On the Calculation of a Microphone Circuit

N. P. Dovganovsky, G. E. Pukhov

The calculation of circuits with variable parameters by the chain method [1] is carried out using the matrix equation

\[ R(t_s)=\zeta(\tau_s)\zeta(\tau_{s-1})\ldots \zeta(\tau_1)R(t_0)=\zeta(t_s)R(t_0), \tag{1} \]

in which \(R(t_k)\) represents the extended vector of continuous variables for the time instant \(t_k(k=s,s-1,\ldots,1)\), \(\zeta(\tau_k)\) are the extended parameters of the process in the circuit for the \(k\)-th interval and, finally, \(\zeta(t_s)\) is the extended parameter of the process with duration

\[ t_s-t_0=\sum_{k=s}^{1}\tau_k . \]

Let the microphone circuit (Fig. 1) be described by the equation

\[ L\frac{di}{dt}+r(t)i=E, \tag{2} \]

where

\[ r(t)=R-r\cos \omega t=R(1-\lambda \cos \omega t), \]

with

\[ R=R_m+R_n \quad \text{and} \quad \lambda=\frac{r}{R}. \]

Fig. 1

For a small time interval \(\tau_k\), equation (2) is written in the form:

\[ L\frac{i(t_k)-i(t_{k-1})}{\tau_k} = -\frac{1}{2}\left[ r(t_{k-1})i(t_{k-1})+r(t_k)i(t_k) \right]+E. \tag{3} \]

Solving (3) with respect to the current \(i_k=i(t_k)\), we obtain

\[ i_k= \frac{1-\dfrac{\delta_{k-1}\tau_k}{2}} {1+\dfrac{\delta_k\tau_k}{2}} \,i_{k-1} + \frac{\tau_k} {1+\dfrac{\delta_k\tau_k}{2}} \cdot \frac{E}{L}, \tag{4} \]

where

\[ \delta_k=\frac{r(t_k)}{L}\quad \text{and}\quad \delta_{k-1}=\frac{r(t_{k-1})}{L}. \]

Consequently, for the microphone circuit the quantities \(R(t)\), \(\zeta(\tau_k)\), and \(\zeta(t_k)\) are equal to:

\[ R(t)= \begin{vmatrix} i\\ 1 \end{vmatrix}, \quad \zeta(\tau_k)= \begin{vmatrix} \dfrac{1-\dfrac{\delta_{k-1}\tau_k}{2}}{1+\dfrac{\delta_k\tau_k}{2}} & \dfrac{\tau_k}{1+\dfrac{\delta_k\tau_k}{2}}\dfrac{E}{L}\\ 0 & 1 \end{vmatrix}, \quad \zeta(t_k)= \begin{vmatrix} \zeta(t_k) & \zeta_0(t_k)\\ 0 & 1 \end{vmatrix}, \tag{5} \]

The vector \(R(t)\) may be called the augmented current vector of the circuit.

Equation (1) is equally valid both for the steady-state and for the transient regime of the circuit. Only in the former case, instead of \(R(t_s)\), one should substitute the vector \(R_0\), equal to the value of \(R(t)\) at the beginning of the steady-state cycle. This vector is calculated by the formula [1]:

\[ R_0= \begin{vmatrix} [1-\zeta(T)]^{-1}\zeta_0(T)\\ 1 \end{vmatrix}, \tag{6} \]

after preliminary determination of the parameters \(\zeta(T)\) and \(\zeta_0(T)\) from the expression

\[ \zeta(T)=\prod_{k=1}^{l}\zeta(\tau_k)= \begin{vmatrix} \zeta(T) & \zeta_0(T)\\ 0 & 1 \end{vmatrix}, \tag{7} \]

where \(l\) represents the number of intervals into which the cycle is divided \(\left(T=\dfrac{2\pi}{\omega}\right)\).

Let us show the application of the given relations by an example of calculating a particular circuit.

Example. Let us calculate the current of a carbon microphone for the transient and steady-state regimes with the following parameters: \(E=1.5\) V, \(R_i=2\) ohm, \(R_m=100\) ohm, \(r=50\) ohm, \(f=800\) cps, \(L=5\cdot10^{-3}\) H, whose values have been taken from [3].

1. We determine the quantities entering into the equation of the microphone circuit:

\[ R=R_m+R_i=102\ \text{ohm},\quad \omega=2\pi f=5\cdot10^3\ \text{sec}^{-1} \]

\[ \lambda=\frac{r}{R}=0.49. \]

1. We divide the period \((T)\) of variation of \(r(t)\) into \(n\) (let us take 12) equal intervals of duration

\[ \tau_k=\tau=\frac{T}{n}=\frac{2\pi}{\omega n}=1.047\cdot10^{-4}\ \text{sec}. \]

1. We determine the augmented parameters of the process in the circuit for all intervals of the cycle

\[ \zeta(\tau_1)= \begin{vmatrix} \dfrac{1-\dfrac{\delta_0\tau}{2}}{1+\dfrac{\delta_1\tau}{2}} & \dfrac{\tau}{1+\dfrac{\delta_1\tau}{2}}\dfrac{E}{L}\\ 0 & 1 \end{vmatrix} = \begin{vmatrix} 0.28201 & 0.01945\\ 0 & 1 \end{vmatrix} \]

and so on (see the second column of the table).

Table

Calculation of the current of a microphone circuit

1. We find the extended parameters for the time intervals

\[ t_1,t_2,\ldots,t_{12}: \]

\[ \zeta(t_1)=\zeta(\tau_1),\quad \zeta(t_2)=\zeta(\tau_2)\,\zeta(t_1)= \]

\[ = \begin{vmatrix} 0{,}21325 & 0{,}01739\\ 0 & 1 \end{vmatrix} \cdot \begin{vmatrix} 0{,}28201 & 0{,}01945\\ 0 & 1 \end{vmatrix} = \begin{vmatrix} 0{,}06014 & 0{,}02154\\ 0 & 1 \end{vmatrix} \]

and so on (see the third column of the table).

1. On the basis of equation (1), we determine the currents at the end of each interval of the first cycle of the transient regime.

Taking

\[ R(t_0)= \begin{vmatrix} 0\\ 1 \end{vmatrix}, \]

since at \(t=0\), \(i_{t=0}=0\), we further note that

\[ R(t_k)= \begin{vmatrix} \zeta(t_k) & \zeta_0(t_k)\\ 0 & 1 \end{vmatrix} \begin{vmatrix} 0\\ 1 \end{vmatrix} = \begin{vmatrix} \zeta_0(t_k)\\ 1 \end{vmatrix}. \]

Thus, the required currents have in fact already been computed and are equal to

\[ i_1=0{,}01945\,a,\quad i_2=0{,}02154\,a \]

and so on (see the third and fourth columns of the table).

1. We determine the currents at the end of each interval of the second cycle, taking into account that for it the vector

\[ R(t_0)= \begin{vmatrix} 0{,}02573\\ 1 \end{vmatrix} \]

is equal to the extended current vector of the twelfth interval of the first cycle of the established regime. We have:

\[ R(t_1)=\zeta(t_1)R(t_0)= \begin{vmatrix} 0{,}28201 & 0{,}01945\\ 0 & 1 \end{vmatrix} \times \begin{vmatrix} 0{,}02573\\ 1 \end{vmatrix} = \begin{vmatrix} 0{,}02671\\ 1 \end{vmatrix}, \quad i_1=0{,}02671a, \]

\[ R(t_2)=\zeta(t_2)R(t_0)= \begin{vmatrix} 0{,}06014 & 0{,}02154\\ 0 & 1 \end{vmatrix} \times \begin{vmatrix} 0{,}02573\\ 1 \end{vmatrix} = \begin{vmatrix} 0{,}02309\\ 1 \end{vmatrix}, \quad i_2=0{,}02309a \]

and so on (see the fifth column of the table).

1. Let us determine the current of the circuit in the established regime. Since \(T=t_{12}\) and

\[ \xi(t_{12})= \begin{vmatrix} 0{,}00000 & 0{,}02573\\ 0 & 1 \end{vmatrix}, \]

then \(\zeta(T)=0{,}00000\) and \(\zeta_0(T)=0{,}02573\,a\).

and therefore

\[ R_0= \begin{vmatrix} [1-0{,}00000]^{-1}\cdot 0{,}02573\\ 1 \end{vmatrix} = \begin{vmatrix} 0{,}02573\\ 1 \end{vmatrix}. \]

Since \(R_0\) is equal to \(R(t_0)\) for the second cycle, the current in the circuit in the established regime will vary according to the same law as in the second cycle (see the fifth column of the table).

The graph of the current \(i=i(t)\) is shown in Fig. 2. We note that if the interval \(\omega\tau\) is taken to be equal to \(\frac{\pi}{18}\) (i.e., \(10^\circ\)), then the results of the calculation coincide with those obtained for this example in [3] by A. I. Yakovlev (see the sixth column of the table).

Fig. 2

References

1. Pukhov G. E. Calculation of an electrical circuit with stepwise-variable resistance (published in the present volume of Izvestiya TPI).
2. Yakovlev A. I. The problem of the current in a microphone circuit. Trudy MEI, 1948.
3. Yakovlev A. I. Calculations of a carbon and condenser microphone. Collected Papers of VZEI, 1951.