Topological Analysis of Transmission Sensitivity

L. A. Naumov, Yu. V. Pikalkin, E. I. Tsimbalist

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

Questions concerning the analysis of the sensitivity of electrical circuits have been examined in sufficient detail in a number of works [1, 2, 3, etc.]. Most often, the transmission sensitivity in the branch of interest, the \(i\)-th branch, is determined by the method of direct differentiation of the found transmission \(T\) with respect to the \(i\)-th branch. Calculations of this kind are cumbersome and, as a rule, contain a considerable number of mutually canceling terms and require simplification operations. It is of definite interest to find the sensitivity of any transmission of an electrical circuit topologically.

In Fig. 1 an electrical circuit is presented, characterized by determinant \(\Delta\), with the distinguished pairs of nodes “\(kg\),” “\(jm\),” and “\(ab\)” for connecting the generator (signal source) \(\Gamma\), the measuring instrument \(\Pi\), and the \(i\)-th branch of interest. As is known [4], the signal transmission from the generator to the instrument is determined by the ratio

Fig. 1.

\[ T_{\Gamma\Pi}=\frac{\Pi}{\Gamma}, \]

where \(\Pi\) is the instrument reading,
\(\Gamma\) is the magnitude of the signal source.

In the present work, the possibility is proved of determining the sensitivity as the product of the transmission of the signal from the generator to a voltmeter \(V_i\), connected to the terminals “\(ab\)” of the \(i\)-th branch, by the transmission of the signal from a current source \(I_i\), connected to the same terminals, to the instrument:

\[ \frac{\partial T_{\Gamma\Pi}}{\partial i} = S_i = \frac{V_i}{\Gamma}\cdot\frac{\Pi}{I_i} = -\,T_{\Gamma V_i}\cdot T_{I_i\Pi}. \tag{1} \]

Let us prove the validity of (1). The transmission \(T_{\Gamma\Pi}\) can be written as

\[ T_{\Gamma\Pi}=\frac{V_i}{\Gamma}\cdot\frac{\Pi}{V_i}. \tag{2} \]

Let us specify the nature of the signal source in order to determine the character of the transmission \(T_{\Gamma V_i}\): let \(\Gamma=e\)—a voltage source. Then the transmission \(V_i/e\) can be determined through the internal total conductance of the circuit \(g_i\) with respect to terminals \(a\) and \(b\) and the conductance of the branch of interest itself \(i\):

\[ T_{\Gamma V_i}=-\frac{V_i}{e}=-\frac{g_i}{g_i+i}, \tag{3} \]

where \(g_i=\dfrac{\Delta_0}{\Delta_i}\), and \(\Delta_0\) and \(\Delta_i\) are elements of the expansion of the circuit determinant with respect to the \(i\)-th branch:

\[ \Delta=i\Delta_i+\Delta_0. \]

The transmission \(\Pi/V_i\) can be determined by Mason’s formula:

\[ \frac{\Pi}{V_i}=\frac{\sum P'\Delta'}{\Delta_i}, \tag{4} \]

where \(P'\) is a path passing through the measuring instrument and the generator \(V_i\) connected to the \(i\)-th branch,

\(\Delta'\) is the algebraic cofactor of the path \(P'\). Substituting (3, 4) into (2), we obtain

\[ T_{\Gamma\Pi}=\frac{V_i}{e}\cdot\frac{\Pi}{V_i} = -\frac{\Delta_0}{\Delta_0+i\Delta_i}\cdot \frac{\sum P'\Delta'}{\Delta_i}. \]

The sensitivity with respect to the \(i\)-th branch is defined as

\[ S_i=\frac{\partial T_{\Gamma\Pi}}{\partial i} = -\frac{\Delta_0}{\Delta}\cdot \frac{\sum P'\Delta'}{\Delta}. \tag{5} \]

The transmission

\[ \frac{V_i}{e}=\frac{\Delta_0}{\Delta} \]

in turn can be calculated through paths passing through the signal source and the voltmeter, and through their cofactors:

\[ \frac{V_i}{e}=\frac{\sum P''\Delta''}{\Delta}. \tag{6} \]

Taking (5, 6) into account, the expression for the sensitivity can be represented in the form

\[ S_i=-\frac{\sum P''\Delta''}{\Delta}\cdot \frac{\sum P'\Delta'}{\Delta} = -T_{\Gamma V_i}\cdot T_{I_i\Pi}, \tag{7} \]

which was to be proved.

Usually, when finding the sensitivity, the circuit determinant \(\Delta\) is known, and it is necessary to compute only the numerator of (7). It is quite obvious that it is equal to the numerators of (4, 6) for any type of instrument and generator connected to the \(i\)-th branch. The obtained relation also extends to circuits containing nullors.

In the most interesting case of analyzing circuits with amplifying elements, when the variable branch is a transconductance (for example, \(s_1\)), the algorithm for finding the numerator of (7) reduces to the following:

1) branches with \(s_1\) are removed from the circuit graph;

2) the voltmeter \(V_i\) is connected between the grid and the cathode (or the base and the emitter), and the numerator of (4) is computed;

3) the source \(V_i\) is connected between the anode and cathode (collector and emitter), and the numerator (6) is computed.

The relative transmission instability \(T_{\mathrm{TT}}\) of the circuit under small changes of the parameters of the \(i\)-th branch can be computed as

\[ \frac{dT_{\mathrm{TT}}}{T_{\mathrm{TT}}} = \frac{1}{T_{\mathrm{TT}}}\sum_{i=1}^{n} S_i \cdot \dot{\partial}_i \cdot i = \frac{1}{\Delta}\sum_{i=1}^{n} i \cdot \Delta_i \cdot \dot{\partial}_i, \]

where the sum is taken over the “\(n\)” varied branches. Example No. 1. Find \(S_i\) for branch \(i=d=r_{ab}\) of the circuit in Fig. 2, a

\[ (k=1,\ r=5,\ j=3,\ m=5,\ a=2,\ b=4). \]

Fig. 2.

The voltage transmission coefficient is equal to

\[ T_{13}=\frac{V}{e}= \]

\[ = \frac{ d[(a+c)(b+f)] + ab(c+f+k) + cf(a+b) }{ d[a(b+f)+(c+k)(b+f)] + a[(c+k)(b+f)+bf] + bf(c+k) } = \]

\[ = \frac{\sum_{\kappa} P_{\kappa}\Delta_{\kappa}}{\Delta}. \]

By the direct method of finding

\[ S_i=\frac{\partial T_{13}}{\partial r_{ab}} \quad \text{gives} \quad S_i= \frac{ak(b+f)[af-b(c+k)]}{\Delta^2}, \]

where, out of 106 terms of the numerator, only 6 are useful, while the remaining ones constitute duplications and cancel.

By the proposed method (Fig. 2, b, 2, c)

\[ S_i= \frac{V_i/e}{\Delta}\cdot\frac{V/V_i}{\Delta} = \frac{ak(b+f)[af-b(c+k)]}{\Delta^2}. \]

Example No. 2. Find \(S_i\) for branch \(i=s_1\) of a transistor amplifier (Fig. 3, a, 3, b)

\[ (k=1,\ r=5,\ j=4,\ m=5,\ a_1=1,\ a_2=2,\ b=3). \]

The voltage transmission coefficient is equal to

\[ T_{14}=\frac{V}{e}= \]

\[ \frac{ s_1[s_2(k+f)+kc] + a[k(b+c)-s_2b] }{ s_1[k(c+s_2)+dc] + a[(b+c)(a+f)+c(b+k)] + } \]

\[ \frac{ s_1[s_2(k+f)+kc] + a[k(b+c)-s_2b] }{ +\,k[c(a+f)+b(a+f+d+c+s_2)] }. \]

Fig. 3.

The direct method for finding \(S_{s_1}=\dfrac{\partial T_{14}}{\partial S_1}\) gives

\[ S_{s_1}= \frac{d\,[f(c+b)+\kappa(b+c)+bc]+\kappa[f(c+b)+b(c+s_2)]}{\Delta} \cdot \frac{[s_2(a+f)+\kappa(s_2+c)]}{\Delta}. \]

By the method of direct differentiation, out of 66 terms, 36 are useful and 30 are duplications.

By the proposed method (Figs. 3г, 3в),

\[ S_{s_1}= \frac{[f(d+\kappa)(c+b)+\kappa d(b+c)+bs_2\kappa+bc(\kappa+d)]\,[s_2(\kappa+f)+\kappa(s_2+c)]}{\Delta^2}, \]

whereas in the transmission (Fig. 3в) two duplications appear.

REFERENCES

1. L. N. Nagornyi. Dependence of the principal parameters of a circuit on variation of the parameters of one of its elements. Elektrosvyaz’, 1961, No. 6.

2. I. Chaika. Sensitivity of the transfer function of an oriented graph to changes in the transmission of any branch of the graph. Izvestiya vuzov SSSR, Radioelektronika, 1971, No. 3.

3. V. F. Borisov. On the application of signal graphs to the calculation of parametric reliability of transistor circuits of radioelectronic equipment. In: Accuracy of Radioelectronic Equipment, Moscow, ROTO, MDNTP, 1971.

4. S. Mason, G. Zimmermann. Electronic Circuits, Signals, and Systems. Moscow, IL, 1963.