SBOA550 October 2022 OPA1671 , OPA2990 , SN74HCS04 , SN74HCS164 , SN74HCS30 , SN74LVC1G00 , SN74LVC1G123 , TLC04 , TLC14 , TS5A9411

A Analytical Solution for Resistance Network Values

Consider an even-length Johnson counter of length N driving weighted resistors (conductors, $G_{i}$ ) as shown into a current summing node at the inverting input of the op amp. For symmetry reasons assume the flip-flops output is ±1 V representing normalized high and low states, see Figure A-1.

Figure A-1 Simplified Johnson Counter With Weighted Resistors Driving a Transconductance Amplifier

When all flip-flops are high (11…11), maximum positive current flows into the virtual ground at the op amp, corresponding to the peak of the cosine function. To normalize the overall Thevenin circuit conductance to 1S (1 Ω) let I_PEAK = 1 A. The contributions of the resistors are weighted to form the steps along the cosine waveform as follows in Figure A-2, summing to 1 A.

Figure A-2 Weighted Current Contributions of Each Pair of R_i Resistors sum up to 1 A (Normalized)

For each $∆_{i}$ when both $G_{i}$ resistors are driven high by their respective flip-flops, see Equation 5.

Equation 5.

∆_{i} = 2 G_{i} = c o s (\frac{π}{N} i) - c o s (\frac{π}{N} (i + 1))

Using a trigonometric identity and simplifying Equation 5 yields Equation 6.

Equation 6.

2 G_{i} = - 2 s i n (\frac{π}{2 N} (2 i + 1)) s i n (\frac{- π}{2 N}) \to G_{i} = s i n (\frac{π}{2 N} (2 i + 1)) s i n (\frac{π}{2 N})

Converting each conductance to a resistance,

Equation 7.

R_{i} = \frac{1}{G_{i}} = \frac{1}{s i n (\frac{π}{2 N})} \times \frac{1}{s i n (\frac{π}{2 N} (2 i + 1))}

Recall $R_{i}$ is normalized to 1 Ω; to get the desired overall Thevenin output resistance R_O, each resistor must be scaled by R_O, see Equation 8.

Equation 8.

R_{i, d e n o r m a l i z e d} = \frac{R_{O}}{G_{i}} = \frac{R_{O}}{s i n (\frac{π}{2 N})} \times \frac{1}{s i n (\frac{π}{2 N} (2 i + 1))}

For an odd-length register, the value for the lone middle resistor is computed separately as Equation 9 and Equation 10 show.

Equation 9.

G_{\frac{N - 1}{2}} = c o s (\frac{π}{N} \frac{N - 1}{2}), f o r o d d N

Equation 10.

R_{\frac{N - 1}{2}, d e n o r m a l i z e d} = \frac{R_{O}}{G_{\frac{N - 1}{2}}} = \frac{R_{O}}{c o s (π \frac{N - 1}{2 N})}, f o r o d d N