High-quality notch filter on transistors. Encyclopedia of radio electronics and electrical engineering

Encyclopedia of radio electronics and electrical engineering / Computers

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The article discusses a simple high-quality narrow-band notch filter on transistors, which works perfectly in the frequency band up to 1 MHz and quite satisfactorily up to 10 MHz. Simple calculation formulas are derived for filter synthesis using the rejection frequency and bandwidth as initial values. Mathematical CAD Maple with the MathSpice extension package [2] and electronic CAD OrCAD [3] were used for calculations.

Analytical tasks are difficult to solve manually. The use of MSpice is a good helper here, sharply shifting the boundary of the complexity of the tasks being solved. It makes available to radio amateurs those tasks that were previously considered academic. The Maple extension package called MathSpice (MSpice) [2] is intended for the analytical solution of electronic circuits and functional diagrams, but can be used as a tool for creating Spice models of signals and electronic devices for various simulators. You can learn more about MSpice by reading "MathSpice - an analytical engine for OrCAD and MicroCAP", MODERN ELECTRONICS Magazine, STA-PRESS, No. 5, No. 6, No. 7, No. 9, No. 10, No. 11, No. 12 2009.

 In some devices in which we are used to seeing op-amps, it is quite possible to get by with transistors. The benefits of using an op-amp to amplify DC signals are undeniable. But on alternating current, the advantages of an op-amp are not as serious as those of a single transistor. An op amp with a unity gain frequency of more than 10 MHz is expensive, while a transistor with a unity gain frequency of up to (100 ... 1000) MHz costs a penny.

Analytical calculations of transistor devices are somewhat more complicated due to the more complex equivalent circuit of an idealized transistor compared to an idealized op-amp. However, at present, this problem is facilitated by the availability of computer calculations [1], [2].

Obviously, the transistor has a much smaller number of zeros and poles, and an extremely large gain per band product. Modern transistors have a large DC gain h21= 300..1000. In many cases this is sufficient.

Resistor-capacitor double T-shaped bridge filters are used as narrow-band notch filters (Fig. 1). Their main advantage lies in the possibility of deep suppression of individual frequency components.

In the frequency domain, well below the unity gain frequency, most parasitic parameters of transistors can be neglected. Therefore, the simplest transistor equivalent circuit shown in Fig. 2 was used for calculations. 1. It is based on a voltage controlled current source (IXNUMX). It is convenient to use it in the calculation of circuits by the method of nodal potentials.

Rice. 1. Scheme of a narrow-band notch filter at a frequency of 6,5 MHz

Compose the Kirchhoff equations for the filter circuit and solve it.

restart: with(MSpice): Devices:=[Same,[BJT,DC1,2]]:

ESolve(Q,`BJT-PSpiceFiles/SCHEMATIC1/SCHEMATIC1.net`):

Solutions

>MSpice v8.43: pspicelib.narod.ru

>Nodes given: {VINP, V12V} Sources: [Vin, VB1, Je]

>V_NET solutions: [V2, V5, V6, V1, V3, VOUT, V4]

>J_NET: [Je, JVin, JReb, JVB1, JR5, JC4, JR4, JR1, JC1, JR6, JR2, JR7, JR3, JC2, JC3, JFt, JJe, Jk, JT]

Find the transfer function of the filter. To simplify the formulas, we take into account that the following relations must hold for a filter with a Wien bridge:

C1:=C: C2:=C: C3:=2*C: R1:=R: R2:=R: R3:=R/2:

VB1:=0: # for linear PCB models

H:=simplify(VOUT/Vin);

(click to enlarge)

It's hard to work with this formula! Then suppose that = oo, C4=oo, R5=oo . Of course, it is somewhat rough to assume that the transistor has infinite gain, but for an emitter follower circuit it is quite appropriate. This allows you to get simple formulas for preliminary calculation. The exact formulas can be obtained using Maple, but they will be very difficult to evaluate the filter parameters (the formulas will take several pages). When setting up, the circuit parameters (quality factor) can be easily adjusted by selecting the resistor R6. After passing to the limit, we obtain a simpler expression for the operator transfer coefficient (1), which is more suitable for analysis.

beta:=x: C4:=x: R5:=x:

H:=collect(limit(H,x=infinity),s): 'H'=%, ` (1)`;

Now find the frequency domain gain, K=K(f), by substituting s=I*2*Pi*f .

Here I is the imaginary unit, f is the frequency [Hz].

K:=simplify(subs(s=I*2*Pi*f,H)): 'K(f)'=%, ` (2)`;

Let us find the rejection frequency (3).

Fp=I*solve(diff(K,f)=0,f)[2]: print(%,` (3)`);

It is convenient to adjust the notch frequency by choosing the resistor R=R1=R2=2*R3.

R:=solve(%,R): print('R'=R,` (4)`);

3 dB level notch

F_3dB:=solve(evalc(abs(K))=subs(f=0,K)/sqrt(2),f):

P:=simplify(F_3dB[4]-F_3dB[2]):

print('P'=P,` (5)`);

The quality factor is defined as Q=Fp/P, hence

Q:=Fp/P: 'Q'=Q,` (6)`;

Let's express the transfer function in terms of the characteristic parameters of the filter by substituting R7=4*Qp*R6-R6, C=1/(2*Pi*R*Fp).

It turns out a very convenient formula (7), which makes it possible to obtain the required Laplace rejector transfer function, without knowing anything about the filter device. Here Hp(s) is the notch operator transfer function, Fp is the rejection frequency, Qp is the quality factor of the notch.

Hp:=simplify(subs(R7=4*Qp*R6-R6,C=1/(2*Pi*R*Fp),H)): 'Hp(s)'=Hp;

Now let's find the modulus of the rejector function in the frequency domain (8).

abs(Kp(f)) = simplify(expand(AVM(Hp,f)),'symbolic'), ` (8)`:

abs(Kp(f)) = Qp*(f^2-Fp^2)/collect(Qp^2*f^4-2*Qp^2*f^2*Fp^2+Qp^2*Fp^4+Fp^2*f^2,f)^(1/2), ` (8)`:

abs(Kp(f)) = Qp*(f^2-Fp^2)/(Qp^2*f^4+collect(-2*Qp^2*Fp^2+Fp^2,Fp)*f^2+Qp^2*Fp^4)^(1/2), ` (8)`;

Kp:=Qp*(f^2-Fp^2)/collect(Qp^2*f^4-2*Qp^2*f^2*Fp^2+Qp^2*Fp^4+Fp^2*f^2,f)^(1/2):

 We have obtained a very convenient formula (8) for the synthesis of the rejector transfer function through the characteristic parameters of the filter. Ue can be used for digital prototypes, when programming filters on microcontrollers.

Example calculation

Suppose we need a filter that provides a rejection of the spectrum of the audio signal of television broadcasting with a center frequency Fp=6,5 MHz in the band P=1 MHz. We choose C=51 pF and, successively using formulas (4) and (6), we calculate the remaining components.

Fp:=6.5e6: R:=1e6: C := 51e-12;

Digits:=5: Q:='Fp/P'=Fp/P; Q:=Fp/P:

R:='1/(2*Pi*Fp*C)'=evalf(1/(2*Pi*Fp*C)); R:=rhs(%):

It is known that the amplifying properties of a transistor depend on the emitter current.

In the emitter follower circuit, the value of the emitter resistor of 1 kΩ will provide an operating current of the transistor of 6 mA at a supply voltage of 12V, which is sufficient to maintain high gain of the transistor at high frequencies.

Let's choose R6+R7=1 kΩ, then R6=(R6+R7)/4/Q=1K/4/Q, and R7=1K-R6.

R6:=1000.0/Q/4: print('R6'=R6); R7:=1000-R6: print('R7'=R7);

Let's plot the frequency response of the frequency gain module of our notch filter.

To do this, we use expression (8) for the transfer function module, substituting the calculated values ​​of the component ratings into it. The same values, rounded to the nearest integer, are indicated on the filter diagram (Fig. 1).

Values(AC,PRN,[]);Digits:=5:

Qp:= '1/4/R6*(R6+R7)'=evalf(1/4/R6*(R6+R7)); Qp:=rhs(%):

П:='4*R6*Fp/(R7+R6)'=evalf(4*R6*Fp/(R7+R6))*Unit([Hz]); П:=evalf(4*R6*Fp/(R7+R6)):

Fp:= '1/(2*Pi*C*R)'=evalf(1/(2*Pi*C*R))*Unit([Hz]); Fp:=evalf(1/(2*Pi*C*R)):

K:=simplify(expand(AVM(H,f))): print('abs(Kp(f))'=Kp); Digits:=10:

HSF([H],f=1e6..10e6,"3) semi[abs(Kp(f))]$500 notch filter |Kp(f)| ");

(click to enlarge)

Download: BJT Filter 6.5MHz

Literature

1. Petrakov O. M. . Analytical calculations in electronics. Magazine SCHEMOTEHNIKA №7, 2006
2. Petrakov O. M. A series of articles "MathSpice - an analytical engine for OrCAD and MicroCAP", MODERN ELECTRONICS Journal, STA-PRESS, No. 5, No. 6, No. 7, No. 9, No. 10 2009. .
3. Razevig VD Design system OrCAD 9.2. SOLON. Moscow 2001
4. Efimov I. P. Design of electronic filters: Guidelines for course design for students studying in the direction 5515.
5. Moshits G., Horn P. Design of active filters: Per. from English. Mir, 1984.- 320 p., ill.
6. Volovich G. I. Analog and digital devices. 2005
7. pspicelib.narod.ru Electronic CAD.
8. pspice.narod.ru Automation of analytical calculations.

Author: Oleg Petrakov, pspicelib@narod.ru; Publication: cxem.net

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