Correction of the frequency-phase response of an amplifier using negative capacitance
Abstract Generated abstract
This article analyzes the use of a negative-resistance converter, operating as a “negative” capacitance, for correcting the frequency and phase response of an amplifier stage. Using an equivalent circuit model, it derives the stage gain and effective impedance for a load formed by an active two-terminal network, and identifies a stability condition relating converter gain and coupling capacitance. The analysis indicates that, for suitable parameter choices, the effective impedance can resemble a low-Q resonant circuit with phase lag, enabling increased bandwidth while limiting phase shift. The method may outperform passive correction for feedback amplifiers at the same gain, although it requires a substantially larger number of circuit elements.
Full Text
Proceedings of the Tomsk Polytechnic Institute
of the Order of the October Revolution and the Order of the Red Banner of Labor
named after S. M. Kirov
Vol. 194 1972
CORRECTION OF THE FREQUENCY-PHASE CHARACTERISTIC OF AN AMPLIFIER BY A “NEGATIVE” CAPACITANCE
Yu. V. PIKALKIN
(Presented by the scientific and technical seminar of the Department of Radio Engineering)
In the literature [1—4] there are indications of the possibility of using negative-resistance converters (NRC) for correction purposes. This article gives an analysis of correction obtained by means of a device of this kind. Let us consider the operation of an amplifier stage loaded by an active two-terminal network (Fig. 1). The magnitude of the input
Fig. 1.
resistance of the two-terminal network is determined [1] as \(Z=-(Z_2+Z_c)/(K_2-1)\),
where
\[
Z_2
\]
— is the output resistance of an amplifier with gain coefficient \(K_2\);
\[
Z_c
\]
— is the coupling resistance.
The circuit of Fig. 1,a can be reduced to Fig. 1,b,
where
\[ e_1 = SZ_1U_{\text{in}} = K_1U_{\text{in}}, \qquad e_2 = K_2e_1. \]
The gain coefficient of interest is determined in the usual way:
\[ \dot K_i = \dot K_1 \frac{Z}{Z_1+Z} = S \frac{Z_1Z}{Z_1+Z} = SZ_3. \]
Assuming \(Z_1=R_1/(1+j\omega C_1R_1)\), \(Z_2=R_2/(1+j\omega C_2R_2)\), \(Z_c=1/j\omega C_c\), \(\dot K_2=K_2>1\), we can express \(Z_3\) in the form
\[
Z_{\mathrm{e}}=R_1\left[1+jx\,\frac{C_2+C_{\mathrm{c}}\dfrac{R_2}{R_1}}{C_1}\right]\cdot
\left\{1-x^2\frac{C_2R_2}{C_1R_1}\left[1+\frac{C_{\mathrm{c}}}{C_2}-\frac{C_{\mathrm{c}}}{C_1}(K_2-1)\right]+\right.
\]
\[
\left.
{}+jx\left[1+\frac{C_2+C_{\mathrm{c}}\dfrac{R_2}{R_1}}{C_1}
-\frac{C_{\mathrm{c}}}{C_1}(K_2-1)\right]\right\}^{-1},
\]
where \(x=\omega C_1R_1\). A sufficient condition for stability of the amplifier \(K_2\) is the relation
\[ K_2 \leqslant 1+\frac{C_1}{C_{\mathrm{c}}}, \]
under which the imaginary parts in the expression for \(Z_{\mathrm{e}}\) are equal and are determined by the constant \((C_2+C_{\mathrm{c}})R_2\); the bandwidth of the stage is then determined by the ratio \(C_{\mathrm{c}}R_2/C_1R_1=R_2/R_1(K_2-1)\). The quantities \(K_2\) and \(R_2/R_1\) may be chosen comparatively small to ensure positive feedback: \(K_2\simeq 2,\ R_2/R_1=1/K_1\). This makes it possible to regard the gain \(K_2\) as frequency-independent.
Analysis of the expression for \(Z_{\mathrm{e}}\) shows that, for \(K_2\simeq 1+C_1/C_{\mathrm{c}}\), \(Z_{\mathrm{e}}\) may have a maximum with phase lag, like a low-\(Q\) \(LC\) circuit.
This makes it possible to use NCF to increase the bandwidth of comparatively low-frequency stages while simultaneously correcting their phase characteristic, which is essential in feedback systems: correction by means of NCF makes it possible to obtain a larger bandwidth at a limiting phase shift of \(90^\circ\) than correction by passive circuits at the same gain. The introduction of NCF into the cathode-correction circuit can provide phase correction with unchanged (and even increasing) attenuation up to the frequency \(x^2=C_1R_1/C_{\mathrm{c}}R_2\). A disadvantage is the considerable increase in the number of elements.
REFERENCES
- L. A. Bessonov. Nonlinear Electric Circuits. “Vysshaya shkola,” 1964.
- B. Ya. Lur’e. Design of Transistor Amplifiers with Deep Feedback. “Svyaz’,” 1965.
- A. D. Artym. Electrical Correcting Circuits and Amplifiers. “Energiya,” 1965.
- Jones, Kaywood, Williams. Use of negative reactive resistance to obtain independent frequency and phase characteristics. Electronics (USA), vol. 37, No. 31, 1964.