A Sensitive Pulse-Width Modulator for Converting DC Voltages and EMF

Yu. M. Ageev, V. P. Kazmin

(Presented by the scientific seminar of the Department of Automation and Telemechanics)

In some cases, because of the low input resistance, a PWM without feedback [1, 2] cannot be used for voltage conversion. Increasing the input resistance by reducing the wire diameter of the control windings cannot be regarded as a satisfactory solution, since in this case difficulties arise in manufacturing the modulator, its cost increases, and the conversion error increases because of a considerable change in the input resistance with changes in the ambient temperature. In addition, increasing the input resistance in this way is accompanied by a decrease in voltage sensitivity.

This same drawback limits the possibilities of using a PWM with negative feedback, where the sensitivity decreases in proportion to the increase in input resistance. However, a reduction in sensitivity can be avoided by using, in the PWM circuit, positive feedback in addition to negative feedback. This principle is taken as the basis of the PWM circuit intended for converting DC voltage. We note that in this case the negative feedback is implemented by the method shown in Fig. 1,a, and the positive feedback by the method shown in Fig. 1,b.

The block diagram of the modulator with positive and negative feedbacks is shown in Fig. 1,c.

If the control voltage \(U_y(s)\) is regarded as the input signal, the transfer function of the PWM will be

\[ W(s)= \frac{\eta_2 S_i(1+sT_{\mathrm{oc}})} {\eta_1(1+sT_y)(1+sT_{\mathrm{oc}})+\beta_1\eta_2 S_i(1+sT_{\mathrm{oc}})-\beta_2\eta_1 S_i(1+sT_y)} . \tag{1} \]

Here

\[ \eta_1=\frac{r_{\mathrm{in}}+r_c}{r_{\mathrm{in}}}; \qquad \eta_2=\sqrt{\frac{Q_y}{Q_y+Q_{\mathrm{oc}}}}; \qquad r_{\mathrm{in}}=4r_y, \]

\(Q_y, Q_c\) are the areas of the core window occupied by the control and feedback windings;

\[ S_i=\frac{W_y}{2f l H_{\min}(p+g-1)\,r_{\mathrm{in}}} \]

is the sensitivity to the control voltage of a PWM without feedback.

Fig. 1

a)

Labels in the diagram: “Input” (−, +), PWM, \(Z_c\), \(Z_k\), \(-E_k\).

b)

Labels in the diagram: “Input,” PWM, \(-E_k\), \(W_{1oc}\), \(W_{2oc}\), \(W_{3oc}\), \(W_{4oc}\), \(Z_k\), \(i_{oc}\), A, B.

c)

Labels in the block diagram: \(U_y(s)\), \(U_k(s)\), \(U_{oc}(s)\), \(W_{\mathrm{in}}(s)\), \(\Phi_y(s)\), \(\Phi_o(s)\), \(\Phi_{oc}(s)\), \(K_i\), \(W_{2H}(s)\), \(\tau_u(s)\), \(\beta\), “LF2.”

\[ T_{\mathrm{oc}}=\frac{2M_0M_{\mathrm{d}}pQ_c}{(p+q-1)l} \left( \frac{\eta_2^2 W_y}{r_{\mathrm{in}}} +\frac{W_{\mathrm{v}}^2}{4r_{\mathrm{v}}} +\frac{W_{\mathrm{cm}}^2}{4r_{\mathrm{cm}}+R_2} +\frac{W_{\mathrm{oc}}^2}{r_{\mathrm{k}}+4r_{\mathrm{oc}}} \right); \]

\[ T_y=\frac{2M_0M_{\mathrm{d}}pQ_c}{(p+q-1)l} \left( \frac{\eta_2^2 W_y^2}{r_{\mathrm{in}}} +\frac{W_{\mathrm{v}}^2}{4r_{\mathrm{v}}} +\frac{W_{\mathrm{cm}}^2}{4r_{\mathrm{cm}}+R_2} \right); \]

\[ \beta_1=\frac{E_k f\cdot r_c}{r_c+r_k};\qquad \beta_2=\frac{E_k f W_{\mathrm{oc}} r_{\mathrm{in}}}{W_y(r_k+4r_{\mathrm{oc}})}. \]

Let us consider the static properties of a PWM with negative and positive feedback. For \(h=0.5\), the input–output characteristic is described by the equation

\[ \tau=\frac{\eta_2 S_i U_y}{\eta_1+\beta_1\eta_2 S_i-\beta_2\eta_1 S_i}. \tag{2} \]

Depending on the ratio of the coefficients of positive feedback and negative feedback, two cases may be distinguished.

In the first, the equality \(\beta_1\eta_2=\beta_2\eta_1\) is ensured, and the static characteristic takes the form:

\[ \tau_i=\frac{\eta_2 S_i U_y}{\eta_1}. \tag{3} \]

In the second case, \(\beta_2S_i=1\), as a result of which we obtain

\[ \tau_i=\frac{U_y}{\beta_1}. \tag{4} \]

In the first case, for values of the coefficients \(\eta_1\eta_2\) close to unity, all characteristics of the PWM with positive feedback and negative feedback are approximately the same as those of a modulator without feedback. The only exception is the input resistance \(R_{\mathrm{in}}\), which, in the presence of feedback, increases by a factor of

\[ \frac{\eta_1}{1-\beta_2S_i}, \]

i.e.,

\[ R_{\mathrm{in}}=\frac{r_{\mathrm{in}}+r_c}{1-\beta_2S_i}. \]

The second case proves to be more interesting; we shall consider it in greater detail.

The static characteristic of the PWM is the straight line (4), whose slope depends only on the parameters of the feedback circuit. A positive feature in this case is also the high input resistance of the PWM, which depends on the resistance of the control winding of the magnetic modulator and which is theoretically equal to infinity. Since the differential sensitivity of a PWM without feedback is a variable quantity, it is first necessary to ensure linearity of the characteristic and then apply feedback. Figure 2 shows the schematic and equivalent circuits of the control, bias, and feedback circuits of a pulse-width modulator for direct voltage.

It is assumed that the polarity of the input voltage does not change. The excitation generator and the power supply are not shown in the figure. The source of the direct bias voltage and the supply of the final stage of the shaping unit is common; its voltage is equal to \(U_{\mathrm{cm}}\).

Fig. 2

Denoting by

\[ R_y' = r_1 + r_{\mathrm{in}};\qquad R_{\mathrm{oc}} = r_{k2} + 4r_{\mathrm{oc}};\qquad R_{\mathrm{cm}}' = R_{\mathrm{cm}}/2, \tag{5} \]

from the equivalent circuit we find

\[ I_y = \frac{E_y\left(r_{k1}-r\right)-U_{\mathrm{out}}r} {R_y' r_{k1}+r_{k1}r+R_y'r}; \qquad I_{\mathrm{oc}}' = \frac{U_{\mathrm{out}}\left(R_y'+r\right)-E_y r} {R_y' r_{k1}+r_{k1}r+R_y'r}; \]

\[ I_{\mathrm{oc}} = \frac{U_{\mathrm{out}}}{R_{\mathrm{oc}}}; \qquad I_{0}'' = \frac{U_{\mathrm{out}}\left(R_{\mathrm{cm}}'+r_p\right)-U_{\mathrm{cm}}r_p} {R_{\mathrm{cm}}'r_{k3}+R_{\mathrm{cm}}'r_p+r_{k3}r_p}; \tag{6} \]

\[ I_{\mathrm{cm}} = \frac{U_{\mathrm{cm}}\left(r_{k3}+r_p\right)-U_{\mathrm{out}}r_p} {2\left(R_{\mathrm{cm}}'r_{k3}+R_{\mathrm{cm}}'r_p+r_{k3}r_p\right)}. \]

If the static characteristic of the PWM without feedback and without correction is described by the equation

\[ \tau_{\mathrm{i}}=\frac{S_{io} I_y}{1+\gamma S_{io} I_y}, \tag{7} \]

where \(\gamma>0\) is a constant coefficient having the dimension of frequency; \(S_{io}\) is the sensitivity of the modulator at \(I_y \to 0\); then the duration of the output pulses in the circuit (Fig. 2) in the steady-state mode will be equal to:

\[ \tau_{\mathrm{i}}= \frac{S_{io}\left(I_y W_y+I_{oc} W_{oc}\right)} {W_y\left[1+\dfrac{\gamma S_{io}}{W_y}\left(I_y W_y+I_{oc} W_{oc}\right)\right]} . \tag{8} \]

After substituting in the last expression the currents \(I_y\) and \(I_{oc}\) according to (6), we write:

\[ \tau_{\mathrm{i}}^2 \gamma U_{\mathrm{cm}} f\left(W_{oc} R_{\mathrm{e}}+W_y R_{oc} r\right) +\tau_{\mathrm{i}}\left[ \frac{W_y}{S_{io}} R_{\mathrm{e}}R_{oc} +\gamma E_y(r_{k1}+r)W_y R_{oc} -U_{\mathrm{cm}} f\left(W_{oc}R_{\mathrm{e}}-W_y R_{oc}r\right) \right] =E_y(r_{k1}+r)W_y R_{oc}, \tag{9} \]

where

\[ R_{\mathrm{e}}=R_y' r_{k1}+r_{k1}r+R_y' r . \]

It is obvious that, in order to obtain a static characteristic of the form

\[ \tau_{\mathrm{i}}=\frac{(r_{k1}+r)E_y}{U_{\mathrm{cm}} f r}, \tag{10} \]

it is necessary that

\[ \tau_{\mathrm{i}}\gamma U_{\mathrm{cm}} f\left(W_{oc}R_{\mathrm{e}}-W_y R_{oc}r\right) +\frac{S_{io}}{W_y}R_{\mathrm{e}}R_{oc} +\gamma E_y(r_{k1}+r)W_y R_{oc} - \]

\[ -U_{\mathrm{cm}} f\left(W_{oc}R_{\mathrm{e}}-W_y R_{oc}r\right) =U_{\mathrm{cm}} f r W_y R_{oc}, \]

or

\[ \frac{S_{io} U_{\mathrm{cm}} f W_{oc}}{R_{oc}W_y}\left(1-\gamma\tau_{\mathrm{i}}\right)=1 . \tag{11} \]

If we now take into account that, at \(p=1\), the sensitivity of the modulator on the initial section of the characteristic is equal to

\[ S_{io}=\frac{W_y}{2 I_{\mathrm{cm}} W_{\mathrm{cm}}}, \]

then condition (11) reduces to the following:

\[ I_{\mathrm{cm}}=\frac{U_{\mathrm{cm}}W_{oc}}{2R_{oc}W_{\mathrm{cm}}}\left(1-\gamma\tau_{\mathrm{i}}\right). \tag{12} \]

Thus, in order to obtain a linear characteristic whose slope would depend only on the parameters of the negative-feedback circuit, condition (11) should be fulfilled. Note that the bias current in the circuit (Fig. 2a) is determined by expression (6); comparing it with (12), one can write two more conditions:

\[ \frac{W_{oc}}{R_{oc}W_{\mathrm{cm}}} = \frac{r_{k3}+r_p}{R_{\mathrm{cm}}' r_{k3}+R_{\mathrm{cm}}' r_p+r_{k3}r_p}; \qquad \frac{r_p f}{r_{k3}+r_p}=\gamma . \tag{13} \]

a)

In the graph: vertical axis \(\Delta \tau_u / \tau_u\); horizontal axis \(\Delta f\,\% / f\).
Annotations: \(S'_E / S_{E\max}=1\), \(S_E / S_{E\max}=3\), \(S'_E / S_{E\max}=0.2\).
Axis marks: \(-40\), \(-20\), \(20\), \(40\).

b)

In the graph: vertical axis \(\Delta \tau_u / \tau_u\); horizontal axis \(\Delta U_{\mathrm{cm}} / U_{\mathrm{cm}}\).
Annotations: \(S'_E / S_{E\max}=1\), \(S'_E / S_{E\max}=0.2\), \(S'_E / S_{E\max}=2\).
Axis marks: \(-40\), \(-20\), \(0\), \(20\), \(40\).

Fig. 3

The resistance \(r_p\) is usually much smaller than the resistance \(R_{\mathrm{cm}}\); therefore, finally we obtain:

\[ I_{\mathrm{cm}}=\frac{U_{\mathrm{cm}}}{R_{\mathrm{cm}}}(1-\gamma \tau_u);\qquad R_{\mathrm{cm}}=\frac{2R_{\mathrm{oc}}W_{\mathrm{cm}}}{W_{\mathrm{oc}}}; \]

\[ \frac{r_p}{r_{k3}+r_p}=\gamma;\qquad S_E=\frac{r_{k1}+r}{U_{\mathrm{cm}}f r}. \tag{14} \]

Fig. 4.

Labels visible in the circuit diagram:

• \(E_y\)
• \(Z_i\)
• \(22\)
• \(W_{1y}\), \(W_{2y}\), \(W_{3y}\), \(W_{4y}\)
• \(Z_{k1}\), \(Z_{k3}\)
• \(I_{\mathrm{dc}}\)
• \(R_{A1}\), \(R_{H1}\)
• Output I
• \(T_1\), \(T_2\), \(T_3\), \(T_4\), \(T_5\), \(T_6\)
• \(R_3\), \(R_4\), \(R_5\), \(R_6\), \(R_7\), \(R_8\), \(R_9\), \(R_{10}\), \(R_{11}\), \(R_{12}\), \(R_{13}\), \(R_{14}\), \(R_{15}\), \(R_{16}\), \(R_{17}\), \(R_{18}\), \(R_{19}\)
• \(R_1\), \(R_2\)
• \(T_r\)
• \(W_{1\mathrm{st}}\), \(W_{2\mathrm{st}}\), \(W_{3\mathrm{st}}\), \(W_{4\mathrm{st}}\)
• \(W_v\), \(W_{2v}\), \(W_{3v}\), \(W_{4v}\), \(W_{5v}\), \(W_{6v}\)
• \(D_1\), \(D_2\)
• \(-U_{\mathrm{st}}\)
• \(-E_k\)
• from the excitation generator
• \(U_c\)
• Output II
• \(I_{\mathrm{dc}}\)
• \(Z_k\)
• \(Z_{k1}\), \(Z_{k3}\)
• \(R_{\mathrm{oc}}\)
• \(R_{\mathrm{st}}\)
• \(I_{\mathrm{out}}\)
• \(I_{\mathrm{st}}\)
• \(R'_y\)
• \(Z/2\)
• \(E_y\)
• \(R^y\)

Subfigure labels:

a)
b)

Let us now consider the errors of a PWM with feedbacks. Since the circuit provides correction of the nonlinearity, the error $\delta_b$ may be neglected. When the circuit parameters change, an error arises which, for $r_{k1} \gg r$,

\[ \frac{\Delta R_{\text{oc}}}{R_{\text{oc}}} = \frac{\Delta R_{\text{cm}}}{R_{\text{cm}}}, \qquad \frac{\Delta r_{k1}}{r_{k1}} = \frac{\Delta r}{r}; \qquad \gamma \tau_y \ll 1 \]

is equal to

\[ \delta_{\Delta} = - \frac{\Delta U_{\text{cm}}}{U_{\text{cm}}} + \frac{ W_{\text{oc}} r_{k1}(r_i+r_{\text{in}}+r)-qW_y R_{\text{oc}}r }{ qW_y R_{\text{oc}}r } \cdot \frac{\Delta f}{f} + \frac{ W_{\text{oc}} r_{k1}(r_i+r_{\text{in}}+r) }{ qW_y R_{\text{oc}}r } \left( \frac{\Delta B_s}{B_s} - \frac{\Delta U_0}{U_0} \right). \tag{15} \]

Taking into account that

\[ \frac{S'_E}{S_E} = \frac{ W_{\text{oc}} r_{k1}(r_i+r_{\text{in}}+r) }{ W_y R_{\text{oc}}r }, \tag{16} \]

we obtain

\[ \delta_{\Delta} = - \frac{\Delta U_{\text{cm}}}{U_{\text{cm}}} + \left( \frac{S'_E}{S_{E\max}}-1 \right) \frac{\Delta f}{f} + \frac{S'_E}{S_{E\max}} \left( \frac{\Delta B_s}{B_s} - \frac{\Delta U_0}{U_0} \right). \tag{17} \]

Here $S'_E$, $S_{E\max}$ are the sensitivity of the PWM with positive feedback and negative feedback, and the maximum sensitivity of the PWM without feedback.

Good agreement between the theoretical results and the experimental ones can be found by comparing the corresponding dependences in Fig. 3, a, b.

The transient process in the PWM under a stepwise change is aperiodic.

Experimental curves for different sensitivities and different coefficients $\eta_2$ are given in Fig. 3, c. As the sensitivity is increased, the duration of the transient process increases practically proportionally.

Analysis of the experimental results shows that, with a sufficient degree of accuracy, the time constant of a PWM with feedbacks can be calculated from the formula

\[ T \simeq T_y \cdot \frac{S'_E}{S_{E\max}}, \tag{18} \]

where $T_y$ is the time constant of the control circuit of the PWM without feedback.

In conclusion, we note that, for the conversion of voltages of different polarity, the PWM can be constructed according to the circuit shown in Fig. 4, a. In it, in contrast to the preceding one, when the polarity of the input signal changes, the polarity of the feedback voltage changes, as a result of which the input resistance of the PWM is always high.

The equivalent circuits of input signals of different polarity at small values, shown in Fig. 4, b, c, do not differ in any way from the one considered earlier (Fig. 2). Therefore, for the modulator (Fig. 4, a) all the conditions and characteristics obtained above are valid.

REFERENCES

1. Yu. M. Ageev. Static characteristics of a magnetic-semiconductor pulse-width modulator. Avtomatika i telemekhanika, No. 5, 1967.

2. Yu. M. Ageev. Theoretical and experimental study of a push-pull magnetic pulse-width modulator. Proceedings of the XI All-Union Conference on Magnetic Elements of Automation and Computer Engineering, Tallinn, 1966.