-parametric sensitivity is as follows:
>> SigmaAlpha=sqrt(SCH)*0.01*8.68588964;
The multiplication by 0.01 defines the standard deviation of all passive elements
>> R1=37e3;R2=37e3;C1=50e-12;C2=50e-12;RG=1e4;RF=1.8e4;
By equating to values, elements changed in the workspace to double and they have become numeric. Substitute those elements into SigmaAlpha.
>> Schoefler1=subs(SigmaAlpha);
Note that in new variable Schoefler1 independent variable is symbolic wd. To calculate its magnitude, we have to define first the frequency range of , as a vector of discrete values in wd. When the frequency in Hz is defined, we have to multiply it by 2. The frequency assumed ranges from 10kHz to 1MHz.
>> fd = logspace(4,6,200);
>> wd = 2*pi*fd;
>> Sch1 = subs(Schoefler1,wd);
>> semilogx(fd, Sch1, 'g-.');
>> title('Multi-Parametric Sensitivity');
>> xlabel('Frequency / kHz'); ylabel('\sigma_{\alpha} / dB');
>> legend('Circuit 1');
>> axis([fd(1) fd(end) 0 2.5])
>> grid;
This is all needed to plot the sensitivity curve of Circuit 1.To add the second example, we set the element values of Circuit 2 in the Matlab workspace:
>> R1=23.1e3;R2=92.4e3;C1=80e-12;C2=20e-12;RG=1e4;RF=1.05e4;
Then we substitute symbolic elements (components) in the SigmaAlpha with the numeric values of components in the workspace to obtain new numeric vales for sensitivity
>> Schoefler2=subs(SigmaAlpha);
>> Sch2 = subs(Schoefler2,wd);
Finally, to draw both curves we type
>> semilogx(fd, Sch1, 'k-', fd, Sch2, 'r--');
>> title('Multi-Parametric Sensitivity');
>> xlabel('Frequency / kHz'); ylabel('\sigma_{\alpha} / dB');
>> legend('Circuit 1', 'Circuit 2');
>> axis([fd(1) fd(end) 0 2.5])
>> grid;
Sensitivity curves of Circuit 1 and Circuit 2 are shown in Figure 6. Recall that both circuits realize the same transfer-function magnitude which is shown in Figure 3a above. Note that only several lines of Matlab instructions have to be repeated, and none of large analytical expressions have to be retyped.In the following Chapter 4, we will use Matlab routines presented so far to construct examples of different filter designs. According to the results obtained from noise and sensitivity analyses we prove the optimum design.
4. Application to second- and third-order LP, BP, and HP filters
4.1. Second-order Biquads
Consider the second-order Biquads that realize LP, HP and BP transfer functions, shown in Figure 7. Those are the Biquads that are recommended as high-quality building blocks; see (Moschytz & Horn, 1981; Jurisic et al., 2010 b,2010c). In (Moschytz & Horn, 1981) only the design procedure for min. GSP is given (and by that providing the minimum active sensitivity design). On the basis of component ratios in the passive, frequency-dependent feedback network of the Biquads in Figure 7, defined by:
the detailed step-by-step design of those filters, in the form of cookbook, for optimum passive and active sensitivities as well as low noise is considered in (Jurisic et al., 2010 b2010c). The optimum design is presented in Table 1 in (Jurisic et al., 2010c) and is programmed using Matlab.Note that the Biquads in Figure 7 shown vertically are related by the complementary transformation, whereas those shown horizontally are RC–CR duals of each other. Thus, complementary circuits are LP (class-4: positive feedback) and BP-C (class-3: negative feedback), as well as HP (class-4) and BP-R (class-3). In class-4 case there is , whereas in class-3 there is
Dual Biquads in Figure 7 are LP and HP (class-4), as well as BP-C and BP-R (class-3); they belong to the same class.
Voltage transfer functions for all the filters shown in Figure 7 in terms of the pole frequency
where numerators n(s) are given by:
Parameters
(a) LP and (c) BP-C | (b) HP and (d) BP-R |
No. | Filter\Design Parameter | |||||||||||
1 | Non Tapered | 1 | 1 | 0.333 | 2.8 | 50 | 50 | 100 | 37 | 37 | 74 | 176.0 |
2 | Capacitively Tapered | 1 | 4 | 0.333 | 1.4 | 80 | 20 | 100 | 46.3 | 46.3 | 92.5 | 102.5 |
3 | Resistively Tapered | 4 | 1 | 0.333 | 5.6 | 50 | 50 | 100 | 18.5 | 74 | 92.5 | 360.9 |
4 | Ideally Tapered | 4 | 4 | 0.444 | 2.05 | 80 | 20 | 100 | 23.1 | 92.5 | 115.6 | 127.7 |
5 | Cap-Taper and min. GSP | 1.85 | 4 | 0.397 | 1.58 | 80 | 20 | 100 | 34.02 | 62.9 | 96.94 | 103.9 |
No. | Filter\Design Parameter | |||||||||||
1 | Non Tapered | 1 | 1 | 0.333 | 2.8 | 50 | 50 | 100 | 37 | 37 | 74 | 201.6 |
2 | Capacitively Tapered | 1 | 4 | 0.333 | 5.6 | 80 | 20 | 100 | 46.3 | 46.3 | 92.5 | 460.1 |
3 | Resistively Tapered | 4 | 1 | 0.333 | 1.4 | 50 | 50 | 100 | 18.5 | 74 | 92.5 | 96.73 |
4 | Ideally Tapered | 4 | 4 | 0.444 | 2.05 | 80 | 20 | 100 | 23.1 | 92.5 | 115.6 | 137.0 |
5 | Res-Taper and min. GSP | 4 | 1.85 | 0.397 | 1.58 | 65 | 35 | 100 | 19.4 | 77.6 | 97.0 | 100.3 |
On the other hand, two 'dual' circuits will have dual sensitivities and dual optimum designs. Dual means that the roles of resistor ratios are interchanged by the corresponding capacitor ratios, and vice versa. It is shown in (Jurisic et al., 2010 c) that complementary Biquads have identical noise transfer functions and, therefore, the same output noise. An optimization of both sensitivity and noise performance is possible by varying the general impedance tapering factors (Eq. 15) of the resistors and capacitors in the passive-RC network of the filters in Figure 7, see (Moschytz, 1999; Jurisic et al., 2010 b). By increasing r>1 and/or >1, the R2 and C2 impedances are increased. High-impedance sections are surrounded by dashed rectangles in Figure 7.For illustration, let us consider the following practical design example as one in (Moschytz, 1999):
As is shown in (Moschytz, 1999), there are various ways of impedance tapering a circuit. By application of various impedance scaling factors in (Eq. 15), the resulting component values of the different types of tapered LP (and BP-C) circuits are listed in Table 3, and the components of HP (and BP-R) filters are listed in Table 4. The corresponding transfer function magnitudes are shown in Figure 8 using Matlab (see Section 3.2). In order to compare the different circuits with regard to their noise performance, the total capacitance for each is held constant, i.e. CTOT=100pF.A multi-parametric sensitivity analysis was performed using Matlab (see Section 3.5) on the filter examples in Table 3 and 4 with the resistor and capacitor values assumed to be uncorrelated random variables, with zero-mean and 1% standard deviation. The standard deviation
The output noise spectral density e
Analysis of the results in Figure 10(b) and the E
4.2. Third-order Bitriplets
The extension to third-order filter sections follows precisely the same principles as those above. Unlike with second-order filters, third-order filters cannot be ideally tapered; instead only capacitive or resistive tapering is possible (Moschytz, 1999).Let us consider the third-order filter sections (Bitriplets) that realize LP and HP transfer functions, shown in Figure 11. Optimum design of those filters for low passive and active sensitivities, as well as low noise, is given in (Jurisic et al., 2010 b, Jurisic et al., 2010c). The optimum design is presented in Table 6 in (Jurisic et al., 2010 c)and is programmed using Matlab. In (Jurisic et al., 2010 a2010c), the detailed noise analysis on the analytical basis is given for the third-order LP and the (dual) HP circuits in Figure 11. Both sensitivity and noise analysis are performed using Matlab routines in Section 3. Voltage transfer functions for the filters in Figure 11are given by:
where numerators n(s) are given by:
Coefficients a
Coefficient | (a) LP |
Coefficient | (b) HP |
The quantity referred to as 'design frequency' is defined by 0=1/(RC) (Moschytz, 1999).The third-order LP and HP filters with the minimum sensitivity to component tolerances as well as the lowest output noise and maximum dynamic range are the circuits designed in the optimum way as presented in Table 6 in (Jurisic et al., 2010 c). The LP filter circuit was designed by capacitive impedance tapering with 2=, 3=2; >1 and 0 chosen to provide r2r3. In the case of the third-order HP filter, the optimum design is dual: circuit has to be designed by resistive impedance tapering with r2=r, r3=r2; r>1 and 0 chosen to provide 23. Thus, the minimum-noise and minimum-sensitivity designs coincide. Comparing the output noise of two third-order dual circuits we see again that HP filter has larger noise than LP filter, c)although their sensitivities are identical, see (Jurisic et al., 2010 .
5. Conclusion
In this paper the application of Matlab analysis of active-RC filters performed regarding noise and sensitivity to component tolerances performance is demonstrated. All Matlab routines used in the analysis are presented. It is shown in (Jurisic et al., 2010 c) and repeated here that LP, BP and HP allpole active-RC filters of second- and third-order that are designed in (Jurisic et al., 2010 b) for minimum sensitivity to component tolerances, are also superior in terms of low output thermal noise when compared with standard designs. The filters are of low power because they use only one opamp. What is shown here is that the second-order, allpole, single-amplifier LP/HP filters with positive feedback, designed using capacitive/resistive impedance tapering in order to minimize sensitivity to component tolerances, also posses the minimum output thermal noise. The second-order BP-C filter with negative feedback is recommended filter block when the low noise is required. The same is shown for low-sensitivity, third-order, LP and HP filters of the same topology. Using low-noise opamps and metal-film small-valued resistors together with the proposed design method, low-sensitivity and low-noise filters result simultaneously. The mechanism by which the sensitivity to component tolerances of the LP, HP and BP allpole active-RC filters is reduced, also efficiently reduces the total noise at the filter output. Designs are presented in the form of optimum design tables programmed in Matlab [see Tables 1 and 6 in (Jurisic et al., 2010 c)].All curves are constructed by the presented Matlab code, and all calculations have been performed using Matlab.
References
- 1.
Jurisic D. Moschytz G. S. 2000 Low Noise Active-RC Low-, High- and Band-pass Allpole Filters Using Impedance Tapering. , Lemesos, Cyprus, (May 29-31, 2000.),591 594 - 2.
Zagreb, Croatia: Ph. D. Thesis, University of Zagreb, April 2002.Jurisic D. 2002 - 3.
Jurisic D. Moschytz G. S. Mijat N. 2008 Low-Sensitivity, Single-Amplifier, Active-RC Allpole Filters Using Tables. ,49 3-4 Nov. 2008),159 173 0005-1144 - 4.
Jurisic D. Moschytz G. S. Mijat N. 2010 Low-Noise, Low-Sensitivity, Active-RC Allpole Filters Using Impedance Tapering. , doi:cta.740,0098-9886 0098 9886 - 5.
Jurisic D. Moschytz G. S. Mijat N. 2010 Low-Sensitivity Active-RC Allpole Filters Using Optimized Biquads. ,51 1 Mar. 2010),55 70 0005-1144 - 6.
Jurisic D. Moschytz G. S. Mijat N. 2010 Low-Noise Active-RC Allpole Filters Using Optimized Biquads. ,51 4 Dec. 2010),361 373 0005-1144 - 7.
Laker K. R. Ghausi M. S. 1975 Statistical Multiparameter Sensitivity-A Valuable Measure for CAD. , April 1975.,333 336 - 8.
Moschytz G. S. Horn P. 1981 . John Wiley and Sons,978-0-47127-850-4 Chichester, UK.: 1981, (IBM Progr. Disk: ISBN 0471-915 43 2) - 9.
Moschytz G. S. 1999 Low-Sensitivity, Low-Power, Active-RC Allpole Filters Using Impedance Tapering. ,CAS-46 8 Aug 1999),1009 1026 1057-7130 - 10.
Schaumann R. Ghausi M. S. Laker K. R. 1990 , Prentice Hall,978-0-13200-288-2 New Jersey 1990 - 11.
Schoeffler J. D. 1964 The Synthesis of Minimum Sensitivity Networks.,11 2 June. 1964),271 276 0018-9324