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 (*R _{E} * = 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 *v _{be} * /

*r*becomes a more general transconductance amplifier with a gain of –

_{m}*G*= –1/

_{m}*R*.

_{m}Another extension is the addition of *Z _{L} * for

*R*. The transfer function is found by applying basic circuit laws, beginning with KCL at the output node:

_{L}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:

*T _{i} * 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,

*G*and a passive path,

_{a}*G*:

_{p}To give the general stage a more concrete instantiation, let *Z _{f} * = 1/

*s*x

*C*, and

_{f}*Z*and

_{i}*Z*be shunt RCs:

_{L}Then some of the fragments in the closed-loop expression are

The forward path, *G* , has two parallel paths, an active path, *G _{a} * , through the amplifier and the other, a passive path,

*G*, around the amplifier through

_{p}*Z*. The two forward-path transmittances are

_{f}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 *i _{i} * and

*Z*so that they are instead an input source voltage of

_{i}*Z*x

_{i}*i*in series with

_{i}*Z*. Then if the transimpedance is expressed for the closed loop,

_{i}while the voltage gain of the loop is

Where

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 *R _{m} * (or more gain for a

*G*amplifier) is shown below. (This is a

_{m}*root contour*and not a root locus plot; a circuit parameter other than quasistatic loop gain is varied. Although loop gain increases as

*R*→ 0 Ω ,

_{m}*R*is not the loop gain and it also affects the value of the RHP zero.)

_{m}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,

*G _{p} * , the passive-path gain, is associated with

*τ*, the input-node

_{p}*τ*associated with the passive path to the output.

*G*is the active path gain and

_{a}*τ*is the input-node

_{a}*τ*associated with it. This

*τ*includes the Miller effect on

*C*x

_{f}*τ*is the output-node

_{L}*τ*, and affects both pole-pair magnitude and angle. None of the three time constants are OCTCs, though

*τ*, in the right grouping as

_{L}*R*x (

_{L}*C*+

_{f}*C*), appears in both

_{L}*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*. When the time constants are equated, the resulting condition is

_{p}which is not realizable, for under the condition that *τ _{a} * =

*τ*,

_{p}*C*= -1(1+1/

_{L}*K*) x

_{v}*C*. For real circuits,

_{f}*C*≥ 0 pF and

_{L}*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.

Dennis:

I think you are burying your reader in far too many equations.

Unless they provide INSIGHT, leave 'em out!! Remember, you

lose (at least) one more reader for every equation! Even the

original paper by Miller had fewer…

The Lone Arranger

Dennis,

Engineering and possessing a thorough understanding of circuit devices is not for the faint of heart.

Gaining insight into circuit operation and analysis requires the theoretical engineer to be as fluent in math as he or she is with a native spoken language. For the math enthusiasts, your use of equations is totally appropriate. Keep up the good work.

Since I know you personally, I may be prejudice.

Dave

Vancouver

Well, at least he filled-in some of the derivation steps rather than “leaving them as an exercise for the student”.

K7JEB

Lone, Dave, K7JEB,

Equations have different purposes. Some of them are simply part of the derivation, which I prefer to offer with more steps than usual, to try to encourage readers to think through the math. A simulator cannot do that for you.

Derivations end with design formulas. I try to both put the equations in circuit-interpretable forms and also relate the equations, by term or factor, to the circuit, and what they mean for circuit behavior.

Being literate in eqns is (as Dave suggests)

necessaryfor engineering competency because algebraic eqns show us what happens when various parameters are varied. This can also be done with SPICE, but it is much faster and more insightful in algebra. We are supposed to be proficient in algebra before we even enter engineering school, so I don't this this is too terrible a burden. And since I do the hard work of deriving the design formulas, all the reader has to do is understand their circuit interpretation.I will keep writing articles with equations, primarily because I don't know how to write something like “Circuits for Dummies”. The exceptions are my articles in the category of “engineering and beyond”. The two articles after the Miller's Theorem series are of this kind. One of them is about depths of understanding, which relates to this topic – and without any equations!