In Seemingly Simple Circuits: The TL431 Voltage Regulator, Part 2, we looked at how voltage gain of the TL431 is measured. Now we plunge deeper in, to look at closed-loop output resistance, derive some formulas, then calculate transconductance and output resistance from manufacturer part data.
The closed-loop incremental output resistance of the TL431 is specified using a test circuit similar to the gain-test circuit but with its divider removed so that νB = νO; R1 = 0
, R2 →
. The changes to the gain-test circuit are
A slightly modified block diagram results. With the Gm amplifier nulled, the open-loop
νG is varied (νg ≠0 V), and νg and νo are measured. As in the gain-test circuit, the open-loop contribution of νg to νo is Τg x νg. The resistance-test circuit equations reduce to
The closed-loop output-port rout is found applying Ohms Law with RL sourcing io = ig , in series with νg ;
and can be calculated from measurement of νo and νg.
From circuit analysis, feedback reduces the open-loop voltage Τg x νg by 1/(1 + G). The divider Τg is effectively
This is the Τg divider with RL and rout(cl) resistances. Equating the second and last expressions and solving,
The quasistatic output resistance of the TL431 is ro in parallel with the 1/Gm source which is dependent on
νe and is also across νg = νb = νo ;
This result also follows directly from the substitution theorem: the voltage on which the current source of Gm depends is also across it, thereby reducing the dependent current source to a resistance, 1/Gm.
rout from Gm Currents
For the resistance-test circuit, the current of the Gm source can be decomposed into currents in ro and RL:
The fraction of Gm current in ro with νb = νo (as in the resistance-test circuit) is
Polarities of current and voltage can be somewhat confusing in this circuit. Because the Gm amplifier sinks current (into the amplifier) as positive, then for KCL to apply, positive current in ro flows upward to a negative νo . The same applies to RL; νo(RL) = io x RL where io and RL are non-negative.
Applying G = Gm x rout while solving for the closed-loop resistance of ro,
The Gm current in RL is
and the closed-loop resistance of RL is
Thus the closed loop effectively reduces the resistance of the branches of the output node by G.
Combining the closed-loop resistances contributing Gm current,
As a circuit, Gm and rout form a dependent current source of resistance 1/Gm in parallel with rout. The same current flows in the parallel loop formed by the two resistances. Gm and rout are independent only if νo across them is 0 V. Without νG (from the test circuits) or a non-static VR, no voltage variation is expected at the output node; νo = 0 V, and rout and 1/ Gm are independent. The above equation implies that νo = 0 V and it also tells us that the 1/ Gm source is effectively equal in resistance to rout/G.
The closed-loop rout derived previously can be expressed in terms of this result as
Thus rout (cl) is reduced from the open-loop rout by the feedback factor, 1 + G, a result consistent with feedback theory; voltage outputs are generally reduced in resistance with a closed loop by feedback factor 1 + G x H.
To measure rout(cl), the output node must be driven with either an independent current source or a voltage source such as νg behind RL. Its contribution to νo causes the resulting current into the node to be that of the closed-loop rout. Substituting the closed-loop νo into the closed-loop io,
The closed-loop contribution of νg to νo is the amplifier closed-loop gain with Τg x νg applied to the output node.