On Correction of the Nonlinearity of the Static Characteristic of a Magnetic Pulse-Width Modulator
Abstract Generated abstract
This paper examines the nonlinearity of the static input, output characteristic of a magnetic pulse-width modulator used for converting direct-current signals. It compares three correction approaches: increasing the magnetic bias field, applying negative feedback, and linking the bias current to the output pulse duration, deriving conditions and circuit relations for their implementation. The analysis indicates that increasing bias and using feedback can substantially reduce principal error but also reduce sensitivity, while bias-current correction can preserve sensitivity if the modulator characteristic satisfies the assumed approximating relation. The paper concludes that combining negative feedback with bias-current correction gives the best linearity, and that simultaneous use of all three methods is preferable when high stability is required.
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Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov
Volume 184 1970
On Correction of the Nonlinearity of the Static Characteristic of a Magnetic Pulse-Width Modulator
Yu. M. Ageev
(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)
A magnetic pulse-width modulator [1], intended for converting direct-current signals, has a certain error, which limits its use as a measuring converter. One of the causes producing the error is the nonlinearity of the static “input—output” characteristic. Investigations have shown that, in the case where the dynamic hysteresis loop is approximated by straight-line segments, the “input—output” characteristic is linear. However, the use for this purpose of high-quality cores with a rectangular hysteresis loop and the reduction of the frequency of the excitation voltage are not always sufficient measures; consequently, it is necessary to apply special correction methods, the principal ones being:
1) increasing the intensity of the bias field in the modulator cores;
2) applying negative feedback;
3) applying a connection between the magnitude of the bias current and the output signal of the PWM.
The purpose of this article is to consider and comparatively analyze each of the listed methods.
1. Correction of the PWM Based on Increasing the Magnetic Bias Field
One of the conditions for normal operation of a pulse-width modulator is that
\[ H_{\mathrm{cm}} > H_{\min} = \frac{B_s}{\mu_0 \mu_d}, \tag{1} \]
where \(H_{\mathrm{cm}}\) is the intensity of the bias field;
\(B_s\) and \(\mu_d\) are parameters of the dynamic hysteresis loop of the magnetic cores.
At the same time, many static and dynamic parameters of the PWM improve with an increase in the coefficient
\[ q = \frac{H_{\mathrm{cm}}}{H_{\min}} . \tag{2} \]
This applies first of all to the linear static “input—output” characteristic. Figure 1 shows the dependence of the value of the os...
Fig. 1
Fig. 2. a—circuit of a PWM with correction of the second and third types;
b—feedback circuit of a PWM with correction of the second type.
the principal error caused by the nonlinearity of the characteristic, on the values of the coefficient \(q\). Obviously, by increasing the bias-field intensity, this error can be substantially reduced and an almost linear characteristic can be obtained. However, this method has significant disadvantages, namely: first, as \(H_{\mathrm{cm}}\) increases, the sensitivity of the modulator decreases (Fig. 1); second, the additional error caused by changes in the frequency of the excitation voltage increases; and, third, the power consumption from the source of this voltage increases.
2. PWM with negative feedback
This method, which consists in applying to the input of the modulator a constant voltage or current proportional to the duration of the pulses at the modulator output, with the opposite sign relative to the control signal, is very effective for correcting nonlinearity. By increasing the feedback coefficient, one can obtain an almost linear static characteristic and substantially reduce the additional errors arising when the bias voltage and the frequency of the alternating excitation voltage change. However, it must be taken into account that increasing the feedback coefficient is accompanied by a proportional reduction in sensitivity, and also imposes stricter requirements on the stability of the voltage of the modulator power-supply source. The introduction of feedback only slightly complicates the PWM circuit, since no special device is required to obtain the feedback signal (Fig. 2b). If, however, the feedback is implemented by means of a separate winding, the latter is connected directly into the collector circuit of the output transistor \(T_2\) of the amplifier—the pulse shaper (Fig. 2a). The current flowing through this winding contains a constant and an alternating component. The constant component, proportional to the duration of the output pulses when the amplitude of the latter is unchanged, creates the feedback signal, while the alternating component has no effect, since for it the alternating-current windings form a short-circuited loop.
3. Correction of a PWM by introducing a connection between the magnitude of the bias current and the output signal
If the actual “input—output” characteristic of a PWM can, with a sufficient degree of accuracy, be approximated by the dependence
\[ \tau_{\mathrm{i}}=\frac{S_0 I_y}{1+S_0\beta I_y}, \tag{3} \]
where \(\beta>0\) is a constant coefficient;
\(S_0\) is the sensitivity of the modulator as \(I_y \to 0\),
then linearization of the characteristic is achieved by introducing into the PWM circuit a connection between the magnitude of the bias current and the output signal. This connection is made in such a way that, as \(\tau_{\mathrm{i}}\) increases, the bias current decreases proportionally:
\[ i_{\mathrm{cm}}=I_{\mathrm{cm}}-k_1\tau_{\mathrm{i}}. \tag{4} \]
Since the sensitivity \(S_0\) depends on the magnitude of the bias current,
\[ S_0=\frac{k_2}{i_{\mathrm{cm}}}, \tag{5} \]
the simultaneous solution of (3)—(5) leads to the following:
\[ \tau_{\mathrm{i}}\left(I_{\mathrm{cm}}-k_1\tau_{\mathrm{i}}\right)=k_2 I_y\left(1-\beta\tau_{\mathrm{i}}\right). \tag{6} \]
Obviously, in order to obtain a linear characteristic of the form
\[ \tau_{\rm i}=K_{2}K_{3}I_{\rm y}, \tag{7} \]
the condition
\[ K_{3}\left(I_{\rm sm}-K_{1}\tau_{\rm i}\right)=1-\beta\tau_{\rm i} \]
must be satisfied, or
\[ I_{\rm sm}=I_{\rm sm0}+\left(K_{1}-\beta I_{\rm sm0}\right)\tau_{\rm i}. \tag{8} \]
Here \(I_{\rm sm0}=\dfrac{1}{K_{3}}\) is the bias current at \(I_{\rm y}=0\). Taking (8) into account, condition (4) can be written in the form
\[ i_{\rm sm}=I_{\rm sm0}\left(1-\beta\tau_{\rm i}\right). \tag{9} \]
Thus, if for a modulator without correction or with correction of the first two types the dependence (3) is valid, then linearity of the “input–output” characteristic is ensured by changing the bias current in the PWM circuit according to law (9). In this case the sensitivity of the modulator will be a constant quantity and equal to the sensitivity of the PWM without correction in the initial section,
\[ S=\frac{K_{2}}{I_{\rm sm0}}=S_{0}. \tag{10} \]
This means that the correction method considered does not lead to a reduction in sensitivity and therefore is the most acceptable in the creation of high-sensitivity measuring converters.
Figure 2 shows the circuit of a PWM with correction of the third type. The direct current flowing through the bias windings in this circuit, in the absence of a signal at the modulator output, is equal to
\[ I_{\rm sm0}= \frac{U_{\rm sm}}{2\left(R_{1}+r_{1}+r_{\rm sm}+r/4\right)} = \frac{U_{\rm sm}}{2R}. \tag{11} \]
If \(r_{1}\ll R\), then, when the output signal changes, the quantity \(i_{\rm sm}\) will change in such a way that condition (9) is fulfilled:
\[ i_{\rm sm}=I_{\rm sm0} \left[ 1-\frac{E_{\rm k}fr_{1}}{U_{\rm sm}\left(R_{2}+r_{1}\right)} \tau_{\rm i} \right]. \tag{12} \]
The dependence of the parameters of the correcting circuit and the bias circuit on the coefficient \(\beta\),
\[ \beta=\frac{E_{\rm k}fr_{1}}{U_{\rm sm}\left(R_{2}+r_{1}\right)}, \tag{13} \]
obtained by comparing (9) and (12), can be used for calculating the correcting circuit. Indeed, for known \(E_{\rm k}\), \(U_{\rm sm}\), \(f\), and \(\beta\), one can determine \(\varepsilon\) by taking \(r_{1}=\varepsilon R_{2}\),
\[ \varepsilon=\frac{\beta U_{\rm sm}}{E_{\rm k}f-\beta U_{\rm sm}}. \tag{14} \]
To choose the remaining parameters of the correcting circuit and the bias circuit, we use dependence (1), which for \(\tau_{\rm i}=\tau_{\rm i\,max}\) can be written in the form
\[ i_{\rm sm}\geq I_{\min}=\frac{H_{\min}l}{w_{\rm sm}}. \tag{15} \]
Taking (7) into account, we obtain
\[ I_{\mathrm{cm0}} \gg \frac{H_{\min}\cdot l}{w_{\mathrm{cm}}\left(1-\beta \tau_{i\max}\right)} . \tag{16} \]
If \(r_1 \ll 0.1R\) and \(\tau_{i\max}=1/2f\), then, taking (11), (14), and (16) into account, one can determine \(R_k=r_1+R_2\):
\[ R_k \ll \frac{E_k(2f-\beta)w_{\mathrm{cm}}}{40\beta H_{\min}\cdot l}, \tag{17} \]
whence
\[ I_k=\frac{E_k}{R_k}\gg \frac{40\beta\cdot H_{\min}\cdot l}{(2f-\beta)w_{\mathrm{cm}}} \tag{18} \]
is the amplitude of the collector current of transistor \(I_2\). Since the permissible collector current \(I_{k,\mathrm{perm.}}\) is determined by the type of transistor and, for high-frequency transistors in switching mode, does not exceed \(50\ \mathrm{mA}\), the value of the resistance must be
\[ R_k \gg \frac{E_k}{I_{k,\mathrm{perm.}}}\simeq 20E_k, \tag{19} \]
and relation (17) can be used to calculate the number of turns of the bias winding.
Fig. 3
Let us now consider the degree of accuracy with which the experimental characteristics of a pulse-width modulator without correction and with correction of the first two types can be approximated by curve 3. Figure 3 shows the errors in approximating the characteristics of the modulator without correction (curve 1), with correction of the first type (curve 2), and with correction of the second type (curve 3).
Analyzing these dependences, one can note that the smallest deviation of the experimental characteristic from the approximated one is observed in the modulator with negative feedback, and the largest in the modulator without correction.
Conclusions
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Three correction methods have been considered that make it possible to substantially reduce the nonlinearity of the static characteristic of the modulator.
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The investigations have shown that an equal reduction in the principal error of a PWM, achieved by the first two methods, leads to an approximately equal decrease in sensitivity.
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The sensitivity does not decrease if the third correction method is used; however, for its implementation it is necessary that the experimental characteristic satisfy condition (3).
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The best linearity of the static characteristic of the PWM can be obtained by the combined use of the second and third correction methods; and in order to achieve high stability it is necessary to use all three methods simultaneously.
References
- Yu. M. Ageev. Magnetic pulse-width DC converter. In: Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov, vol. 141, TSU Publishing House, 1966.