Extending Blackman's Formula to
Multiple-TransistorFeedback Circuits
Eugene Paperno
AbstractThe Blackman's formula for the impedance seen from an arbitrary terminal of a single-transistor feedback circuit is extended to the case of multiple-transistor circuits. To reach this aim, we revisit the proof of the Blackman's formula for single-transistor circuits, proof it in a similar way for double-transistor circuits, and then extend for the case of multiple-transistor circuits.
Index TermsBlackman's formula,feedback circuits,impedance evaluation, single-, double-, triple-, multiple-transistor circuits,return ratio.
I.Introduction
To examine the effect of negative feedback [1]-[17] on terminal impedances, it is important to describe the impedances analytically as a function of the feedback partial gains, such as return ratios. The impedance of an arbitrary terminal of a single-transistor circuit can be described by the Blackman's formula [3]:
,
whereRt is the closed-loop impedance seen by a test source vt connected to an arbitrary terminal of the circuit [see Fig. 1(a)], R't is the open-loop terminal impedance, RRsc is the return ratio for the short-circuited terminal, and RRoc is the return ratio for the open-circuited terminal.
The aim of the present work is to extend the Blackman's formula to the case of multiple-transistor circuits. To reach this aim, we revisit the proof of the Blackman's formula for single-transistor circuits, proof it in a similar way for double-transistor circuits, and then extend for the case of multiple-transistor circuits.
II. Single-Transistor Feedback Circuits
Let us consider a linear equivalent model of a generic single-transistor circuit (see Fig. 1). To find the impedance of an arbitrary terminal, we connect to it a voltage test source in Fig. 1(a) and find by applying superposition the control signal
,(1)
where the input transmittance for the voltage test source
,(2)
and the return ratio for the short-circuited (vt=0) terminal
.(3)
Note that the signals in (2) and (3) with the single and double prime symbolscorresponds to the case, where the only active sources are vt and aOLs, respectively.
From (1),vt can be obtained as:
.(4)
Let us now replace in Fig. 1(b) source vt withit, such that it = vt/Rt, to keep the same conditions of the test source brunch. Keeping the same brunch voltage and current leaves the signal sunchanged. As a result,
,(5)
where the input transmittance for the current test source
,(6)
and the return ratio for the open-circuited (it=0) terminal
.(7)
From (5),it can be obtained as:
.(8)
Hence, the terminal impedance
.(9)
Gv and Gi in (9) can be found from Figs. 1(c) and (d):
.(10)
,(11)
where R't is the open-loop terminal impedance (for aOL=0).
Note that it=vt/R't in (11) helps keeping the same conditions of the test source brunch for both vt and i't sources, as a result s'in (10) and (11) has the same value.
Considering (9)(11), we obtain the Blackman's formula for single-transistor circuits:
.(12)
III.Double-Transistor Feedback Circuits
Following similar approach, we extend in this section the Blackman's formula for double-transistor circuits (see Fig. 2).
Figs. 2(a) and (b) suggest that the control signals of the dependent sources
,(13)
.(14)
Equations (13) and (14) can be solved forvt and it:
,(15)
,(16)
Dividing (15) by (16) gives
,(17)
where
.(18)
The first term in (17) represents the open loop impedance seen by vt:
Fig. 1. Finding the impedance seen by a test sourceconnected to anarbitrary terminal of a generic single-transistor feedback circuit.
,(19)
where i't=vt/R't [see Fig. 2(d)].
To find the second term in (17), let us consider Fig. 3, where vt=1, it=1/R*t, s1=1, and s2=0, and, hence:
,(20)
.(21)
From (20) and (21),
.(22)
Considering that in (22) in accordance with (19),
,(23)
(22) can be rewritten as follows:
.(24)
Considering (17), (19), and (24),Rt can eventually be obtained as:
.(25)
IV.Multiple-Transistor Feedback Circuits
Following the approach given in the previous sections, the terminal impedance can be obtained in accordance with (25) for multiple-transistor feedback circuits. For example, for triple-transistor circuits the short- and open-circuit return ratios in (25) are as follows:
,(26)
.(27)
Fig. 2. Finding the impedance seen by a test sourceconnected to anarbitrary terminal of a generic double-transistor feedback circuit.
Appendix
Double-Transistor Example Circuit
Let us solve a double-transistor circuit (see Fig. A.1) for the impedance seen by the input source. From Fig. A.1(c),
.(A.1)
From Fig. A.1(d),
,(A.2)
,(A.3)
.(A.4)
From Fig. A.1(e),
,(A.5)
Fig. 3. Ageneric double-transistor feedback circuit withvt=1, i*t=1/R*t, s1=1, and s2=0.
,(A.6)
,(A.7)
.(A.8)
.(A.9)
By substituting (A.1)‒(A.9) into (18) and (25), one can obtain the impedance seen by the input source.
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Fig. A.1. Finding the input impedance of a double-transistor circuit. (a) Example circuit. (b) Original equivalent small-signal circuit. (c) The "prime" circuit, where the independent source is the only active one. (d) The "double-prime" circuit, where hfe1ib1 is the only active source. (e) The "triple-prime" circuit, where hfe2ib2 is the only active source. Note that the dependent sources in (d) and (e) are controlled by the corresponding signals of the original circuit (b).
E. Paperno is with the Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel (e-mail: ).