On the Phase Transient in a Resonant Amplifier
Abstract Generated abstract
This paper analyzes phase transients in a single-tuned resonant amplifier, motivated by errors in measuring the phase shift between radio pulses when the carrier frequency or amplifier tuning is imperfect. Using an equivalent circuit and Laplace-transform treatment of a sinusoidal pulse input, it derives expressions for the output voltage and for the time-dependent phase shift during detuned operation. The analysis shows that at exact resonance only the amplitude settles, whereas detuning produces both amplitude and phase transients. For small detunings, simplified formulas indicate that phase settling depends on the equivalent quality factor and detuning, may take longer than amplitude settling, and in more general cases can have an oscillatory, nonmonotonic character consistent with reported experimental observations.
Full Text
On the Phase Transient in a Resonant Amplifier
F. I. Peregudov
(Presented by the scientific seminar of the Radio Engineering Faculty)
In radio-engineering devices one often has to deal with the measurement of the phase shift between two radio pulses. The accuracy of the measurement depends on the stability of the frequency of the high-frequency filling. The phase distortions to which the pulses are subjected during their formation, and then also during passage through the phasemeter channel, will introduce an error into the measurement results.
In this connection, analysis of phase transients in a single-tuned resonant amplifier (Fig. 1) is of considerable interest.
In deriving the basic relations, its equivalent circuit shown in Fig. 2 is used. The transfer coefficient of the amplifier
Fig. 1. Circuit of a single-tuned resonant amplifier.
\[ Y(i\omega) + \frac{U_{\mathrm{m\,out}}}{\frac{1}{\mu} U_{\mathrm{m\,in}}} = \frac{\mu Z(i\omega)}{R_i + Z(i\omega)}, \tag{1} \]
where
\[ Z(i\omega) = \frac{ \left(r + i\omega L\right)\frac{1}{i\omega C} }{ r + i\omega L + \frac{1}{i\omega C} }. \]
At the input of the circuit under consideration a voltage is applied
\[ \left. \begin{aligned} f(t) &= E_{\mathrm{m}}\sin(\omega_0 t+\beta), \qquad t>0,\\ f(t) &= 0, \qquad\qquad\qquad\quad t<0. \end{aligned} \right\} \tag{2} \]
As is known, if the frequency of the input signal \(\omega_0\) differs from the resonant frequency of the circuit \(\omega_p\), then, after the end of the transient process, the output voltage will be shifted in phase by the amount
\[ \varphi=\operatorname{arctg}\frac{x}{r\left(1+\dfrac{R_{oe}}{R_i}\right)}, \tag{3} \]
where
\[ x=\omega L-\frac{1}{\omega C}; \qquad R_{oe}=\frac{L}{Cr}. \]
Thus,
\[ U_{\text{out}}(t)=U_{m\,\text{out}}\sin(\omega_0 t+\beta-\varphi). \tag{4} \]
Fig. 2. Equivalent circuit of a single-tuned resonant amplifier.
It follows from (2) and (4) that during the transient process a change occurs in the constant component of the phase.
In connection with this, it becomes necessary to establish the character of this transient process.
Using the direct Laplace transform, the input voltage can be written in the form
\[ E_p=E_m\frac{p\sin\beta+\omega_0\cos\beta}{p^2+\omega_0^2}, \tag{5} \]
and the transfer coefficient, if the quantity \(r\) in the numerator is neglected in comparison with \(pL\), is
\[ Y(p)=\frac{\mu}{R_i C}\, \frac{p}{p^2+2\alpha_э p+\dfrac{1}{LC}}, \tag{6} \]
where
\[ \alpha_э=\frac{1}{2}\left(\frac{r}{L}+\frac{1}{R_i C}\right). \]
For the output voltage we obtain
\[ U_{\text{out}}(t)= \frac{\mu E_m}{R_i C}\, \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{p\left(p\sin\beta+\omega_0\cos\beta\right)e^{pt}} {\left(p^2+\omega_0^2\right)\left(p^2+2\alpha_э p+\dfrac{1}{LC}\right)} \,dp . \tag{7} \]
From (7), after transformations, we have
\[ U_{\text{out}}(t)= \frac{\mu E_m L}{R_i C q}\,\sin(\omega_0 t+\beta-\varphi) - \frac{\mu E_m L}{R_i C q}\,e^{-\alpha_{\mathrm{e}}t} \left\{ \sin(\beta-\varphi)\times \right. \]
\[ \left. {}\times \cos\omega_p t+ \left[ \frac{\cos(\beta-\varphi)}{\omega_0\omega_pLC} - \frac{\alpha_{\mathrm{e}}}{\omega_p}\sin(\beta-\varphi) \right]\sin\omega_p t \right\}, \tag{8} \]
where
\[ q=\sqrt{r_{\mathrm{e}}^{\,2}+x^2}; \qquad \omega_p^2=\frac{1}{LC}-\alpha_{\mathrm{e}}^2. \]
For \(\omega_0\simeq\omega_p\) and \(\dfrac{\alpha_{\mathrm{e}}}{\omega_p}\ll 1\), i.e., in the case of resonance,
\[ U_{\text{out}}(t)= \frac{\mu U_0 R_{0e}}{R_{0e}+R_i} \left(1-e^{-\alpha_{\mathrm{e}}t}\right)\sin(\omega_0t+\beta). \tag{9} \]
Thus, when tuned to resonance, only the amplitude is established in the amplifier.
In the presence of detuning, the output voltage changes during the transient process both in amplitude and in phase. Thus, for the output voltage one may write
\[ U_{\text{out}}(t)=A(t)\sin[\omega_0t+\beta-\varphi(t)], \tag{10} \]
where \(A(t)\) is the amplitude of the output voltage, depending on time; \(\varphi(t)\) is the phase change during the transient process.
The analysis of amplitude establishment in transient processes has been studied in detail by a number of authors [1, 2, 3].
The value of \(\varphi(t)\) can be found by equating expressions (8) and (10). After simple transformations we obtain
\[ \varphi(t)=\beta-\operatorname{arctg}\times \]
\[ {}\times \frac{ \operatorname{tg}(\beta-\varphi) - e^{-\alpha_{\mathrm{e}}t} \left\{ \operatorname{tg}(\beta-\varphi)\cos\Delta\omega t - \left[ \frac{1}{\omega_0\omega_pLC} - \frac{\alpha_{\mathrm{e}}}{\omega_p}\operatorname{tg}(\beta-\varphi) \right]\sin\Delta\omega t \right\} }{ 1 - e^{-\alpha_{\mathrm{e}}t} \left\{ \operatorname{tg}(\beta-\varphi)\sin\Delta\omega t - \left[ \frac{1}{\omega_0\omega_pLC} - \frac{\alpha_{\mathrm{e}}}{\omega_p}\operatorname{tg}(\beta-\varphi) \right]\cos\Delta\omega t \right\} }, \tag{11} \]
where
\[ \Delta\omega=\omega_0-\omega_p. \]
If one takes into account that
\[ p=\frac{\Delta\omega}{\omega_0}; \qquad \omega_p=\omega_0(1-p) \]
and
\[ \frac{1}{\omega_0\omega_pLC} \simeq (1-p)\left(1+\frac{1}{4Q_{\mathrm{e}}}\right) \simeq (1-p) \quad \text{for } Q_{\mathrm{e}}>3, \]
then (11) can be written in the following form:
\[ \varphi(t)=\beta-\operatorname{arctg}\times \]
\[ \times \frac{ \operatorname{tg}(\beta-\varphi) - e^{-\frac{\omega_0}{2Q_{\mathrm{э}}}(1-p)t} \left\{ \operatorname{tg}(\beta-\varphi)\cos\Delta\omega t - \left[(1-p)-\frac{1}{2Q_{\mathrm{э}}}\operatorname{tg}(\beta-\varphi)\right]\sin\Delta\omega t \right\} }{ 1 - e^{-\frac{\omega_0}{2Q_{\mathrm{э}}}(1-p)t} \left\{ \operatorname{tg}(\beta-\varphi)\sin\Delta\omega t + \left[(1-p)-\frac{1}{2Q_{\mathrm{э}}}\operatorname{tg}(\beta-\varphi)\right]\sin\Delta\omega t \right\} }. \tag{12} \]
Of practically the greatest interest is the character of the transient process for small detunings, which occur in the amplifier after it has been tuned with the aid of a tube voltmeter or an oscilloscope. The accuracy of tuning can be estimated by the expression
\[ \frac{U_{\mathrm{out}}}{U_{\mathrm{res.\,out}}}<\kappa, \tag{13} \]
where \(\kappa=0.85\div0.9\).
Because of inaccurate tuning, a phase transient will occur in the amplifier, giving a steady-state value of the phase due to the detuning
\[ \varphi<\operatorname{arctg}\sqrt{\frac{1-\kappa^2}{\kappa^2}}, \tag{14} \]
which follows directly from the known relation
\[ \frac{U_{\mathrm{out}}}{U_{\mathrm{res.\,out}}} = \frac{1}{\sqrt{1+\operatorname{tg}^2\varphi}}. \tag{15} \]
It follows from expression (14) that
\[ p<\frac{1}{2Q_{\mathrm{э}}}\operatorname{arctg}\sqrt{\frac{1-\kappa^2}{\kappa^2}}. \tag{16} \]
For \(Q_{\mathrm{э}}>5\) and \(\kappa<0.8\), \(p<0.05\).
Obviously, for detunings of less than 5%, expression (12) may be simplified:
\[ \varphi(t)=\beta-\operatorname{arctg}\times \]
\[ \times \frac{ \operatorname{tg}(\beta-\varphi) - e^{-\frac{\omega_0}{2Q_{\mathrm{э}}}t} \left\{ \operatorname{tg}(\beta-\varphi)\cos\Delta\omega t - \left[1-\frac{1}{2Q_{\mathrm{э}}}\operatorname{tg}(\beta-\varphi)\right]\sin\Delta\omega t \right\} }{ 1 - e^{-\frac{\omega_0}{2Q_{\mathrm{э}}}t} \left\{ \operatorname{tg}(\beta-\varphi)\sin\Delta\omega t + \left[1-\frac{1}{2Q_{\mathrm{э}}}\operatorname{tg}(\beta-\varphi)\right]\cos\Delta\omega t \right\} }. \tag{17} \]
The function \(\varphi(t)\) describes the change of phase from the initial value \(\beta\) to the value \((\beta-\varphi)\). It may be assumed that the character of the transient process does not depend on the magnitude of the initial phase \(\beta\). This is confirmed by verification calculations. Therefore one may calculate \(\varphi(t)\) for the particular case \(\beta=\varphi\) and extend it to all possible variants. Under this condition formula (17) is considerably simplified. Indeed,
\[ \varphi(t)=\varphi-\operatorname{arctg} \frac{ e^{-\frac{\omega_0}{2Q_{\mathrm{э}}}t}\sin\Delta\omega t }{ 1-e^{-\frac{\omega_0}{2Q_{\mathrm{э}}}t}\cos\Delta\omega t }. \tag{18} \]
It follows from the expression obtained that the process of phase settling for the operating frequency \(\omega_0\) depends on the equivalent quality factor \(Q_{\mathrm e}\) and on the magnitude of the detuning \(\Delta \omega\).
In the case where, over the amplitude settling time,
\[ T_A \approx \frac{4.6 Q_{\mathrm e}}{\omega_0}. \tag{19} \]
(measured at the level \(0.9\) of the established amplitude value), the value of the argument \(\Delta \omega T < 0.2\), then it may be assumed that during this time
\[ \cos \Delta \omega t \simeq 1; \qquad \sin \Delta \omega t \simeq \Delta \omega t. \]
Taking into account the approximations obtained, expression (18) may be written in the simpler form
\[ \varphi(t)=\frac{\Delta \omega}{\alpha_{\mathrm e}} \left(1-\frac{e^{-\alpha_{\mathrm e}t}\alpha_{\mathrm e}t}{1-e^{-\alpha_{\mathrm e}t}}\right). \tag{20} \]
Fig. 3. Settling of the amplitude and phase of oscillations in a resonant amplifier at small detunings.
In Fig. 3 the phase-settling curve
\[ \frac{\varphi(t)}{\dfrac{\Delta \omega}{\alpha_{\mathrm e}}} \]
is shown. For comparison, the amplitude-settling curve is plotted there as well.
It is not difficult to see that the phase settling time \(T_\varphi\) to the level \(0.9\) of the steady-state value is greater than the time \(T_A\):
\[ T_\varphi \approx \frac{7.2 Q_{\mathrm e}}{\omega_0}. \]
In the general case, phase settling in the system may proceed nonmonotonically.
In Fig. 4 the phase-settling curves are given for several cases.
The amplitude build-up curves are plotted according to the formula obtained by Gonorovskii [2] for the case when \(\dfrac{\Delta\omega}{\omega_p} \ll 0.1 \div 0.15\)
\[ \frac{A(t)}{A_{cm}}=\sqrt{1-2e^{-\alpha_3 t}\cos \Delta\omega t+e^{-2\alpha_3 t}} . \tag{21} \]
Fig. 4. Establishment of the amplitude and phase of oscillations in a resonant amplifier.
Essman [4] carried out an experimental investigation of phase establishment during the transient process.
The experimental results confirm the possibility of an oscillatory character of phase establishment.
REFERENCES
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Gonorovskii I. S., Radio Signals and Transient Phenomena in Radio Circuits, Svyazizdat, 1954.
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Gonorovskii I. S., Frequency Modulation and Its Application, Svyazizdat, 1948.
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Evtyanov S. I., Transient Processes in Receiving-Amplifying Circuits, Svyazizdat, 1948.
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Essman D., Zeitschrift für angewandte Mathematik und Physik, 1954, Bd. 6, 3, ss. 115—120.