On the Calculation of Transient Processes in Circuits with Incandescent Lamps
Abstract Generated abstract
The paper develops calculation formulas for transient processes in circuits where an incandescent lamp is used as a controlled light source rather than only as a nonlinear resistance. Starting from a filament heat balance that relates temperature to lamp and circuit parameters, it introduces practical approximations for tungsten properties and derives expressions for heating under voltage-source and current-source step excitation, as well as a linearized first-order transfer function near an operating point. The analysis shows that the lamp time constant and transfer coefficient depend on both filament temperature and the external switching regime, with greatest inertia in the current-source case. Measurements on NSM 6×20 lamps are reported to agree reasonably with the calculated transients and time constants, supporting the proposed formulas for practical estimates.
Full Text
ON THE CALCULATION OF TRANSIENT PROCESSES IN CIRCUITS WITH INCANDESCENT LAMPS
V. F. ANIKIN
(Presented by the scientific and technical seminar of the Department of Automation and Telemechanics)
A number of works [1–3] are devoted to questions of the dynamics of incandescent lamps (L). Most often, an L is considered as a two-terminal element in current–voltage coordinates. For the linearized regime, the incremental impedance, an equivalent substitution circuit [2], etc., are determined.
However, often an incandescent lamp is used not as a nonlinear resistance, but as a controlled light source. In this case the output quantity of the L is the light radiation, directly related to the temperature of the filament by the Stefan–Boltzmann law. In this connection it is of interest to obtain the necessary calculation dependences relating the change in filament temperature to the parameters both of the lamp itself and of the external circuit.
- Let us consider a circuit consisting of a vacuum incandescent lamp and a resistance \(R_i\) connected in series with it. It can be shown that the heating equation in this case has the form
\[ 4.187 m_{\mathrm{n}}(T)c_p(T)\cdot \frac{dT}{dt} + \delta \varepsilon \tau l_F(T)\cdot T^4(1-e^{-\beta T}) = \frac{4l_\rho(T)}{\pi d^2}\cdot \rho(T)\cdot I^2(t), \tag{1} \]
for \(I>0\). \(T=T_0\) for \(I=0\),
where
\(m_{\mathrm{n}}\) — mass of the filament;
\(c_p\) — mean heat capacity of the metal;
\(\delta\) — coefficient of incompleteness of radiation, taking into account the shape of the incandescent body;
\(\varepsilon\) — Boltzmann constant;
\(d\) — filament diameter;
\(\rho\) — specific resistance;
\(l_F, l_\rho\) — equivalent lengths of the filament of the lamp L, respectively, for radiation and dissipated power;
\(\beta\) — coefficient taking into account the difference between the radiation of metals and the radiation of an absolutely black body;
\(I\) — current in the circuit of the L;
\(T\) — maximum temperature of the filament;
\(T_0\) — ambient temperature.
Owing to cooling by the electrodes and holders, the temperature of the incandescent body is distributed nonuniformly along the filament. To take this circumstance into account, corrections are introduced for the length of the real filament having the same maximum temperature as the ideal one, under the condition of equivalence of the corresponding parameters [5]. Naturally, in this case equation (1) includes three equivalent lengths (besides \(l_F\) and \(l_\rho\), also \(l_T\)—the equivalent length with respect to heat capacity, entering into \(m_{\mathrm n}(T)\)); moreover, \(l_F(T) \ne l_\rho(T) \ne l_T(T)\).
The nonlinear coefficients entering into (1) not only depend on the temperature in a rather complicated way [5, 6], but are also determined, in addition, by a number of technological factors: the annealing regime [7], the wire diameter [8], etc.
- Integration of expression (1) in general form does not appear possible. For further analysis we introduce the following simplifications:
a) we take the coefficients \(c_\rho(T)\), \(l_F(T)\), \(l_\rho(T)\), \(l_T(T)\) to be constant, equal to their mean value in the operating temperature interval \((900 \div 2200^\circ\mathrm K)\);
b) we approximate the dependence \(\rho(T)\) by the expression
\[ \rho(T) \simeq b_1 \cdot T,\qquad b_1 \simeq 27 \cdot 10^{-9}\ \Omega\cdot \mathrm{cm}/\mathrm{deg}; \tag{2} \]
c) the expression for the energy luminosity of tungsten is represented in the form
\[ \varepsilon T^4\left(1-e^{-\beta T}\right) \simeq \varepsilon a_F T^5,\qquad a_F = 1.29\cdot 10^{-4}\ \mathrm{deg}^{-1}. \tag{3} \]
Taking the simplifications into account, equation L will be written in the following form:
\[ \kappa_{\mathrm p}\, m_{\mathrm{n\,sr}}\, c_{\mathrm{sr}}\,\frac{dT}{dt} +\gamma T^5 = \frac{U^2 b_\rho T}{(R_1+b_\rho T)}, \tag{4} \]
\[ T=T_0\quad \text{at } U=0, \]
where
\[ b_\rho=\frac{4l_{\rho\mathrm{sr}}}{\pi d^2}\,b_1;\qquad \gamma=\delta \varepsilon a_F \pi d l_{F\mathrm{sr}},\qquad \kappa_{\mathrm p}=4.187 . \]
Let us consider transient processes in several characteristic regimes. For \(R_1=0\) (supply from a voltage source), the solution of equation (4) for a voltage step equal to \(A_u\) can be obtained in implicit form:
\[ t=b_u\left[ \frac{1}{6}\, \frac{(g^2+gT^2+T^4)(g-T_0^2)^2} {(g^2+gT_0^2+T_0^4)(g-T^2)^2} + \frac{1}{\sqrt{3}}\arctg \frac{2g\sqrt{3}(T^2-T_0^2)} {3g^2-(2T^2+g)(2T_0^2+g)} \right], \tag{5} \]
where
\[ b_u=\frac{\kappa_{\mathrm p} m_{\mathrm{n\,sr}} c_{\mathrm{sr}} b_\rho^{1/3}} {(2\gamma A_u)^{2/3}};\qquad g=\sqrt[3]{\frac{A_u^2}{\gamma b_\rho^2}} . \]
Equation (5) makes it possible to find the time for heating L to a specified temperature.
For the regime of supply from a current source \(\left|R_1 \gg R_{\mathrm L}\right|\), for a current step \(A_I\),
\[ T(t)=T_0\sqrt[4]{ \frac{A_I b_\rho} {\gamma T_0^4+\left(A_I b_\rho-T_0^4\gamma\right)e^{-t/b_I}} }. \tag{6} \]
where
\[ b_1=\frac{4.187\,m_{\mathrm{n,avg}}\,C_{\mathrm{avg}}}{4A_l b_\rho}. \]
The heating time of the lamp to the required temperature in the given regime is
\[ t=b_1\ln \frac{A_l b_\rho-T_0^4\gamma}{T_0^4\left(\dfrac{A_l b_\rho}{T^4}-\gamma\right)}. \tag{7} \]
Usually the radiation indicator of an incandescent lamp is an element with a limited spectral characteristic, and expressions (5), (7) make it possible to determine the delay time of the output signal of the element (from the known long-wavelength boundary of its spectral characteristic).
- If the operating regime of the lamp admits linearization, then it is meaningful to speak of the transfer function of the incandescent lamp. From (1), for a sufficiently small neighborhood of the operating point \(T_p, U_p\), we obtain
\[ W_l(p)=\frac{\Delta \bar T(p)}{\Delta \bar U(p)} =\frac{\kappa_l(T_p)}{\tau_l(T_p)p+1}, \tag{8} \]
where
\[ \Delta \bar T=\Delta T/T_p,\qquad \Delta \bar U=\Delta U/U_p, \]
\(W_l(p)\) is the transfer function of the lamp.
Since
\[ \rho_l(T)=a_\rho T^{\alpha_R},\qquad (\alpha_R\approx 1.2), \]
the relative transfer coefficient \(\kappa_l\) and the time constant of the incandescent lamp \(\tau_l\) are determined, respectively, as
\[ \kappa_l(T_p)=\frac{\Delta \bar T}{\Delta \bar U} =\frac{2(m+1)}{(5-\alpha_R)(m+1)+2\alpha_R}; \tag{9} \]
\[ \tau_l(T_p)= \frac{\kappa_p m_n(T_p)c_n(T_p)} {\gamma T_p^4\left(5-\dfrac{\alpha_R(m-1)}{m+1}\right)}. \tag{10} \]
where \(m=\dfrac{R_1}{R_{\mathrm{lp}}}\), and \(R_{\mathrm{lp}}\) is the resistance of the lamp at the operating point.
It is evident from (9) and (10) that \(\kappa_l\) and \(\tau_l\) depend not only on \(T_p\), but also on the method of switching on the lamp. In particular, the maximum inertia and the largest transfer coefficient of the incandescent lamp occur at \(m=\infty\) (in the current-source regime).
- Verification of the results obtained above was carried out on incandescent lamps of type NSM 6×20. Figure 1 gives calculated curves and oscillograms of transient processes for the current-source regime.
Comparison of the curves shows good agreement between the calculated and experimental data. Indeed, for the curve \(I_l=14\ \mathrm{mA}\), at the 0.9 level the duration of the transient process according to the oscillogram is \(135\ \mathrm{ms}\), and according to calculation it is \(138\ \mathrm{ms}\). Similar results are also obtained for other current values.
The calculated and experimental data for the time constants of the lamp are given in Table 1.
It is evident from the table that, as the operating current decreases, the calculation error increases. This is explained by the fact that average values were adopted
equivalent filament lengths (1), corresponding to fairly high temperatures on the order of 1500°K, which, at low currents, reduces the accuracy of the calculation, which nevertheless remains acceptable for a practical estimate of the inertia of a circuit with an incandescent lamp.
Fig. 1. Transient characteristics of an incandescent lamp
- Thus, the material presented above indicates that the expressions found describe the dynamics of circuits with incandescent lamps sufficiently correctly.
In [4], on the basis of experimental data, Hauert asserts that an incandescent lamp is described by a transfer function of the form of an aperiodic second-order element. This is not consistent with the theory of incandescent lamps and with the experimental data obtained by us. Familiarity with the experimental procedure in [4] allows one to regard as one of the possible reasons for the discrepancy in the results the fact that Hauert assigned to the incandescent lamp the time constant of the light indicator illuminated by it.
Table 1
| $I_\ell$, mA | 6 | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|
| $\tau_\ell$ exp., msec | $450 \pm 50$ | $200 \pm 20$ | $100 \pm 10$ | $80 \pm 5$ | $50 \div 55$ | $40 \div 45$ |
| $\tau_\ell$ calc., msec | 510 | 93 | 53 | 42 |
REFERENCES
-
L. A. Sinitskii. Some questions in the calculation and application of nonlinear circuits in measurement technology. Dissertation. Lviv, 1954.
-
E. I. Tsimbalist. Dissertation. Tomsk, TPI, 1967.
-
M. S. Roitman, E. I. Tsimbalist, A. I. Lysov. A photoelectric converter as a controlled element of an electrical circuit. Izv. vuzov, “Priborostroenie,” L., No. 6, 1966.
-
Hauert. The incandescent lamp as an element of a control system. (USA). Pribory dlya nauchnykh issledovanii, No. 5, 1965.
-
A. P. Ivanov. Electric Light Sources. GEI, Moscow, 1955.
-
Thermophysical Properties of Substances. Handbook, Moscow, 1959.
-
K. A. Gel’d. Tungsten and Molybdenum. Energiya Publishing House, Moscow, 1964.
-
S. Ipatova. Some Summary Properties of Wires Made of Tungsten, Molybdenum, and Their Alloys. Moscow, 1952.