Circuit poles from the Cochrun-Grabel method
In the first of this series of related articles on circuit dynamics, “Circuit Dynamics for Design”, the groundwork was laid in polynomial algebra for the development of some simple methods for finding circuit poles and zeros. One of them, the Cochrun-Grabel method, is presented here. It is a tabular method that in manual use is often simple enough to not even require a table. Though more powerful methods have emerged, this one is so easy to use it is worth knowing. It is quite useful for multistage amplifiers with interacting stages.Cochrun-Grabel Method
An improvement in accuracy over the OCTC method was introduced by Cochrun and Grabel in 1973 (“A Method for the Determination of the Transfer Function of Electronic Circuits”, Basil L. Cochrun, Arvin Grabel, IEEE Transactions on Circuit Theory, Vol. CT-20, No. 1, JAN73). It proceduralizes determination of the polynomial coefficients of D(s) for circuits with n capacitors but does not include zeros. It is based on an expansion of s-domain pole polynomial D(s) as in the OCTC method, where the linear term is the sum of the OCTCs. However, the method does not stop there and includes the coefficients of all the powers of s. In normalized form,
a0 = 1
Designate the OCTCs as τi, i = 1,..., n. Then the coefficient of the linear term is
The coefficient of s2 is
where τm;k is τ of the Cm port with the port of Ck shorted: τm;shorted with the remaining ports open. For the τ1 factor, find the time constants of the other ports with port 1 shorted. Then short port 2 and do the rest of the ports. For each successive τi, the number of product terms decreases by one. For n = 2, it is the product of two time constants, an OCTC and the other with the OCTC port shorted.
For n = 3,
By the third degree, the general procedure is quite cumbersome and non-intuitive:
For n = 3, it is not nearly so daunting;
For a large (> 3) value of n, keeping track of the combinations of terms in each coefficient can be arduous, and Sol Rosenstark streamlined the method by tabulating it in his book, Feedback Amplifier Principles (Macmillan, 1986). (I also describe it in Designing High-Performance Amplifiers (D. Feucht, SciTech, 2010); www.scitechpub.com). The use of Rosenstark tables is central to a refined Cochrun-Grabel procedure. The tables are triangular in form; the general second- and third-degree tables are shown below.
The columns are added to give the am after multiplying each column entry by the time constants in the columns to its left. The first (leftmost) column is a1, the sum of the OCTCs. The second collects the combinations of a2. Each column entry is then multiplied by the time constants in each of the columns to its left before adding it to the other (similarly-multiplied) column entries. Rosenstark truncates the tables at a2 or a3 for tractable approximations to D(s). While the Cochrun-Grabel method is not difficult to learn using Rosenstark tables, it is easy to retain intuitively in the case of quadratic or cubic polynomials. Most of the work is in finding the OCTCs. Because the order of choosing capacitors is arbitrary there are two possible ways of computing the quadratic coefficient and they are equivalent. It is demonstrated here for the textbook CE stage (with RE = 0 Ω), and using rπ = (β0 + 1)·re. First, find the OCTCs for both Ce and Cc:
Rbe = RB||rπ ; Rbc = RL + Rbe·(1 + RL/rm)
Now short Cc and find
Re;c = Rbe||(RL||rm)
A shorted Cc places the dependent current source across the voltage upon which it is dependent. By the substitution theorem, if a current source of vbe/rm is placed across vbe, then it is equivalent to a resistance of vbe/(vbe/rm) = rm.
Next, short Ce instead and find the resistance across the b-c port with b-e port shorted:
Rc;e = RL
We only need to short one of the two capacitances for the procedure, but because the order of the capacitors does not matter, there are two ways of working the method and both are given here.
The linear coefficient, b, is the sum of the OCTCs:
The quadratic coefficient can be found in two ways:
After much algebra, the two expressions for a are found to be equivalent. In both cases, the OCTC of the first C is multiplied by the SCTC of the second capacitor with the first C shorted. For either alternative, the resulting pole polynomial is
s2·(τe·τc;e) + s·(τe + tc) + 1 = s2·a + s·b + 1