Design Feature: August 3, 1995

Improved circuit-analysis techniques require minimum algebra

Vatché Vorpérian,
Jet Propulsion Laboratory, California Institute of Technology

When designing circuits, most analog engineers rely largely on experience and trial-and-error methods, eschewing much of the matrix algebra they learned in college. Using the Extra Element Theorem, however, is a helpful analytical technique that can yield insight into network design.

Systematic nodal or loop analysis is the universally adopted formal method of teaching network theory at the undergraduate level. Although the matrix algebra of formal network analysis is ideal for numerical computation on a computer, it fails when you're looking for analytical answers or trying to acquire more insight into a circuit's operation.

Consequently, for most analog-design engineers, the only route to a better understanding of circuits (with a goal of designing improved ones) is the slow process of cut and try. As you follow this route, you'll find yourself either breadboarding circuits or simulating them using CAD tools—but not using the analytical methods you learned in school.

Professor RD Middlebrook, a consultant and educator at California Institute of Technology (Caltech), recognized this problem. He developed efficient analytical methods that derive faster and better analytical answers and avoid the horrors of runaway algebra (Refs 1, 2, and 3). One analytical method, which is called the Extra Element Theorem (EET), cuts through the algebra (see box, "The EET specialized to an impedance function"). For a general discussion and derivation of the EET, see Ref 2.

The circuit in Fig 1 is borrowed from a well-known textbook by LO Chua and Pen-Min Lin (Ref 4). In Fig 1, input resistance is to be determined as a function of the transconductance (g_m) using the parameter-extraction method. Because the parameter-extraction method requires a considerable amount of matrix manipulation, which would become prohibitively complex if all elements were in symbolic form, Chua and Lin assigned numerical values to the resistors: R₁=1 Ohm, R₂=0.2 Ohm, R₃=0.5 Ohm, R₄=10 Ohm, and R_B=0.1 Ohm. The resulting input resistance is

(1a)

which can be rewritten by making the leading term in the numerator and the denominator unity:

(1b)

Note from Eq 1b that if g_mis 0, then R_in=0.7 Ohm.

The result of Eq 1b illustrates the shortcomings of tedious traditional analysis techniques. Matrix manipulation yields results for specific numerical values of network elements but fails to provide symbolic expressions for circuit parameters such as input resistance. As a result, matrix manipulation provides no insight into circuit operation. For instance, Eq 1b provides no information on how changes in R₁, R₂, R₃, R₄, and R_B affect R_in.

However, you can determine such parameters in entirely symbolic form in a few simple steps using the EET twice in succession. The EET allows you to remove impedance elements and dependent sources from a circuit so you can restrict your analysis to a simpler circuit.

The two elements that most complicate the analysis of this circuit are g_mand R_B. So, first take out the dependent current source by letting g_m=0. This step produces the circuit in Fig 2, which is an ordinary bridge circuit with input resistance R_in'. In trying to determine R_in', note that R_Bis the only element left that causes difficulty, so take it out, too. Thus, you derive the circuit in Fig 3a. Note that

R'_in= R_in | g_m= 0 and R''_in= R'_in | R_B= [infinity]

Looking into the input port of Fig 3a immediately shows that R₁ and R₃ parallel R₂and R₄, so that

R''_in= (R₁+ R₃) || (R₂ + R₄).

Now, two more simple calculations are required to determine R_in'. First, determine the resistance looking into port B, with the input port shorted as shown in Fig 3b. Note the parallel combination of R₁ and R₃ in series with the parallel combination of R₂ and R₄, so that

R_S^(B) = R₁ || R₃+R₂ || R₄.

Second, determine the resistance looking into port B with the input port open (Fig 3c). Now, if you look into port B, you see R₁+R₂ in parallel with R₃+R₄ so that

R₀^(B) = (R₁+ R₂) || (R₃+ R₄).

According to EET, R_in' is

Substituting for R_in'', R_S^(B), and R_O^(B) in Eq 6, you obtain the following expression for the input resistance of a bridge circuit in which the influence of R_Bon R_inis very clear:

(7)

Important features of Eq 7 are that R₁ through R₄ appear in series and parallel combinations and that R_B appears in a ratio that can easily be compared to unity. These features allow you to simplify an expression by ignoring the smaller of two series impedances or the larger of two parallel impedances. Similarly, in the expressions of the form 1+x/R_Bin Eq 7, you can ignore the x/R_B term if it's small with respect to unity, or you can ignore the unity term if x/R_B>>1. Eq 7 is called a low-entropy expression (Ref 1) because of its highly ordered nature.

In a second application of EET, reinstate g_mby performing two additional simple calculations with the dependent current source g_mv₁replaced by an independent current source, i_m, pointing in the opposite direction (Figs 4a and 4b). First, determine the inverse gain (v₁/i_m) (transresistance) with the input port shorted (Fig 4a). The current i_m of Fig 4a divides between R_Band the R₁ || R₃ and R₂ || R₄ combinations according to the current division formula:

R₁ and R₃ are in parallel because of the short on the input, so they have the same voltage drop (i times R₁ in parallel with R₃) across them. Therefore, the reverse transresistance with the input shorted is

Next, repeat the same procedure with the input port open (Fig 4b). Using the current division formula between branches R₁+R₂ and R_B in parallel with R₃ + R₄, you obtain

Because the voltage across R₁ is v₁=iR₁, you get

The EET now shows that R_in is

(10)

Substituting for R_in', A_S^(m), and A₀^(m) in Eq 10 yields the desired result:

(11)

The result in Eq 11 is superior to the result in Eq 1b, because it contains useful symbolic information about all the circuit elements. It is important to realize that not all symbolic expressions contain useful information unless their elements are grouped together in series and parallel combinations and ratios (Eq 11). If you were to use nodal or loop analysis, the result would have come out as a single numerator and a single denominator, each containing the sum of products of four resistances at a time. And the coefficient of g_m would contain the sum of products of five resistances at a time.

Such a high-entropy answer not only would have been prohibitively unpleasant to obtain, but also would have contained no recognizable information about the circuit's operation. Obtaining a high-entropy result is precisely how engineers discover that nodal- or loop-analysis techniques fail to help them understand circuit performance, even if they were to carry out the analysis and survive the algebra.

In another example of how the EET can obtain a simple low-entropy result, the bridge circuit in Fig 1 can be made reactive by replacing R_B with a capacitor C_B (Fig 5). This circuit illustrates three important points:

A reactive circuit can be analyzed with the same ease as a resistive circuit without ever having to deal with the reactive impedance term (1/(sC_B).
Single extraction of the parameter g_m is not useful because g_m affects the low-frequency asymptote as well as the pole and the zero of Z_in(s).
By using the EET, you can automatically determine Z_in(s) in factored pole-zero form.

You designate the capacitive-reactance Z_B(s)=1/sC_B as the extra element in Fig 5. Removing it from the circuit yields the circuit in Fig 6. Obtain the input impedance of this circuit from the input impedance of the original circuit in Fig 1 by letting R_B approach infinity in Eq 11. Hence,

(12)

Now reinstate Z_B using the EET, which, in turn, requires that you determine port impedances R_S^(B) and R_O^(B), shown in Figs 7a and 7b, respectively. In Fig 7a, the current i_T consists of the sum of g_mv₁ and the current through the branch made up of R₁ in parallel with R₃ plus R₂ in parallel with R₄. Therefore,

In Fig 7a, v₁ is related to v_T by the voltage division between R₁ in parallel with R₃ and R₂ in parallel with R₄. Therefore,

Hence, for R_S^B = v_T/i_T,

To determine R_O^(B), refer to Fig 7b, where, in this case, i_T consists of the sum of g_mv₁ and the currents through the branches R₁ + R₂ and R₃ + R₄, so that

(15a)

Because v₁ is related to v_T by the voltage division between R₁ and R₂, you can rewrite Eq 15a as

Hence, for R_O^(B)=v_T/i_T,

Using the EET, you obtain Z_in(s):

which you can write in pole-zero form as

where R_O is the low-frequency asymptote given in Eq 12, and

The elegance and simplicity of this derivation illustrate how, when the extra element is a reactance in an otherwise- resistive circuit, the pole and zero can each be determined independently. The EET can be applied to the process of determining any kind of transfer function, such as voltage gain, current gain, or loop gain, and can be extended to two or more extra elements (Refs 3 and 5). Note that this work was executed by the Jet Propulsion Laboratory at Caltech, under a contract with NASA.

The EET specialized to an impedance function
In general, the Extra Element Theorem (EET) can be applied to the determination of any kind of transfer function, such as voltage, current, or loop gain, and can be extended to two or more extra elements. The following derivation is for an impedance function. In Fig A, part a shows a circuit for which Z_in is to be determined. Let Z₁ be an impedance element connected across port 1 such that if Z₁ were taken out of the circuit, the determination of Z_in would be greatly simplified (as in the example of the bridge circuit in the main text). In essence, the EET is a "tool" that helps you do just that. Here's how it works: Take Z₁ out and determine Z_in as Z₁ goes to infinity (see part b). With the input port shorted, look into port 1, across which Z₁ was connected, and determine Z_S⁽¹⁾ (part c). With the input port open, look into port 1, across which Z₁ was connected, and determine Z₀⁽¹⁾ (part d). To obtain Z_in, assemble the three calculations above using the following formula, which is the EET for impedance functions: You could have made Z₁ a short circuit instead of an open circuit, as in part b of the Fig A, in which case the EET would take the form You can also apply the EET when the extra element is a dependent source (any one of the four possible kinds, such as voltage-controlled voltage source, voltage-controlled current source, and so on). Fig B, part a shows a dependent current source controlled by i₁=Au₂ controlled by u₂, which can be either a voltage or a current at some other point in the circuit. Now, apply the EET by performing the following three calculations: Determine Z_in as A goes to zero (part b). Connect an independent source (i_T) across port 1 pointing in the opposite direction of i₁ with the input port shorted; then, determine the inverse gain A_S⁽¹⁾=u₂/i_T (part c). Connect an independent source (i_T) across port 1 pointing in the opposite direction of i₁with the input port open; then, determine the inverse gain A₀⁽¹⁾=u₂/i_T (part d). To determine Z_in, assemble the above calculations using this formula: You also could have set A to infinity, as is common for op-amp circuits, so that the EET would take the form

The EET specialized to an impedance function

In general, the Extra Element Theorem (EET) can be applied to the determination of any kind of transfer function, such as voltage, current, or loop gain, and can be extended to two or more extra elements. The following derivation is for an impedance function.

In Fig A, part a shows a circuit for which Z_in is to be determined. Let Z₁ be an impedance element connected across port 1 such that if Z₁ were taken out of the circuit, the determination of Z_in would be greatly simplified (as in the example of the bridge circuit in the main text). In essence, the EET is a "tool" that helps you do just that. Here's how it works:

Take Z₁ out and determine Z_in as Z₁ goes to infinity (see part b).
With the input port shorted, look into port 1, across which Z₁ was connected, and determine Z_S⁽¹⁾ (part c).
With the input port open, look into port 1, across which Z₁ was connected, and determine Z₀⁽¹⁾ (part d).
To obtain Z_in, assemble the three calculations above using the following formula, which is the EET for impedance functions:

You could have made Z₁ a short circuit instead of an open circuit, as in part b of the Fig A, in which case the EET

would take the form

You can also apply the EET when the extra element is a dependent source (any one of the four possible kinds, such as voltage-controlled voltage source, voltage-controlled current source, and so on). Fig B, part a shows a dependent current source controlled by i₁=Au₂ controlled by u₂, which can be either a voltage or a current at some other point in the circuit. Now, apply the EET by performing the following three calculations:

Determine Z_in as A goes to zero (part b).
Connect an independent source (i_T) across port 1 pointing in the opposite direction of i₁ with the input port shorted; then, determine the inverse gain A_S⁽¹⁾=u₂/i_T (part c).
Connect an independent source (i_T) across port 1 pointing in the opposite direction of i₁with the input port open; then, determine the inverse gain A₀⁽¹⁾=u₂/i_T (part d).
To determine Z_in, assemble the above calculations using this formula:

You also could have set A to infinity, as is common for op-amp circuits, so that the EET would take the form

Author's biography

Vatché Vorpérian holds a PhD from the California Institute of Technology (Caltech) Pasadena, CA, and is a technical-staff member of the Jet Propulsion Laboratory (JPL) at Caltech. For four years, he has assisted in developing spacecraft power supplies. Vorpérian is responsible for developing new technology and providing analytical support for the JPL's power-electronics group. His interests include reading and attending concerts.

Reference

Middlebrook, RD, "Low-Entropy Expressions: The Key to Design-Oriented Analysis," IEEE Frontiers in Education, Twenty-First Annual Conference, Purdue University, Sept 21 to 24, 1991, pg 399 to 403.
Middlebrook, RD, "Null Double Injection and the Extra-Element Theorem," IEEE Transactions on Education, Volume 32, No. 3, August 1989, pg 167 to 180.
"Structured Analog Design," a professional-development course prepared by RD Middlebrook and offered through Ardem Associates (San Dimas, CA).
Chua, Leon O, and Pen-Min Lin, Computer Aided Analysis of Electronic Circuits: Algorithms and Computational Techniques, Prentice Hall, 1975, pg 568 to 569.
Middlebrook, RD, "The Two Extra Element Theorem," IEEE Frontiers in Education, Twenty-First Annual Conference, Purdue University, September 21 to 24, 1991, pg 702 to 708.