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RC Networks

While perusing through volume one of Chestnut and Mayer's Servomechanisms and Regulating System Design (John Wiley & Sons, New York, second edition, 1959) I came across an interesting table on page 560 (the same table is also in Korn and Korn's Electronic Analog and Hybrid Computers from 1964, another McGraw-Hill publication, as Table A-1 on page 553). At the bottom of the table was the attribute to the origin of this table: an article in the April 1952 edition of Electronics magazine, published by McGraw-Hill. As luck would have it, the American Radio History website has a scanned copy of that magazine: Electronics, McGraw-Hill, April 1952. And there on page 147 (155 of the PDF) is the original table (Table 1) at the end of an article on Driftless DC Amplifiers by Frank Bradley and Rawley McCoy of Reeves Instrument Corp, New York, NY, the table itself apparently the work of their colleague S. Godet.

So, what's all the fuss about something from 66 years ago? Take a look: here's the table from that magazine article: Table 1: Transfer Functions of R-C Input and Output Networks.

What an interesting treasure trove of RC networks. Some familiar, some not so familiar. And all catalogued complete with expressions describing their function.

In Godet's table the networks were grouped by transfer impedance functions. We could also group them by the network configuration and their conjugate, swapping R and C. For example, take the single R - it's conjugate would be the single C. Some network configurations would not have a natural conjugate, for example the R and C in parallel or in series - swapping the R and C would produce the exact same network. It is also interesting to note that Godet categorised the networks into those with a pure resistive path, using $$A$$ as the expression for pure resistance (assume all capacitors open circuit), and using $$B$$ as the expression for the pure reactance (assume all resistors short circuit). In active circuits where DC bias conditions are required, then at least one of the networks must have a DC path to the bias voltage (either ground or some other bias voltage source).

The full title of these transfer functions is "short-circuit transfer impedance functions". This means that the functions relate the input voltage to the output current into a short-circuit. Which is exactly what is required for circuit blocks based on the virtual-earth inverting op-amp circuit:

Here, the input pin to the amplifier is driven to be zero volts (virtual earth) by the feedback path around the amplifier through $$Z_o$$, precisely cancelling any currents flowing in through $$Z_i$$.

The transfer function $$\frac{V_{out}}{V_{in}}$$ is the well-known $$\frac{-Z_o}{Z_i}$$. Given a desired transfer function we can then split it into two parts for $$Z_o$$ and $$Z_i$$ and then consult the table to find suitable RC networks.

It is interesting to note that when the paper was published the notation of the time was to use $$p$$ (or sometimes $$P$$) as the complex frequency variable, compared to today where $$s$$ (the Laplace operator) is used. In this case they can be used interchangably, so for $$p$$ read $$s$$ to get to something that looks more familiar.

On this page, over time, I will be cataloguing each of these circuits, as well as providing further information about them, such as DC paths (useful for setting up bias conditions).

In the following table, using the same notation as the original, time constants are denoted $$T_n$$, DC path resistance is denoted $$A$$, AC path reactance is denoted $$B$$, and, where applicable, time constant scaling factor $$\theta$$.

Note that the table is split into two sections: the first section covers DC path networks, while the second section covers DC blocking networks.

IDDC Path
Transfer Impedance Function Network Relations Inverse Relations
I $$A$$
$$A = R$$ $$R = A$$
II $$\frac{A}{1+sT}$$
$$A = R$$
$$T = RC$$
$$R = A$$
$$C = \frac{T}{A}$$
III $$A(1+sT)$$
$$A = 2R$$
$$T = \frac{RC}{2}$$
$$R = \frac{A}{2}$$
$$C = \frac{4T}{A}$$
IV $$A \left( \frac{1+s{\theta}T}{1+sT} \right)$$
$$\theta < 1$$
$$A = R_1 + R_2$$
$$T = R_2 C$$
$$\theta = \frac{R_1}{R_1 + R_2}$$
$$R_1 = A \theta$$
$$R_2 = A(1-\theta)$$
$$C = \frac{T}{A(1-\theta)}$$
$$A = R_1$$
$$T = (R_1 + R_2)C$$
$$\theta = \frac{R_2}{R_1 + R_2}$$
$$R_1 = A$$
$$R_2 = \frac{A\theta}{1-\theta}$$
$$C = \frac{T(1-\theta)}{A}$$
V $$A \left( \frac{1+sT}{1+s{\theta}T} \right)$$
$$\theta < 1$$
$$A = \frac{2 R_1 R_2}{2 R_1 + R_2}$$
$$T = \frac{R_1 C}{2}$$
$$\theta = \frac{2 R_1}{2 R_1 + R_2}$$
$$R_1 = \frac{A}{2(1-\theta)}$$
$$R_2 = \frac{A}{\theta}$$
$$C = \frac{4T(1-\theta)}{A}$$
$$A = 2R_1$$
$$T = \left( R_2 + \frac{R_1}{2}\right) C$$
$$\theta = \frac{2 R_2}{2 R_2 + R_1}$$
$$R_1 = \frac{A}{2}$$
$$R_2 = \frac{A\theta}{4(1-\theta)}$$
$$C = \frac{4T(1-\theta)}{A}$$
$$A = 2R$$
$$T = \frac{R}{2}(C_1 + C_2)$$
$$\theta = \frac{2 C_2}{C_1 + C_2}$$
$$R = \frac{A}{2}$$
$$C_1 = \frac{2T(2-\theta)}{A}$$
$$C_2 = \frac{2T\theta}{A}$$
VI $$A \left[ \frac{1+sT_2}{(1+sT_1)(1+sT_3)} \right]$$
$$T_1 < T_2 < T_3$$
$$A = R_1 + R_2$$
$$T_1 = R_1 C_1$$
$$T_2 = \left( \frac{R_1 R_2}{R_1 + R_2} \right) ( C_1 + C_2 )$$
$$T_3 = R_2 C_2$$
$$R_1 = \frac{A(T_2 - T_1)}{T_3 - T_1}$$
$$R_2 = \frac{A(T_3 - T_2)}{T_3 - T_1}$$
$$C_1 = \frac{T_1(T_3 - T_1)}{A(T_2 - T_1)}$$
$$C_2 = \frac{T_3(T_3 - T_1)}{A(T_3 - T_2)}$$
$$A = R_2$$
$$T_2 = R_1 C_1$$
$$T_1 T_3 = R_1 R_2 C_1 C_2$$
$$T_1 + T_3 = R_1 C_1 + R_2 C_2 + R_2 C_1$$
$$R_1 = \frac{A {T_2}^2}{(T_3 - T_2)(T_2 - T_1)}$$
$$R_2 = A$$
$$C_1 = \frac{(T_3 - T_2)(T_2 - T_1)}{A T_2}$$
$$C_2 = \frac{T_1 T_3}{A T_2}$$
$$A = R_1 + R_2$$
$$T_2 = \left( \frac{R_1 R_2}{R_1 + R_2} \right) C_2$$
$$T_1 T_3 = R_1 R_2 C_1 C_2$$
$$T_1 + T_3 = R_1 C_1 + R_2 C_2 + R_2 C_1$$
$$R_1 = \frac{A {T_2}^2}{T_1 T_2 + T_2 T_3 - T_1 T_3}$$
$$R_2 = \frac{A(T_3 - T_2)(T_2 - T_1)}{T_1 T_2 + T_2 T_3 - T_1 T_3}$$
$$C_1 = \frac{T_1 T_3}{A T_2}$$
$$C_2 = \frac{(T_1 T_2 + T_2 T_3 - T_1 T_3)^2}{A T_2 (T_3 - T_2)(T_2 - T_1)}$$
$$A = 2R$$
$$T = \frac{R}{2}(C_1 + C_2)$$
$$\theta = \frac{2 C_2}{C_1 + C_2}$$
$$R = \frac{A}{2}$$
$$C_1 = \frac{2T(2-\theta)}{A}$$
$$C_2 = \frac{2T\theta}{A}$$
IDDC Blocking
Transfer Impedance Function Network Relations Inverse Relations
I $$\frac{1}{sB}$$
$$B = C$$ $$C = B$$
II $$\frac{1}{sB}(1+sT)$$
$$B = C$$
$$T = RC$$
$$R = \frac{T}{B}$$
$$C = B$$
III $$\frac{1}{sB} \left( \frac{1+sT}{sT} \right)$$
$$B = \frac{C}{2}$$
$$T = 2RC$$
$$R = \frac{T}{4B}$$
$$C = 2B$$
IV $$\frac{1}{sB} \left( \frac{1+sT}{1+s{\theta}T} \right)$$
$$\theta < 1$$
$$B = C_1$$
$$T = R(C_1 + C_2)$$
$$\theta = \frac{C_2}{C_1 + C_2}$$
$$R = \frac{T(1-\theta)}{B}$$
$$C_1 = B$$
$$C_2 = \frac{B\theta}{1-\theta}$$
$$B = C_1 + C_2$$
$$T = RC_2$$
$$\theta = \frac{C_1}{C_1 + C_2}$$
$$R = \frac{T}{B(1-\theta)}$$
$$C_1 = B\theta$$
$$C_2 = B(1-\theta)$$
V $$\frac{1}{sB} \left( \frac{1+s{\theta}T}{1+sT} \right)$$
$$\theta < 1$$
$$B = C_2$$
$$T = RC_1\left( \frac{2C_2 + C_1}{C_2} \right)$$
$$\theta = \frac{2C_2}{2C_2 + C_1}$$
$$R = \frac{T\theta^2}{4B(1-\theta)}$$
$$C_1 = \frac{2B(1-\theta)}{\theta}$$
$$C_2 = B$$
$$B = \frac{C_1^2}{2C_1+C_2}$$
$$T = RC_2$$
$$\theta = \frac{2C_1}{2C_1 + C_2}$$
$$R = \frac{T\theta^2}{4B(1-\theta)}$$
$$C_1 = \frac{2B}{\theta}$$
$$C_2 = \frac{4B(1-\theta)}{\theta^2}$$
$$B = \left(\frac{R_1}{R_1 + R_2}\right)C$$
$$T = R_2C$$
$$\theta = \frac{2R_1}{R_1 + R_2}$$
$$R_1 = \frac{T\theta^2}{2B(2-\theta)}$$
$$R_2 = \frac{T\theta}{2B}$$
$$C = \frac{2B}{\theta}$$
VI $$\frac{1}{sB} \left[ \frac{(1+sT_1)(1+sT_3)}{1+sT_2} \right]$$
$$T_1 < T_2 < T_3$$
$$B = C_2$$
$$T = RC_1\left( \frac{2C_2 + C_1}{C_2} \right)$$
$$\theta = \frac{2C_2}{2C_2 + C_1}$$
$$R = \frac{T\theta^2}{4B(1-\theta)}$$
$$C_1 = \frac{2B(1-\theta)}{\theta}$$
$$C_2 = B$$
$$B = \frac{C_1^2}{2C_1+C_2}$$
$$T = RC_2$$
$$\theta = \frac{2C_1}{2C_1 + C_2}$$
$$R = \frac{T\theta^2}{4B(1-\theta)}$$
$$C_1 = \frac{2B}{\theta}$$
$$C_2 = \frac{4B(1-\theta)}{\theta^2}$$
$$B = \left(\frac{R_1}{R_1 + R_2}\right)C$$
$$T = R_2C$$
$$\theta = \frac{2R_1}{R_1 + R_2}$$
$$R_1 = \frac{T\theta^2}{2B(2-\theta)}$$
$$R_2 = \frac{T\theta}{2B}$$
$$C = \frac{2B}{\theta}$$
$$B = \left(\frac{R_1}{R_1 + R_2}\right)C$$
$$T = R_2C$$
$$\theta = \frac{2R_1}{R_1 + R_2}$$
$$R_1 = \frac{T\theta^2}{2B(2-\theta)}$$
$$R_2 = \frac{T\theta}{2B}$$
$$C = \frac{2B}{\theta}$$

Note: all circuit diagrams were produced using a modified version of the CircDia package by Stefan Krause, together with scripts to convert circuit descriptions into PNG files.