Output Voltage of a Ring Frequency Converter for the Case of Small Input Voltages
Abstract Generated abstract
This paper analyzes the output voltage of a ring frequency converter operating with small input voltages. It compares two exponential approximations for the current, voltage characteristic of identical nonlinear diode elements, derives Kirchhoff equation systems for the equivalent circuit, and solves them by an approximate Newton, Raphson method for cases where either input voltage predominates. The resulting formulas show that the two diode approximations give nearly identical first approximation results for small signals, although one leads to more complicated general expressions. A simplified expression is obtained for the regime of pure voltage multiplication, providing conditions for calculating ring mixers under different circuit parameters and signal voltages.
Full Text
Output Voltage of a Ring Frequency Converter for the Case of Small Input Voltages
G. I. Levashkin
(Presented by the scientific seminar of the Radio Engineering Faculty)
An analysis of the operation of the circuit of a ring frequency converter (Fig. 1) has been considered in a number of works by Soviet \([1 \div 4]\) and foreign scholars \([6]\).
In the present work, a comparison is made of the obtained expressions for the output voltage of a ring mixer when the characteristics of the nonlinear element are approximated in the form
\[ i = \frac{1}{R_0} U e^{cU} \tag{1} \]
and
\[ i = \frac{1}{cR_0}\left(e^{cU} - 1\right), \tag{2} \]
where \(R_0 = \dfrac{dU}{di}\) at \(U = 0\), and \(c = 1.5 \div 4 \dfrac{1}{\text{V}}\) for point-contact germanium diodes. With a corresponding choice of the value \(c\), approximation (1) gives a close representation of the reverse conductance of the diode, while (2) better reflects the region of forward conductance for small values of signal voltages.
Fig. 1.
In solving the problem, we assume that all nonlinear elements are identical and that the transformers are ideal.
The equivalent circuit of the mixer is shown in Fig. 2. The quantities indicated on it:
\[ \varepsilon_1 = i_1 \frac{R_a}{4}; \qquad \varepsilon'_1 = i_5 \frac{R_a}{4}; \qquad \varepsilon_2 = i_9 \frac{R_e}{4}; \]
\[ \varepsilon'_2 = i_3 \frac{R_e}{4} \]
are the e.m.f.’s of mutual induction between the halves of the windings of transformers having brought-out midpoints. The Kirchhoff equations, written according to Fig. 2, have the form:
\[ U_1+U_2+U_3+U_4=0; \]
\[ U_1+U_2+\frac{R_a}{2}(i_1+i_2-i_3-i_4)=e_1; \]
\[ U_1+U_3+2R_i(i_1-i_2+i_3-i_4)=2e_2; \tag{3} \]
\[ U_1+U_4+\frac{R_e}{2}(i_1-i_2-i_3+i_4)=0. \]
Here \(U_1,\ U_2,\ U_3,\ U_4\) are the voltages on the corresponding diodes.
Fig. 2.
Case \(e_1>e_2\)
In approximating the current–voltage characteristic of the diode (1), from (3) we obtain the following system of equations:
\[ v_1+v_2+v_3+v_4=0; \]
\[ v_1+v_2+\frac{\beta}{2} \left(v_1 e^{v_1}+v_2 e^{v_2}-v_3 e^{v_3}-v_4 e^{v_4}\right)=e_{\mathrm I}; \]
\[ v_1+v_3+\gamma \left(v_1 e^{v_1}-v_2 e^{v_2}+v_3 e^{v_3}-v_4 e^{v_4}\right)=2e_{\mathrm{II}}; \tag{4} \]
\[ v_1+v_4+\frac{\alpha}{2} \left(v_1 e^{v_1}-v_2 e^{v_2}-v_3 e^{v_3}+v_4 e^{v_4}\right)=0, \]
where
\[ v_j=cU_j;\qquad \alpha=\frac{R_e}{R_0};\qquad \beta=\frac{R_a}{R_0};\qquad \gamma=\frac{2R_i}{R_0};\qquad e_{\mathrm I}=ce_1;\qquad e_{\mathrm{II}}=ce_2. \tag{5} \]
We find the expression for the control voltage on the diode by solving system (4) by the approximate Newton–Raphson method [5]. Restricting ourselves to the second approximation, we have:
\[ v_1=\frac{cE_1}{2(1+\beta)} \left[1-\frac{3c^2E_1^2\beta}{32(1+\beta)^3}\right]\cos\omega_1 t -\frac{\beta c^3E_1^3}{64(1+\beta)^4}\cos 3\omega_1 t . \tag{6} \]
Assuming further that the resistance of the nonlinear elements can be replaced by linear ones, but varying in time according to the law
\[ r_1=\frac{R_0}{1+v_1}e^{-v_1}, \]
we compose, in accordance with the equivalent circuit (Fig. 3), a system of Kirchhoff equations. The solution of the system gives the expression for the output voltage in the form
\[ U_{\text{out}}= \frac{2e_2}{1+4\dfrac{R_i}{R_e}}\cdot \frac{\operatorname{sh}v_1+v_1\operatorname{ch}v_1} {a_1+\operatorname{ch}v_1+v_1\operatorname{sh}v_1}, \tag{7} \]
where
\[ a_1=\frac{1+2\alpha\gamma(1-v_1^2)}{\alpha+2\gamma}. \]
Fig. 3.
If the voltages \(e_1\) and \(e_2\) are such that \(v_1=cU\ll 1\), then the value of \(U_{\text{out}}\) can be written more simply:
\[ U_{\text{out}}= \frac{cE_1E_2} {\left(1+4\dfrac{R_i}{R_e}\right)(1+\beta)(1+\alpha)} \cos\omega_1 t\cdot \cos\omega_2 t, \tag{8} \]
where
\[ a=\frac{1+2\alpha\gamma}{\alpha+2\gamma}. \]
Representing the diode characteristic in the form (2), and using the notation adopted earlier, we obtain the system of Kirchhoff equations in the form
\[ v_1+v_2+v_3+v_4=0; \]
\[ v_1+v_2+\frac{\beta}{2}\left(e^{v_1}+e^{v_2}-e^{v_3}-e^{v_4}\right)=e_1; \]
\[ v_1+v_3+\gamma\left(e^{v_1}-e^{v_2}+e^{v_3}-e^{v_4}\right)=2e_{II}; \]
\[ v_1+v_4+\frac{\alpha}{2}\left(e^{v_1}-e^{v_2}-e^{v_3}+e^{v_4}\right)=0. \]
Solving analogously to the preceding case and taking into account that \(r_1=R_0 e^{-v_1}\), we obtain the expression for \(v_1\) and \(U_{\mathrm{out}}\) in the form
\[ v_1=\frac{cE_1}{2(1+\beta)} \left[ 1-\frac{\beta c^2E_1^2}{32(1+\beta)^3} \right]\cos\omega_1t -\frac{\beta c^3E_1^3}{92(1+\beta)^4}\cos3\omega_1t\ldots \]
\[ U_{\mathrm{out}}= \frac{2e_2}{1+4\dfrac{R_i}{R_e}}\cdot \frac{\operatorname{sh}v_1}{a+\operatorname{ch}v_1}. \tag{9} \]
For small values of \(e_1\) and \(e_2\), we obtain expression (8).
Case \(e_2 \gg e_1\)
When approximating the diode current–voltage characteristic by (1), using the system of equations (4), we obtain the expression for the control voltage on the diode in the form
\[ v_2=\frac{cE_2}{1+2\gamma} \left[ 1-\frac{3c^2E_2^2\gamma}{4(1+2\gamma)^3} \right]\cos\omega_2t -\frac{\gamma c^3E_2^3}{4(1+2\gamma)^4}\cos3\omega_2t, \]
and the expression for \(U_{\mathrm{out}}\):
\[ U_{\mathrm{out}}= \frac{e_1}{1+\dfrac{R_a}{R_e}}\cdot \frac{\operatorname{sh}v_2+v_2\operatorname{ch}v_2} {a_2'+\operatorname{ch}v_2+v_2\operatorname{sh}v_2}, \tag{10} \]
where
\[ a_2'=\frac{1+\alpha\beta(1-v_2^2)}{\alpha+\beta}. \]
When solving the problem using (2),
\[ v_2=\frac{cE_2}{1+2\gamma} \left[ 1-\frac{c^2E_2^2\gamma}{4(1+2\gamma)^3} \right]\cos\omega_2t -\frac{\gamma c^3E_2^3}{12(1+2\gamma)^4}\cos3\omega_2t. \]
In this case
\[ U_{\mathrm{out}}= \frac{e_1}{1+\dfrac{R_a}{R_e}}\cdot \frac{\operatorname{sh}v_2}{a_2+\operatorname{ch}v_2}, \]
where
\[ a_2=\frac{1+\alpha\beta}{\alpha+\beta}. \]
Conclusions
Comparing the formulas obtained for the effective expressions at the diodes and for \(U_{\text{out}}\), one can see that both approximations in the first approximation give approximately identical values.
The approximation of the characteristic of the nonlinear element by the dependence (1) gives a more complicated general formula for \(U_{\text{out}}\) (see 7 and 10) than the approximation by expression (2).
The \(U_{\text{out}}\) values of the mixer when operating with small input voltages do not differ from one another.
Expression (8) can be used in calculations of a ring mixer in which “pure” multiplication of voltages is carried out.
The results of the work make it possible to determine the conditions for “pure” multiplication of voltages for various values of the circuit parameters and signal voltages.
Literature
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Andreev V. S., On the operation of a ring converter with small input signals. Radiotekhnika, vol. 12, No. 12, 1957.
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Dobrovol’skii G. V., Analysis of nonlinear multi-terminal networks. Publishing House of the Academy of Sciences of the USSR, 1947.
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Kharkevich A. A., Multiplying circuits, Radiotekhnika, vol. 9, No. 3, 1954.
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Scarborough A. A., Numerical Methods of Mathematical Analysis, 1934.
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Weaver D. K., “A Third Method of Generation and Detection a Single Side-band Signals,” PJRE, v. 12, 1956.