In the undergraduate active-circuits course in electronics engineering, the analysis of circuits is simplified, as it must be for introductory presentation. One of the simplifications that is often not true in engineering beyond school is the “textbook common-emitter (CE) amplifier stage” analysis with zero emitter resistance (RE = 0 Ω ), as shown below.
The textbook CE stage can be generalized to the single-stage amplifier, shown below, where the BJT dependent current source of vbe /rm becomes a more general transconductance amplifier with a gain of –Gm = –1/Rm .
Another extension is the addition of ZL for RL . The transfer function is found by applying basic circuit laws, beginning with KCL at the output node:
where transconductance-amplifier output current is defined as positive coming out of the amplifier. Then the amplifier current on the right side of the equation is going into the amplifier and out of the output node. This equation boils down to the voltage gain,
This is the open-loop forward-path voltage gain. Applying KCL at the input node,
Combining the two KCL equations,
Rearranging, the closed-loop transfer function is
The closed-loop transimpedance is in a form that makes the feedback blocks explicit for the feedback gain formula:
Ti precedes the feedback loop and is not in it, yet is entangled in the gain formula, as so often happens for feedback circuits. The forward path consists of two parallel paths, an active path, Ga and a passive path, Gp :
To give the general stage a more concrete instantiation, let Zf = 1/s x Cf , and Zi and ZL be shunt RCs:
Then some of the fragments in the closed-loop expression are
The forward path, G , has two parallel paths, an active path, Ga , through the amplifier and the other, a passive path, Gp , around the amplifier through Zf . The two forward-path transmittances are
The complete forward path is thus
The closed-loop gain can be expressed as a voltage gain and the amplifier reconfigured as a voltage amplifier by thevenizing the Norton input of ii and Zi so that they are instead an input source voltage of Zi x ii in series with Zi . Then if the transimpedance is expressed for the closed loop,
while the voltage gain of the loop is
The left grouping of the linear coefficient, b has its terms separated according to open-circuit time constants (OCTCs) per capacitance; the right grouping is by node resistances and their time constants. The pole-zero migration for decreasing Rm (or more gain for a Gm amplifier) is shown below. (This is a root contour and not a root locus plot; a circuit parameter other than quasistatic loop gain is varied. Although loop gain increases as Rm → 0 Ω , Rm is not the loop gain and it also affects the value of the RHP zero.)
Examination of the a and b coefficients of D (s ) leads to identification of three time constants, as shown in the table below. (|| is a math operator, not a topological descriptor.) D (s ) takes the form,
Gp , the passive-path gain, is associated with τ p , the input-node τ associated with the passive path to the output. Ga is the active path gain and τ a is the input-node τ associated with it. This τ includes the Miller effect on Cf x τ L is the output-node τ , and affects both pole-pair magnitude and angle. None of the three time constants are OCTCs, though τ L , in the right grouping as RL x (Cf + CL ), appears in both a and b . This does not necessarily mean that it is an OCTC. (Note: OCTCs are a basic concept in dynamic circuit analysis. Refer to Circuit Dynamics: Design-Oriented Analysis at http://www.innovatia.com for a more complete, step-by-step development of the subject.)
Though the three time constants are not OCTCs, D (s ) does determine the poles of the circuit and is in the same form as the CE stage. Their separation in the above s -plane plot can be expressed by the damping, ζ , of the generalized single-stage amplifier:
Maximum bandwidth occurs when both poles are as high in frequency as possible which occurs when they are equal, at ζ = 1. Bandwidth is maximum when ζ is minimum. Then τ L = τ a = τ p . When the time constants are equated, the resulting condition is
which is not realizable, for under the condition that τ a = τ p , C L = -1(1+1/Kv ) x Cf . For real circuits, C L ≥ 0 pF and τ a > τ p .
In the next part of this series, we return to Miller’s theorem, and consider what the consequences are when the amplifier is not ideal and has a finite frequency response.