ANALYSIS OF THE DYNAMICS OF FLIP-FLOP TRANSISTOR SWITCHES BY THE STATE-VARIABLE METHOD USING A DIGITAL COMPUTER

G. P. TROFIMOV, V. V. PFENING

(Presented by the scientific seminar of the Department of Radio Engineering)

For experimental allowance for the influence of the shape of the measured voltage on voltmeter readings, it is necessary to have sources calibrated for all parameters of signals of some typical form. Convenient for this purpose is an alternating rectangular voltage [1]. Among other requirements imposed on a voltage of this kind [2] is the requirement of providing a minimum edge and a flat top, without overshoots. As a consequence, a rigorous mathematical analysis is necessary of transient processes in devices that realize the specified curve shape. Since these devices are built with transistors and are complex nonlinear systems, the solution of the problem posed is possible only by numerical methods with the use of a digital computer. The use of a digital computer makes it possible to eliminate a whole series of restrictions and simplifications that are usually introduced in calculating general-purpose equipment, and to use a mathematical model that rather fully reflects the real physical processes, which makes it possible to obtain the most accurate and reliable results.

Investigation of devices suitable for generating rectangular voltage [2] has shown that the closest approximation to the required curve shape is provided by a flip-flop transistor switch, the circuit of which is shown in Fig. 1. Alternate switching of the transistors is accomplished by applying unipolar voltage steps to the input, for example from a trigger.

Fig. 1. Schematic circuit of a flip-flop switch with ungrounded power supplies

By choosing the magnitudes of the base currents, it is not difficult to ensure that one transistor turns on more rapidly than the other. This leads to two characteristic points.

First, the open transistor, by shunting the load, sharply reduces voltage spikes due to the parasitic passage of the control signal through the collector circuit of the transistor being turned on. Second, for a certain interval of time a low-resistance loop arises, consisting of the sources \(E_1\), \(E_2\) and the closed transistors, in which a significant through current appears. The latter promotes rapid resorption of charge carriers accumulated in the previously saturated transistor and reduces the delay of the output signal relative to the input. In addition, since at any instant one of the transistors is open, a small equivalent circuit time constant and a short leading edge of the output voltage are obtained.

To obtain a mathematical model of the device under analysis, we shall use the ideas of the state-variable method [3]*). This method leads to a system of normally connected first-order differential equations of the form

\[ \dot{x}=f(x,y,t), \]

where \(x(t)\) is the vector of state variables with respect to which the equations are integrated;

\(y(t)\) is the vector of input disturbances, and \(\dot{x}\) is the derivative of the variable with respect to time.

The state-variable method relies essentially on concepts from circuit topology [5]. To obtain the equations of the dynamic state, a so-called normal tree is constructed, i.e., a tree containing the maximum number of capacitances and the minimum number of inductances. The vector of state variables \(x(t)\) is composed of the voltages of the capacitive branches of the tree and the currents of the inductive links.

The equivalent circuit of the switch under consideration with the selected tree is shown in Fig. 2. For the transistors a modified Ebers–Moll model is adopted, with capacitances \(C_{\mathrm{e}}\) and \(C_{\mathrm{k}}\) added in parallel to the junctions; each of these is determined by the sum of the charge and diffusion capacitances of the corresponding junction:

\[ C_{\mathrm{e}}=C_{\mathrm{ze}}+C_{\mathrm{de}}= \frac{A_{\mathrm{e}}}{\sqrt{\varphi_{\mathrm{oe}}-U_{\mathrm{e}}}} +\tau_{\omega N}\frac{i_1+I_{\mathrm{et}}}{\varphi_{\mathrm{te}}}, \tag{2} \]

\[ C_{\mathrm{k}}=C_{\mathrm{zk}}+C_{\mathrm{dk}}= \frac{A_{\mathrm{k}}}{\sqrt{\varphi_{\mathrm{ok}}-U_{\mathrm{k}}}} +\tau_{\alpha I}\frac{i_2+I_{\mathrm{kt}}}{\varphi_{\mathrm{tk}}}, \tag{3} \]

where \(A_{\mathrm{e}}, A_{\mathrm{k}}\) are empirical coefficients,

\(\varphi_{\mathrm{oe}}, \varphi_{\mathrm{ok}}\) are the contact potential differences for the emitter and collector junctions,

\(U_{\mathrm{e}}, U_{\mathrm{k}}\) are the voltages across the junctions,

\(\tau_{\omega N}, \tau_{\alpha I}\) are the time constants of the emitter and collector current-transfer coefficients, respectively,

\(\varphi_{\mathrm{te}}, \varphi_{\mathrm{tk}}\) are diffusion (thermal) potentials,

\(I_{\mathrm{et}}, I_{\mathrm{kt}}\) are thermal currents,

\(i_1, i_2\) are the currents of the emitter and collector diodes,

\[ i_1=I_{\mathrm{et}}\left(\exp\frac{U_{\mathrm{e}}}{\varphi_{\mathrm{te}}}-1\right), \qquad i_2=I_{\mathrm{kt}}\left(\exp\frac{U_{\mathrm{k}}}{\varphi_{\mathrm{tk}}}-1\right). \]

The tree branches in Fig. 2 are indicated by solid lines, the links by dashed lines. Since the circuit contains a loop consisting of voltage sources and capacitances, one capacitance (\(C_{\mathrm{k2}}\)) is included in a link as a redundant element.

*) The static regime was calculated using a program whose basic principles are set forth in [4].

Fig. 2. Equivalent switching circuit with clipping diodes.

For convenience in obtaining the state equations, we compose the matrix of fundamental loops \(B\) (Table 1). The emitter and collector junctions of the transistors are represented in the matrix by a single branch, i.e.

Table 1

Matrix of fundamental loops of the flip-flop switch

\[ i_{\э}=i_1-a_N i_2+i_{C\э}=i_{a\э}+i_{C\э}, \tag{4} \]

\[ i_{\к}=i_2-a_I i_1+i_{C\к}=i_{a\к}+i_{C\к}, \tag{5} \]

where \(i_{a\э}\) and \(i_{a\к}\) are the static components of the junction currents of the selected model. The matrix \(B\) establishes the following relations between the currents and voltages of the circuit branches:

\[ I_t=BI_l, \tag{6} \]

\[ U_e=-B^{*}U_t, \tag{7} \]

where the index \(t\) corresponds to the branches of the tree, \(l\) to the links, and \(B^{*}\) is the transposed matrix \(B\).

In our case \(y(t)=U_{\mathrm{in}}(t)\); the vector \(x(t)\) will consist of the voltages of the capacitive branches of the tree, i.e.

\[ x=[u_{C_1},u_{C_2},u_{K_1},u_{\Э_1},u_{\Э_2}]^{*},\quad \text{and}\quad \dot{x}=C^{-1}i_C, \]

since for a capacitance

\[ \frac{\partial u_{C_\nu}}{dt}=\frac{i_{C_\nu}}{c_\nu}. \]

Here \(C\) is a diagonal matrix composed of the capacitances, and \(i_C\) is the vector of the corresponding capacitive currents. Consequently, it is necessary to find the currents of the capacitive branches of the tree.

Using the matrix \(B\) and expressions (4) and (5), we find a system of equations of the form (6):

\[ \begin{aligned} i_{C_1} &= i_{r b_1} - i_{R_1} + i_{R_3},& i_{C_{k_1}} &= i_{k_2} - i_{R_H} - i_{\alpha k_1},\\ i_{C_2} &= i_{r b_2} - i_{k_2} - i_{R_4},& i_{C_{e_1}} &= i_{r b_1} - i_{R_H} - i_{k_2} - i_{\alpha e_1},\\ &&i_{C_{e_2}} &= i_{r b_2} - i_{k_2} - i_{\alpha e_2}. \end{aligned} \tag{8} \]

Analogously, relations of the form (7) will be written as the following system:

\[ \begin{aligned} u_{R_1} &= u_{C_1},& u_{R_H} &= u_{k_1} - u_{e_1} + E_1,\\ u_{R_2} &= u_{C_2},& u_{k_2} &= u_{e_1} + u_{e_2} - u_{k_1} - E_1 - E_2,\\ u_{r b_1} &= u_{\text{in}} - u_{C_1} - u_{e_1},& u_{R_3} &= u_{\text{in}} + E_3 - u_{C_1},\\ u_{r b_2} &= -u_{C_2} - u_{C_1} - u_{\text{in}},& u_{R_H} &= u_{C_2} + u_{\text{in}} + E_3 . \end{aligned} \tag{9} \]

Dividing the resistive components of system (9) by the values of the corresponding resistances, we determine the resistor currents of system (8). Since the variables \(u_{e_1}\), \(u_{e_2}\), \(u_{k_1}\) enter the vector \(x\), and \(u_{k_2}\) is determined from (9), the currents \(i_1\) and \(i_2\) for the emitter and collector diodes, and consequently also the currents \(i_{\alpha e_1}\), \(i_{\alpha e_2}\), \(i_{\alpha k_1}\), \(i_{\alpha k_2}\), can be found. Thus, on the right-hand side of equations (8) all quantities are known except the current \(i_{k_2} = i_{\alpha k_2} + i_{C_{k_2}}\).

To find \(i_{C_{k_2}}\), we integrate the expression for \(U_{k_2}\); we obtain

\[ i_{C_{k_2}} = C_{k_2} \left( \frac{i_{C_{e_1}}}{C_{e_1}} + \frac{i_{C_{e_2}}}{C_{e_2}} - \frac{i_{C_{k_1}}}{C_{k_1}} \right). \tag{10} \]

Substituting into the last relation the values \(i_{C_{e_1}}\), \(i_{C_{e_2}}\), and \(i_{C_{k_1}}\) from (8), and solving it with respect to \(i_{k_2}\), we finally obtain

\[ i_{C_{k_2}} = \left( \frac{i_{r b_1}+i_{R_H}-i_{\alpha e_1}}{C_{e_1}} + \frac{i_{r b_2}-i_{\alpha e_2}}{C_{e_2}} + \frac{i_{R_H}+i_{\alpha k_1}}{C_{k_1}} + \frac{i_{\alpha k_2}}{C_{k_2}} \right) \Bigg/ \left( \frac{1}{C_{e_1}} + \frac{1}{C_{e_2}} + \frac{1}{C_{k_1}} + \frac{1}{C_{k_2}} \right). \tag{11} \]

Thus, all expressions are available in order to represent the circuit equations in the form \(x' = f(x, y, t)\).

The obtained system of equations was solved by the fourth-order Runge–Kutta method with a constant step. The input signal was taken to be trapezoidal in shape, with voltage levels of \(-0.3\) V and \(-9.5\) V, a negative-front duration of 100 ns, and a positive-front duration of 50 ns (a signal from a high-frequency trigger). The transient process in the switch was calculated on an M-220 digital computer for the following values of the parameters of the passive elements and transistors:

\[ R_1 = R_2 = 3\,\text{k}\Omega,\quad R_3 = R_4 = R_5 = 1\,\text{k}\Omega,\quad E_1 = E_2 = 10\,\text{V}, \]

\[ E_3 = 2.4\,\text{V},\quad C_1 = C_2 = 1000\,\text{pF}, \quad \text{transistor MP16B }(\alpha_N = 0.987,\ \alpha_I = 0.714, \]

\[ \tau_{\alpha N} = 0.08\,\mu\text{s},\quad \tau_{\alpha I} = 0.3\,\mu\text{s},\quad A_e = A_k = 70\,\text{pF}\cdot\text{V}^{1/2},\quad \varphi_{0e} = \varphi_{0k} = 0.6\,\text{V}, \]

\[ I_{eT} = 1.3\,\mu\text{A},\quad I_{kT} = 1.8\,\mu\text{A},\quad \varphi_{Te} = \varphi_{Tk} = 0.03\,\text{V}), \quad \text{transistor MP11A} \]

\[ (\alpha_I = 0.667,\ I_{eT} = 6\,\mu\text{A},\ I_{kT} = 9\,\mu\text{A},\ \text{the remaining parameters as for MP16B}) \]

The calculated waveform of the output voltage, obtained for \(r_{b1} = r_{b2} = 200\,\Omega\), practically coincides with the experimentally recorded one and is represented in Fig. 3 by the curve \(U_{\text{out}}\). As the resistance of the base layers decreases, in the calculated output voltage before and after the transient pro-

significant overshoots appear in the process (for \(r_{b1}=r_{b2}=100\ \Omega\) they exceed \(3\ \mathrm{V}\)), which in practice does not occur. This reveals a shortcoming of the adopted equivalent circuit. In all other respects it proves quite satisfactory. The experimentally observed overshoots in individual switches reach \(0.5\ \mathrm{V}\), which

Fig. 3. Time diagrams of the input and output voltages

is inadmissible for a reference signal. They can be eliminated by connecting resistors of about \(100\ \Omega\) in series with the accelerating capacitors.

In conclusion, let us note that, if one disregards the indicated criticality with respect to the values of the base resistances, the adopted equivalent transistor circuit proves quite satisfactory and provides results that practically coincide with those observed experimentally.

For relatively simple circuits with up to 10 nodes and one capacitive loop, compiling the matrix \(B\) and solving the system of equations by the method considered are not difficult. The required amount of computer core memory is also small, and any medium-sized digital computers may be used.

The computer time for solving the dynamics problems of the switch considered did not exceed 3.5 minutes.

The authors express their gratitude to Docent M. S. Roitman for valuable advice that made it possible to improve the content of the article.

REFERENCES

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3. Kuh, Rohrer. Application of the state-variable method to circuit analysis. TIIER, 1965, No. 7.

4. V. V. Pfening. Program for calculating the static operating mode of circuits with semiconductor diodes and triodes. Tomsk, Izvestiya Tomskogo politekhnicheskogo instituta, 1971, vol. 231.

5. S. Seshu, M. B. Reed. Linear Graphs and Electrical Networks. Moscow, “Vysshaya shkola,” 1971.