On the Transient Characteristics of Incandescent Lamps
Abstract Generated abstract
This paper examines transient processes in low-power tungsten incandescent lamps, emphasizing effects neglected in simpler first-order models, especially heat removal through supports and current leads and reverse radiation from the bulb. The authors propose a linearized dynamic model with three coupled first-order equations for filament temperature, heat loss to holders, and absorbed bulb radiation, and derive the corresponding transfer function. Experiments on aged NSM 6×20 lamps used small step changes in supplied power and photodiode monitoring of luminous flux to identify the model parameters from staged exponential transients. The calculated transient characteristics based on the identified coefficients differed from the measured oscillograms by no more than about 2 percent, supporting the adequacy of the proposed model within the tested linear approximation.
Full Text
On the Transient Characteristics of Incandescent Lamps
V. P. Zibalov, M. S. Roitman
Low-power incandescent lamps (IL) with a tungsten filament have found application in various areas of measurement technology, in automatic control and regulation devices, and in amplitude-stable self-oscillators. Very often the technical parameters of the corresponding devices depend substantially on the characteristics of the transient processes occurring in incandescent lamps. However, the dynamics of these processes has not yet been studied sufficiently fully. In works [1, 2] some transient characteristics are given for tungsten filaments heated by an electric current, but without taking into account heat transfer by the filament to the current leads. Attempts to take the latter circumstance into account are made in a number of works [3, 4, 7]. In these works the electrical and thermal processes occurring in an IL, for relatively small temperature changes, are represented by a linearized first-order differential equation:
\[ \tau_1 \frac{d(\Delta T)}{dt} + \Delta T = \kappa_1 \Delta P_{\mathrm{e}}, \tag{1} \]
where \(\tau_1\) and \(\kappa_1\) are IL parameters, depending on its design and technological data, usually determined by calculation or experimentally at the selected operating point;
\(\Delta T\) is the change in the filament temperature “averaged over the length” [5];
\(\Delta P_{\mathrm{e}}\) is the change in the electric power supplied to the IL.
In this case, power losses due to heat removal by the current leads and holders were taken into account by introducing corrections to the length of the real filament.
Such an idealization of the process of heat transfer by the filament is valid only when considering steady-state regimes in an IL [5] and is very crude for characterizing nonstationary regimes. If, in the dynamics, heat transfer from the more heated parts of the filament to the less heated ones due to thermal conductivity is taken into account, then for the simplest case it is necessary to write a one-dimensional equation of the following form [6]:
\[ c\rho \frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial x^2} + \frac{1}{S} F(x,t), \tag{2} \]
where \(c\) is the specific heat capacity of the incandescent body (filament);
\(\rho\) is its density;
\(K\) — coefficient of thermal conductivity;
\(S\) — cross-sectional area of the filament;
\(F(x, t)\) — a function expressing the presence of internal heat sources;
\(x\) — the running coordinate along the longitudinal axis of the filament. We assume that the filament of low-power ILs is straight, homogeneous, and so thin that at every instant the temperature \(T\) of all points of a given cross section of the filament will be the same. Otherwise we would have to write the heat-conduction equation for three-dimensional space. From equations (2) it is seen that the temperature \(T\) is a function of the coordinate \(x\) and time \(t\). If the coordinate \(x\) is fixed, then the function \(T(x, t)\) expresses the law of temperature variation in the given cross section of the filament with time.
As is known, there is a simple dependence between the energy of thermal motion of a body and temperature:
\[ Q = c \rho V T, \tag{3} \]
where \(Q\) is the energy of thermal motion or, equivalently, the amount of heat in calories;
\(T\) — absolute temperature;
\(V\) — volume of the body.
Using (3), one can determine how much heat is transferred in a given cross section of the filament, i.e., for fixed \(x\). If, however, the heat is summed over the entire length of the filament at constant \(t\), then we shall find the law of heat (energy, power) transfer as a function of time. This is, in general form, the path for solving the problem posed, but along this path there are, in our opinion, insurmountable difficulties. In order to obtain the function \(T(x, t)\), we must integrate equation (2) under specified initial and boundary conditions.
The boundary conditions, i.e., the law of cooling of the ends of the filament by the holders, are unknown; for low-power ILs this question has not been investigated by anyone. Therefore it is not possible to obtain the dependence \(T = T(x, t)\) in analytical form; consequently, we cannot express the power transferred to the holders as a function of time.
But from (4) it is seen that the differential heat-conduction equation is a first-order equation in the variable \(t\); obviously, the law of heat transfer in time must obey an exponential dependence. Based on such a hypothesis, we shall take as the mathematical model of the process of heat (power) transfer from the filament to the holders an equation of the form:
\[ \tau_2 \frac{d(\Delta P_{\text{д}})}{dt} + \Delta P_{\text{д}} = k_2 \Delta T, \tag{4} \]
where \(\tau_2\) and \(k_2\) are IL parameters depending on its design and technological data and determined experimentally;
\(\Delta P_{\text{д}}\) — change in the power removed by the holders.
In the works cited earlier, the process of reverse radiation of energy by the walls of the IL bulb onto the filament surface was not taken into account. Without analyzing the complex picture of absorption and emission of thermal power by the bulb, we shall, by analogy with (4), write the following equation:
\[ \tau_3 \frac{d(\Delta P_{\text{к}})}{dt} + \Delta P_{\text{к}} = k_3 \Delta T, \tag{5} \]
where \(\tau_3\) and \(k_3\) are IL parameters determined experimentally;
\(\Delta P_{\text{к}}\) — change in the power radiated by the bulb.
In general, for an incandescent lamp we write the heat-balance equation:
\[ \Delta P_{\mathrm{t}}+\Delta P_{\mathrm{d}}+\Delta P_{\mathrm{g}} = \Delta P_{\mathrm{e}}+\Delta P_{\mathrm{k}}, \tag{6} \]
where
\(\Delta P_{\mathrm{t}}\) is the power expended on changing the heat content of the filament and on radiation;
\(\Delta P_{\mathrm{d}}\) is the power removed from the filament by cooling it through its supports and current leads;
\(\Delta P_{\mathrm{g}}\) is the power transferred through the gas medium (for vacuum incandescent lamps it is usually neglected [5]);
\(\Delta P_{\mathrm{e}}\) is the power released by the electric current;
\(\Delta P_{\mathrm{k}}\) is the radiation power absorbed by the filament from the walls of the bulb.
The power \(\Delta P_{\mathrm{k}}\) is, as it were, supplied to the filament from outside; therefore it is written on the right-hand side of equation (6). This corresponds well to the real physical process: if, for some reason, the temperature of the bulb changes, then naturally the temperature of the filament will also change, owing to the emission of energy from the walls of the bulb.
Fig. 1. Structural diagram of the transmission of disturbances in an incandescent lamp.
Fig. 2.
Thus, the transient processes occurring in an incandescent lamp are described by the following system of equations:
\[ \left. \begin{aligned} \tau_1 \frac{d(\Delta T)}{dt}+\Delta T &= k_1\left(\Delta P_{\mathrm{e}}+\Delta P_{\mathrm{k}}-\Delta P_{\mathrm{d}}\right),\\[6pt] \tau_2 \frac{d(\Delta P_{\mathrm{d}})}{dt}+\Delta P_{\mathrm{d}} &= k_2 \Delta T,\\[6pt] \tau_3 \frac{d(\Delta P_{\mathrm{k}})}{dt}+\Delta P_{\mathrm{k}} &= k_3 \Delta T . \end{aligned} \right\} \tag{7} \]
If, as the input quantity causing the transient process, we take the change in the supplied electrical power \(\Delta P_{\mathrm{e}}\), and as the output quantity the change in the “integral” temperature of the filament \(\Delta T\), then we can construct the structural diagram for the transmission of disturbances in the incandescent lamp (Fig. 1).
The diagram is constructed so that it reflects the physics of the phenomena: a change in the supplied electrical power from its steady-state value causes a change in the filament temperature. This leads to a corresponding change in the power consumed by the incandescent lamp through the feedback circuits. The lower feedback branch (Fig. 1) reflects the power consumption by the supports, and the upper branch the influx of power due to reverse radiation by the bulb.
On the basis of the structural diagram, the transfer function of the IL can be obtained:
\[ \frac{\Delta T(p)}{\Delta P_{\mathrm{e}}(p)} = \frac{\dfrac{K_1}{\tau_1 p+1}} {1+\dfrac{K_1}{\tau_1 p+1} \left( \dfrac{K_2}{\tau_2 p+1} - \dfrac{K_3}{\tau_3 p+1} \right)} , \tag{8} \]
where
\[ p \equiv \frac{d}{dt}. \]
Naturally, the question arises: how well does the transfer function (8) reflect the behavior of the IL? Only experiment can answer this question. The study of transient processes in the linear approximation was carried out on ILs of type NSM 6×20, of ordinary and soffit design. The incandescent lamps were preliminarily aged with a current of 20 mA for 48 hours. Fig. 2 shows the diagram of the measuring apparatus used to carry out the necessary investigations. Step-like (small) changes in the supplied power
Fig. 3.
to the IL are set by shorting resistor \(R_1\) with contact K of relay P, controlled from the NGPK generator. From this same generator, a slow sweep is supplied to the S1-19B oscilloscope. Monitoring of the change in the temperature of the IL is carried out with the aid of a photodiode (PD) of type KFDM, which reacts practically without delay to the increment of the luminous flux of the IL, and correspondingly to the change in the filament temperature. Under the action of light, the PD changes its resistance (in the circuit its reverse connection is used), and consequently a signal proportional, on some scale, to the change in the temperature of the IL filament is applied to the oscilloscope from resistor \(R_3\).
The experimental transient characteristic shown in Fig. 3 gives a general idea of the change in filament temperature, from which it is seen that at the initial moment the temperature changes relatively rapidly up to a certain maximum value (in Fig. 3—the jump, because of the slow sweep; let us call it the 1st stage); then the tempe-
Fig. 4a, b, c. Transient characteristics of incandescent lamps.
the temperature falls to a certain value (stage II), and subsequently it slowly tends to a new steady-state value (stage III). In Fig. 4a, stage I is photographed with a fast sweep; the value of each mark is 10 msec. The second and third stages are presented, respectively, on a reduced scale along the vertical axis in Figs. 4b and 4c. The value of the marks for Fig. 4b is 1 sec, and for Fig. 4c, 10 sec.
The characteristics obtained were used to determine the coefficients of the transfer function of the incandescent lamp by the interpolation method [8]. In a first approximation, the time constants \(\tau_1\), \(\tau_2\), and \(\tau_3\) can be found from the formula:
\[ \tau = t_1 \ln \frac{\Delta T_1}{\Delta T_2 - \Delta T_1}, \]
where \(\Delta T_1\) is the deviation of the output quantity of the incandescent lamp at an arbitrarily chosen time \(t_1\);
\[ \Delta T_2 \]
is the deviation of the output quantity at the time \(t_2 = 2t_1\).
This possibility exists because, for each of the curves (Figs. 4a, b, and c), the exponential property is well satisfied [9].
The quantities \(\Delta T_1\), \(\Delta T_2\), and \(t_1\) must be taken from the oscillogram of the corresponding stage.
For the incandescent lamp whose transient characteristics are given in the article, the coefficients of the transfer function have the following values:
\[ k_1 = 59 \frac{^\circ K}{\text{mW}}; \qquad k_2 = 0.00063 \frac{\text{mW}}{^\circ K}; \qquad k_3 = 0.00013 \frac{\text{mW}}{^\circ K}; \]
\[ \tau_1 = 0.1 \text{ sec}; \qquad \tau_2 = 1.8 \text{ sec}; \qquad \tau_3 = 67 \text{ sec}. \]
On the basis of these data, calculations of the transient processes in the incandescent lamp were made by stages. The calculated characteristics differ from the experimental ones by no more than 2%. The experiments and calculations presented sufficiently well confirmed the mathematical model of transient processes in incandescent lamps adopted in the article.
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