ï~~Analyzing the Moog VCF for Digital Implementation
Tim Stilson (stiltimccrma. stanf ord.edu) Julius Smith (josmccrma. stanf ord. edu)
CCRMA (http" //www-ccrma. stanford. edu/) Music Department, Stanford University
Abstract
Various alternatives are explored for converting the Moog four-pole Voltage Controlled Filter
(VCF) to discrete-time form for digital implementation in such a way as to preserve the usefulness of its control signals. The well known bilinear transform method and backward-difference
method yield a delay-free loop and cannot be used without introducing an ad-hoc delay. Methods
from digital control theory help guide the changes made to compensate for this delay
1 Introduction
The Voltage-Controlled Filter (VCF) designed and
implemented by Robert Moog is an influential filter in the history of electronic music. In this paper, the filter is analyzed in continuous time (CT)
and then several transformations of the filter into
discrete time (DT) are analyzed for various properties such as efficiency, ease of implementation, and
the retention of certain of the original filter's good
properties, such as constant-Q, and separability of
the Q and tuning controls. The Root-Locus, a particularly useful tool from control systems, is used
extensively in the analysis of the VCFs and provides hints for finding new filters.
In this work, Root-Locus techniques were found
to be useful. The Root-Locus comes from controlsystems analysis and has particular usefulness in
the analysis of systems with sweepable control inputs (inputs intended to have sampling-rate updates) The rules of how the root locus works also
give the designer new tools and hints for sweepable
filter design.
2 The Moog VCF
The VCF used in Moog synthesizers employs the
filter structure shown in Fig. 1.
x(1) E G(ts))G1(s) G1(.0 Gj) yQ)
Figure 1: The Moog VCF.
The transfer function of each section is
Gi(s) = 1
1 + s wC
The four real poles at s = -w, combine to provide a lowpass filter with cut-off frequency (-3 dB
point) at w = w,. The overall transfer function with
feedback as shown is
H(s) o Y(s) _ G4(s) _ 1
X(s) - 1 Â~ k G(s) - k + (1 + s/w )4
where g is the feedback gain which is varied between 0 and 4. Each real pole section can be implemented as a simple (buffered) RC section. Moog
implemented the RC sections using a highly innovative discrete analog circuit known as the "Moog
ladder" [Moog 1965, Hutchins 1975].
At w = w,, the complex gain of each pole section
is
G1(jw c) = 1 _=7-=eJ'4
Therefore, the gain and phase of all four sections
together is
1. 1
G (jwc) = e - (-1)
4 4
I.e., the total gain is 1/4 and the phase is -180
degrees (inverting). In contrast, at w = 0, the gain
is 1 and the phase is 0 degrees (non-inverting), while
at w = oo, the gain is 0, and the phase is -360
degrees (also non-inverting). In summary, the four
one-pole sections comprise a lowpass filter with cutoff frequency w = we, which is inverting at cut-off.
Therefore, the use of inverting feedback provides
resonance at the cut-off frequency. This is called
"corner peaking" in analog synthesizer VCF design
[Hutchins 1975, p. 5d(12)]. As the feedback gain k
approaches 4, the total loop gain approaches 1, and
the gain at resonance goes to infinity.
20.
m
c= C -...........-.0.......
40............................ i i.................!
100 101 102
Frequency (rad/sec)
Figure 2: Amplitude response of the analog Moog VCF
for different levels of feedback (wc = l0rad/sec). At
k = 0, the filter has no corner peaking. Also shown are
k = 4[0.3,0.6,0.9,0.99]. As k increases, corner peaking
develops at the cut-off frequency. At k = 4, the lowpass
filter oscillates at its cut-off frequency.
Stilson & Smith
398
ICMC Proceedings 1996
