Calculation of non-linear circuits
Course work
Calculation of non-linear circuits
Problem 1
VAC of initials nonlinear element (resistance) is given in Fig.1,2
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Fig. 1 |
Fig. 2
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Operate point on VAC is defined by shift voltage Uo. On non-linear element influence harmonic voltage.
Uin = Umcos(ωt)
Given amplitude Um and frequencyf.
Find:
- Defined and denote an operate point on VAC.
- Approximate in this region VAC by analytical expression in form of full Taylor series of the fourth order.
- Find numerically meanings of approximating coefficients by method of select point.
- Define current in non-linear element, construct its form
- Construct amplitude spectrum of current
- Simulate current i(t) using date of spectrum by computer. (Harmonic syntax signal).
Initial numerical data take from table. Choose the variant “n”correspondingly to the last number of test book.
Table
| Task | Variant number | |||||||||
| 1 | ||||||||||
| VAC form | 2d | |||||||||
| U0,V | 0.6 | |||||||||
| Um,V | 0.25 | |||||||||
| f,MHz | 1.5 | |||||||||
Methodical Guidelines
Counting of voltages is more convenient to produce from shift voltage U0, instead of zero. Thus, the working section of the VAC approximated in the input voltage swing Uinlimits. Cosine input voltage is symmetrically relate to the operating point, and takes both positive and negative values. So, the Taylor series of fourth power for the current has the form:
When Uin=0 we immediately obtain the coefficient a0, it is equal to value of current in the operating point.
To determine other four approximation coefficients we should select four points in the field of approximation and make the equations for the four selected points. You can select the point at equal distances from each other. However, the accuracy of the approximation will increase if more densely arranged points on the more difficult sections of the VAC (on areas of large curvature near the extreme, etc.). Four equations with four unknown coefficients of approximation a1, a2, a3, a4 are solved by next Gaussian exceptions or Cramer determinants. It should be emphasized, that here are needed high accuracy of approximation (accuracy of calculations).
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After determining the coefficients a1, a2, a3, a4 write the polynome (I) and check the accuracy of approximation by substituting the polynome (I) some numerical values of voltage and calculate the corresponding values of the current between two points.
To determine the spectral components can be used next relationships:
In constructing the spectrograms on the abscissa should be set numerical values of the absolute frequency f.
Course work
Calculation of non-linear circuits
Problem 2
VAC of nonlinear element is approximated by means of piecewise-linear dependence and includes 2 line sections.
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Ubeg- voltage at which NE no holds current - voltage lock
S - steepness slope segment VAC of NE.
On NE effects voltage, which instantaneous value is next:
u=U0+uin=U0+Umcoswt, where
U0– shift voltage
uin - instantaneous value of input voltage signal
Um- amplitude input voltage signal
w - cyclic frequency of input signal.
Find:
- Calculate shift voltage UOto obtain cut-off angleθ.
- Draw position of input signal respectively VAC.
- Draw the form of nonlinear electric current.
- Using Berg’s functions, calculate components of nonlinear electric current, limiting constant component and 5 harmonics.
- Write analytical expression for this spectrum.
- Construct diagrams of amplitude-frequency and phase-frequency spectrums.
- Model signal i(t) on computer by results of spectrum.
Choose the variant “n” in corresponding to the penultimate number of test book.
Table
| Task | Variant number | |||||||||
| 1 | ||||||||||
| Ubeg,V | 0.2 | |||||||||
| S,mA/V | 20 | |||||||||
| Um,V | 0.5 | |||||||||
| θ° | 60 | |||||||||
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Methodical Guidelines
Shift voltage U0, which provides a given cutoff angle is determined by the formula:
Berg’s functions can be found with the help of graphs (КуринирВ.Ф., ФерсманБ.А. Теориянелинейныхэлектрическихцепей. -М: Связь, 1974, стр.71) Knowing that some of them can be negative. These ratios are also calculated by the known formulas described in textbooks. Forexample, Баскаков С.И. Радиотехнические цепи и сигналы. -М.: Высш.шк., 1983.-536с.
Current spectrum must be represented with taking into consideration that for the first harmonic phase shift is equal to zero.
Course work
Calculation of non-linear circuits
Problem 3
Range of resistive circuit shown in Figure 1.
VAC nonlinear element is not listed in Figure 2.
Parameter range specified in the table. Forms input signals raised diagrams in Figure 3.
Amplitude of the input signal:
Variant number selected by the last two digits of the record book. The last digit corresponds to the number in the table, and the penultimate - the number of the signal.
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You have to:
- to determine and to construct transmission characteristics:
- to construct the graph of the input signal 
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- to determine the spectra of input and output signals of the circuit by decomposing signals into the harmonic series. Find the coefficients of the series. To construct the spectra of input and output signals.
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Methodical Guidelines
Equivalent circuit in Figure 1 can be represented in Figure 4.
In fig.4 U=0.5V (Fig 2)
Obviously, while input voltage will not exceed the total voltage E + U in the current is not exist in branch a b c. Thus input voltage:
For the calculation circuit 4 with U>E+Uac we’ll use the method of node voltages.
In this equation it is necessary to substitute
To calculate Uout and to construct straight line of circuit’s transmission characteristics.
- points of fracture of characteristics.
Taking into account transfer coefficient for two straight sections of transfer characteristics:
To construct the graph of the input signal.
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