Performance Analysis of Tunable Frequency Sinusoidal Oscillator Employing CCII+ Based AD844

Performance Analysis of Tunable Frequency Sinusoidal Oscillator Employing CCII+ Based AD844

Abstract

This paper presents realization of canonic tunable fre-

0 1 0

quency sinusoidal oscillator and its physical implemen-

= 1 0 0

tation using commercially available IC AD844. These oscillators employ second generation current conveyor as there functional block. State variable equation matrix are used to determine values of passive elements and

0 ±1 0

give tuning law for oscillation. THD and DC component of generated waveform are used to determine the quality of sinusoidal wave.

1. Introduction

   Single element controlled, specifically a resistor or in some cases a grounded capacitor, oscillator finds appli- cation in numerous of measurement and instrumenta- tion. Such oscillators are also used to generate low fre- quency sinusoidal signal.

   Current feedback operational Amplifiers (CFOAs) such as AD844 having four terminals, particularly the com- pensation terminal, have become more popular than traditional voltage mode op-amps (VOA). Advantages provided are such as nearly constant bandwidth inde- pendent of gain, higher slew rate, ease of designing cir- cuits with generalized second order deferential equation for oscillators, low distortion level and least no. of ex- ternal elements is attracting prominent circuit designers. The CFOAs used has a terminal characteristics that of second generation current conveyor CCII+ [1-6] charac- terized by hybrid matrix

   Where x, y are input terminal and z is output terminal.

   	CCII+	Z
   Y+

   X-

   Fig.1. Second Generation Current Conveyor.

   Although a no. of CFOAs based sinusoidal oscillators have been evolved[8-11,13-18] none of them employ both grounded resistors as well as capacitor. Here pro- vided two CFOAs, condition for sinusoidal oscillators are characterized using state variable equation [11], in such a way that there is non-interacting control between condition of oscillation (CO) and frequency of oscilla- tion(FO). The second order generalized state variable equation can be stated using eq. (1). Characteristics equ- ation (CE) for oscillators is equal to zero so from (1) and (2) [i.e. for loop gain L(s)] we have eqn. (3).

   1 = 11 12 1 (1)

   2

   21 22 2

   1-L(s) = 0 (2)

   2 11 + 22 + 11221221 = 0

   (3)

   From (3) condition of oscillation and frequency of oscilla- tion can be given by (4) and (5)

   (11 + 22) = 0 (4)

   Circuit 1:-

   Assuming the matrix [A] as

   1

   1

   [A1] = 1.3

   2.1

   1

   1.3

   1

   2.2

   Or 11 = 22

   = 1122 1221 (5)

   Now from eqn. (4) and eqn. (5) we have condition of os- cillation (CO) and FO are given by

   The proposed methodology involved selecting the pa-

   1

   1.3

   1

   2.2

   = 0 (7)

   rameter where i = 1, 2 and j = 1, 2, inaccordance with the required features, converting [A] matrices into node equations, and, finally, synthesizing the resulting node equations by physical circuits using CFOA and RC ele- ment.

2. Configuration of Oscillators

   Most of the SRCOs are based on the tuning law giv-

   1 = 1

   1.3 2.2

   3 = 2

   2 1

   Frequency is given by

   0

   = 1

   13

   . 1

   22

   + 1

   13

   . 1

   21

   (8)

   (9)

   en by eq. (6-7)

   CO: R1= R3 (6)

   1

   1

   = 2

   1213

   (10)

   FO: 0

   = 1

   1.2.2.3

   (7)

   1 < 1 (From sensitivity calculation)

   2

   Characteristic eqn. for the above state variable equation

   From equation (6), (7) condition of oscillation is controlled by R1 and that of frequency of oscillation

   can be stated as

   S2 S 1 1 + 1 . 1 1 . 1 = 0

   by R3 respectively. However oscillators presented in this paper have been configured with a novel ap- proach as of given by (6)& (7). This approach re-

   13

   22

   13

   22

   13

   21

   (11)

   quires following steps-

   Nodal eqn. for the state variable equation can be stated as

   1. Determining the elements of [A] matrices.

      1 1 = 12 = 1 2

      (12)

   2. Converting [A] matrices in node equation.

      3

      3

      3

   3. Synthesizing the node equation into physical

      2 2 = 1 2 (13)

      circuits using corresponding elements and

      1

      2

      CFOAs

      Here we have discussed two circuits with different arrangement of elements of matrix [A], hence giving out different circuit configuration.

      Here, X1 and X2 are the voltages across C1 and C2

      1 , 2 , 1 , 2 are the branch currents. Realizing the corre-

      3 3 1 2

      sponding circuit using CCII+ port characteristic Fig 2 gives the equivalent circuit diagram.

      Passive Sensitivity:Frequency sensitivity of the oscil- lator is defined as deflection produced in the operating frequency due to change in any of the passive components and is given as:

      Circuit 2:

      In this circuit 5 passive elements are used. The state vari- able matrix is given as

      =

      (14)

      1 ( 1 1 ) 1

      [A2] = 1

      2

      3

      13 (19)

      1 ( 1 1 ) 1

      2 3 1 23

      CE can be stated as

      S2 1 1 1 1

      + 1 1 1 . 1 +

      1

      2

      3

      23

      1

      2

      3

      23

      1 . 1 1 1 = 0 (20)

      13 2 3 1

      2 1 1 1 1

      + 1

      1 + 1 1

      1

      2

      3

      23

      123

      2

      3

      3

      1 = 0 (21)

      1

      From above eqn. (21) FO and CO can be stated as

      CO: 1 1 1 1

      = 0 (22)

      1

      2

      3

      2.3

      1 1 1 = 1

      (23)

      1

      2

      3

      23

      Fig 2: Circuit-1corresponding to eq. (12) and (13).

      3 1 1 = 1

      (24)

      where, Rx is any passive component on which frequency

      2

      3

      2

      depends.Frequency sensitivity of different component is

      3 = 1 + 1

      (25)

      0 =

      found to be as follows

      2

      2

      1 = 2 = 3 = 1

      (15)

      FO: 1

      1 + 1 (26)

      2

      1.2.3

      2

      1

      For R1, R2 sensitivity is found to be

      1 = 1 1

      (16)

      0 2

      1 1

      =

      1.2.3.1

      (27)

      1

      2 1

      2

      Passive sensitivity:

      Hence for system to be stable -1 < < 1 , therefor1 <

      1

      2

      3 1

      for1 < 1

      2

      Similarly for 2 = 1 . 1

      (17)

      2

      0 = 0 = 0 = 2 (28)

      As frequency of oscillation is given by same expression so sensitivity of different passive elements is same as that

      1

      2 2

      1

      of found in circuit 1

      If R1 and R2 are taken in such a way that 2 = then

      2 = . 1

      frequency stability factor is stated as:

      1

      1

      0 2

      1 = 1

      2

      1

      1

      (29)

      (30)

      SF =-2 1 ( 1) (18)

      0

      1

      2 1

      2

      = -2 ; for n >> 1

      Hence for system to be stable -1 < < 1 , therefor1 <

      ble 1. The values are selected in accordance to eq. (9),

      0

      1

      2

      (10) for circuit 1and eq. (25), (27) for circuit 2. The slight variation is for initialization of oscillation. Fig.4 gives the

      Now taking 3 = 2 the stability of the circuit can be stated

      2

      as

      schematic generated in SPICE for circuit 1 for which gen- erated sinusoidal output waveform is given by in Fig 5. To detect the harmonics generated Fourier transform for

      the generated signal is taken which indicates a very nar-

      SF = 8 1

      2 1

      = 2 ; 1

      (31)

      row spectrum given by Fig 6. Table 2 and 3 shows the normalized component, Fourier component and normal- ized phase component for circuit 1 and circuit2 respec- tively. Overall deviation from the main frequency compo-

      The node equation corresponding to state variable matrix can be stated as

      C1 1 = 1 1+2 (32)

      nent is given by THD and the DC component which are quit low as compared to conventional oscillators.

      Table 1 : Values of elements used in

      2

      3

      simulation

      PARAMETERS	Values
      	Circuit 1	Circuit 2
      VDD	5V	5V
      VSS	-5V	-5V
      R1	3.2K	4K
      R2	11.6K	4.9K
      R3	10K	10K
      RL	100k	100K
      C1	1E-9 F	1nF
      C2	1E-9 F	1Nf

      C2 2 = 1 + 12 (33)

      1

      3

      Realizing the circuit corresponding node equations with x1 and x2 as node voltage across capacitor C1 and C2, is shown in fig 3

      Fig 3: Circuit-2 corresponding to eq. (32) and (33).

      3 Performance Analysis and Experimental Re- sults

      The above synthesized circuit have been realized using AD844 ICs model files in SPICE. The value of circuit elements used for each of the two circuits is listed in Ta-

      Fig 4: Sinusoidal waveform generated for circuit 1

      Fig : 5 FFT for Circuit-1

      Fig :6 Variation of frequency f0 with R1

      Table 2: Fourier component for the transient response of circuit-1

      HAR- MONICNO	FRE- QUENCY(HZ)	FOURIERCOM- PONENT	NORMALIZED- COMPONENT	PHASE(D EG)	NORMALIZED- PHASE (DEG)
      1.	2.200E+04	7.969E-01	1.000E+00	4.743E+01	0.000E+00
      2.	4.400E+04	1.876E-02	2.354E-02	-1.541E+02	-2.490E+02
      3.	6.680E+04	2.926E-03	3.671E-03	-6.298E+01	-2.053E+02
      4.	8.800E+04	8.019E-03	1.006E-02	-6.298E+01	-3.546E+02

      Table 3: Fourier component for the transient response of circuit-2

      HARMONICNO	FREQUENCY(HZ)	FOURIER COMPONENT	NORMALIZED COMPONENT	PHASE (DEG)	NORMALIZEDPHASE (DEG)
      1.	1.033E+04	1.359E-03	1.000E+00	8.792E+01	0.000E+00
      2.	2.066E+04	1.355E-03	9.964E-01	8.584E+01	-9.000E+01
      3.	3.099E+04	1.346E-03	9.904E-01	8.377E+01	-1.800E+02
      4.	4.132E+04	1.335E-03	9.820E-01	8.171E+01	-2.700E+02

   4. Practical feasibility and significance

Both of the discussed oscillators contain a difference term in the expression for their FO which could be generalized to give eq. (34)

0

= 1

(34)

Where is the ratio of frequency controlling resistor and thus qualify to be used for generating very low frequency oscillations. These oscillators could be practically imple- mented using commercially available AD844 ICs which nearly accurate and stable frequency output as shown in fig 6 for circuit 1 the output of sinusoidal waveform from a Digital oscilloscope.

Table 4: Output parameters for circuit 1 and 2.

5.Conclusion

Two new sinusoidal circuits employing CFOAs have been analysed each having grounded capacitor. The circuits have been derived by employing new tuning laws. Several others laws may be undertaken to real- ize sinusoidal oscillators. More over these circuits

Fig 7: Sinusoidal Waveform from DSO for circuit-1

can be physically realized with bare minimum four passive elements and are suitable for is generation of VLF oscillation. Output waveform for such a circuit is shown in fig 7. It believed that aforementioned oscillators would found several applications in fur- ther to those discussed here.

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