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Multi-frequency Functional Generator
Section 8. Technical sciences
Dubrovin Viktor Stepanovich, Ogarev Mordovia State University, Candidate of Engineering Sciences, Associate Professor at the Department of communication networks and relay systems
E-mail: [email protected]
Zyuzin Alexey Mikhailovich, Non-state educational institution of supplementary professional education «Saransk House of Science and Technology of the Russian Union of scientific and engineering public organizations», Director
E-mail: [email protected]
Multi-frequency Functional Generator
Abstract: the author describes a multi-frequency functional generator, which can be used to form the grid of frequency in communications equipment, measure and compute devices, as well as to generate quadrature signals of several harmonic frequencies and signals of different forms of the same frequency.
Keywords: functional generator, block diagram, quadrature signals, frequency synthesizer, error signal.
Introduction
Virtual laboratory works (VLWs) are widely applied during the courses «General theory of network», «Digital processing of signals» and «Circuit engineering of telecommunication devices» at the Department of Info-communication technologies and network systems of Ogarev Mordovia State University. The main advantages of VLWs include [1]:
• virtual laboratory works are safer than bench laboratory works;
• virtual works ensure universality and multi-functionality as well as flexibility and easy adaptation to different object;
• there is an opportunity to conduct an experiment, which is not possible in ordinary conditions, or its conduct can be time-consuming or costly;
• reduction of costs for creation of laboratory works allows expanding their base within a short span of time and ensuring a big flexibility in learning.
It should be noted that the use of virtual laboratory works in the educational process can significantly improve its quality, but it is not recommended to replace bench laboratory works (BLWs) completely.
A lot of measuring equipment is required in the multiphase systems of converter equipment, quadrature modulators-demodulators of network systems, frequency multipliers with phase-locked loop systems and other similar devices for the synthesis of complex oscillatory mode and creation of text signals.
If, in case of the usage of VLWs, there are usually no problems with nomenclature and number of used blocks
and devices, in case of the usage of BLWs these issues come to the fore. For instance, the synthesis of complex signals requires simultaneous application of a big number of generators for creation of harmonic signals of different frequencies. As a result, the cost of the equipment required for BLWs will rise sharply.
The ways of composition of functional generators are described in [2; 3], and separate solutions are described in [4-6]. The task lies in the composition of a quite simple multi-frequency functional generator (MFFG), which can be used to form frequency grids as well as quadrature harmonic signals of several frequencies and signals of different forms of same frequency.
Main part
Structural scheme of a multi-frequency functional generator is depicted in Fig.1.
MFFG includes: a regulated source of quadrature harmonic signals (SQS); synthesizer of frequency (SF); two module computers (MC-1 and MC-2); phase modulator (PM) that includes control pulse generator (G-1) and commutator (C); bipolar rectangular pulse generator (G-2) and two summers.
To form quadrature signals Vt(t) and V2(t), one can use either quadrature signals generator (QSG) [7-9], or different phase-shift networks [10], on the basis of which different quadrature signal generators are built (QSG) [11-14].
To form a functional generator, the works [15-21] suggest using additive signal generator of triangular shape. In the generator (Fig. 1), the additive generator is made of two module computers MC-1, MC-2 and the summer.
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Section 8. Technical sciences
Fig. 1. Structural scheme of a functional generator
Multi-frequency generator forks as follows. V2(t) = A ■ cos(®0t),
When control energy Ec is supplied to the inlet of the where A — is an amplitude, a ®0 — is a circular frequency source of quadrature signals, harmonic signals shifted relative of signals V, (t) and V2(t).
to one another by 90 electric degrees are set at its outlets after Module computer MC-2 is inverting, hence, signals are
the end of transition process. formed (Fig. 2) at the outlets of MC-1 and MC-2.
V2(t) = A■ sin(rn0t), Ml(t) = mod[V,(t)] and M2(t) = -mod[V2(t)].
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Multi-frequency Functional Generator
As a result of summarizing of signals Ml(t) and M2(t), a synthesized signal of quasitriangular form is formed:
Sint(t) = k • mod[V1(t)]-k2 • mod[V2(t)], where k and k2 — are coefficients of transfer of the first summer through the first and second inlets respectively.
At к = k2 = 1 the amplitude of the signal Ssinl(t) will be equal to the amplitude value A of the signals V2(t)andV2(t).
In Fig.2 the diagrams are constructed for the normalized value A = 1. The value of the actual angles x = rn0t is expressed in radians. The period T0 of the fundamental harmonics is determined with the frequency®,,:
T0 = 1/ f0 = 2n / ®0 ,
consequently, the frequency of the fundamental harmonics П0 of the synthesized signal of triangular shape Ssint (t) is equal to the doubled value of frequency ®0 of quadrature signals V2(t) and V2(t):
Q0 = Щ.
At the sites of «forward line» (from zero to n/2) and «reverse line» (from n/2 to n) the signal Ssinl(t) has S - like
characteristics, i. e. it is «quasilinear». To evaluate non-linear nature of the synthesized signal Ssint (t), let’s calculate the value of error e 0(t), i. e. deviation of the synthesized signal Ssinl(t) from the reference signal Set(t):
e 0(t) = Set(t) - Ss,nt(t) .
The diagram of dependence of e0(t) from the actual angle value x is shown in Fig.2. The maximal deviation of e0(t) in module exceeds 4 % (E0max = 42,5 mV. under normalized value of the amplitude A = 1000 mV).
The works [15, 17, 20-23] suggest various ways of linearization of the synthesized signal of triangular shape with the help of a correction signal.
If the correction signal Sk (t) that exactly coincides in form and value with error signal e 0 (t) is formed, and then one adds it up to the signal Ssint(t) in the reversed phase, the result signal N2(t) will be equal to the reference signal Sel(t). The task of formation of the correction signal Sk (t) is performed by a synthesizer of frequency and phase modulator (Fig.2). The synthesizer of frequency includes five frequency doublers FD-1... FD-5 (Fig.3).
At the outlet of the first frequency doubler, the signal is formed:
S,(t) = m, ■V1(t)-V2(t) = (1)
= m 1-sin(®0t)• cos(a0t) = (m, /2)• sin(2®0t), where m2 — is a scale coefficient of FD-1.
At m2 = 2 at the outlet of FD-1, the frequency signal S:(t) = sin(2a>0t) = sin(Q0t) will be formed (Fig.4) at the first outlet, the frequency ofwhich H0will be equal to the frequency of the first harmonics of the synthesized signal 5sint (t).
At the outlet of the fourth frequency doubler FD-4, the signal is formed:
S4(t) = [V2(t )f-[V1(t)] = cos2Kf) - sin2 (^t) = (2) = cos(2®0t) = cos(Q0t),
which is supplied (Fig.4) to the respective outlet of the frequency synthesizer.
The second frequency doubler operates according to the algorithm specified in the equation (1):
S2(t) = m2 • S2(t) • S2(t) = m 2-sm(Q0f) • cos(Q0f) =
= (m2 / 2) • sin(2Q0t).
At m2 = 2 at the outlet of FD-2, the frequency signal S2(t) = sin(2Q0t) = sin(4®0t )will be formed (Fig.4) at the second outlet, the frequency of which will exceed the frequency of the first harmonics of the synthesized signal Ssint (t) by two times.
The frequency doubler FD-5 operates according to the algorithm specified in the equation (2):
S5(t) = [St(t)] -[S2(t)] = cos2 (Q0t) - sin2 (Q0t) = (4)
= cos(2Q0t) = cos(4®0t).
At the outlet of the fourth frequency doubler FD-3, the signal is formed:
S3(t) = m3 • S2(t)• S5(t) = m3-sin(2Q0t)• cos(2Q0t) =
= (m3/2) • sin(4Q0t). (5)
At m2 = 2 at the outlet of FD-3, the frequency signal S3(t) = sin(4Q0t) = sin(8®0t) will be formed (Fig.4) at the third outlet, the frequency of which will exceed the frequency ®0 of quadrature signals V2(t) and V2(t) by eight times.
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Section 8. Technical sciences
As a result of summarizing the signals S2(t) and S3(t) at the outlet of the second summer, the signal is formed (Fig.4): S0(t) = k ■ S2(t) + k ■ S5(t), (6)
where k4 and k5 — are coefficients of transfer of the second summer through the first and second inlets respectively.
The coefficient of transfer k4 is selected to ensure equation k4 ■ A = E0max; herewith, the coefficient of transfer k5 shall be approximately seven times smaller than the coefficient k4.
Control pulse generator G-1 under the effect of the signal S2(t) produces (Fig.4) the signal of rectangular form Sy(t) affecting the control inlet of the commutator K, at the outlet of which the correction signal Sk (t) supplied to the third inlet of the first summer is generated.
The correction signal Sk(t) is almost same as the error signal e 0(t); hence, «quasilinear» signal of triangular form N2(t) with insignificant deviation from the reference signal Set(t) is formed at the outlet of the first summer.
The qualitative evaluation of the residual error of linearization can be calculated with the help of the equation e2(t) = Set (t) - N(t), where N(t) — is the signal of triangular form obtained in the result of correction.
Results of the analytical calculations and mathematical modeling (in program PSIM-9) on determination of the residual error are shown in Fig.5, whence it follows that at correction the value of the residual error e2(t) reduced by approximately 24 times and accounted for 0,18 % of the normalized value of the amplitude A .
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Multi-frequency Functional Generator
Fig. 6. depicts outlet signals of multi-frequency func- are formed; at the third outlet (Out3) and the sixths tional generator. outlet (Out6) — quadrature harmonic signals Sl(t) and
At the first outlet of the generator (Out1) signals of S4(t) are formed, the frequencies of which are two times
triangular form Nl(t) are formed; at the second outlet higher than the main frequency of inlet signals V1(t)and
(Out2) — the signals of bipolar rectangular form N2(t) V2(t).
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Section 8. Technical sciences
The quadrature harmonic signals S2(t) and S5(t) are formed at the respective outlets (Out4) and (Out7); herewith, their frequency exceeds the main frequency a0 by four times.
At the fifth outlet of the generator (Out5) the harmonic signal S3(t) is formed, the frequency of which is eight times higher than the main frequency of inlet signals.
Quadrature harmonic signals V2(t) and V2(t) are supplied directly to the eights (Out 8) and ninths (Out 9) outlets of MFFG from the outlet of SQS.
Conclusion
1. Multi-frequency functional generator allows obtaining quadrature harmonic signals with frequencies in correlation 1:2:4, and also a harmonic signal the frequency of which exceeds the main frequency by eight times.
2. Implementation of correction block enabled to significantly improve the linear nature of a formed signal of triangular form simultaneously reducing the residual error by approximately 24 times.
3. The results of calculations and mathematical modeling in the program PSIM-9 showed good coincidence.
4. Proposed functional generator can operate in wide diapason of frequencies on retention of high metrological characteristics of formed signals.
5. Multi-frequency functional generator can be applied in precision devices of radio electronics, automatics and communication networks.
6. The generator can be made in integral or hybrid way with the use of modern operational amplifier and multipliers that do not require trimming elements.
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