An interesting graph can be drawn from the motor analysis in the previous section.
The motor's closed loop poles both lie on the real axis of the s-plane, to the left of the imaginary axis (i.e., the motor is an inherently stable system). The 'electrical' pole is very fast, and lies way out to the left at around s=-336. The dominant 'mechanical' pole is much slower, and lies at around s=-0.23 i.e., where the root's curve (bold line) intersects with the factory platter mass (red line).

The graph above shows how this pole moves (bold line, left Y-axis) as the platter mass is altered over the range 0 to 8kg. The feint line (and right Y-axis) shows that the platter mass has almost no effect on the locus of the 'electrical' pole. As the platter mass is increased, the mechanical pole moves closer and closer to s=0, i.e, the mechanical pole becomes progressively slower, as one expects.

A time simulation of the 'naked' motor with 2.9kg platter shows this behaviour (below).

The motor is energised at time=1 second by a constant armature voltage. Evidently, the time to reach desired running speed is very long. Theoretically, the motor NEVER reaches the setpoint speed, but comes ever closer as time proceeds to infinity. Practically, the error becomes vanishingly small after about half a minute. Not shown is the response to torque disturbances. The performance is poor.

Inspection of the simplified motor block diagram that wil be used ('black block' in the diagram below), shows that any motor inherently exhibits negative velocity feedback by virtue of the back-emf generated. As the motor accelerates, so the back-emf increases, opposing the armature current flow that the terminal voltage causes. The back-emf rises to a level where it is only marginally lower than the impressed terminal voltage: this diffference, impressed on the armature resistance (R_a) gives rise to the armature current, which generates the torque. Slow the motor down by mechanically loading it and the back-emf drops, increasing the voltage drop across R_a, hence increasing the current, thence the torque, counteracting the mechanical load effect.

The problems in direct voltage control of the motor is due to the amature resistance, which makes the load regulation 'soggy'. This is analogous to the degradation in damping of a loudspeaker, by having a high voice coil resistance, which Philips solved with Motional Feedback^TM.

And alternate approach is to drive the motor from a negative impedance source. It's effective, but does not attain the accuracy that can be achieved with a full servo system

A simple velocity feedback loop (proportional control) improves the dynamics of the motor significantly (below). Disturbance rejection (not shown here) is also improved.

However, proportional control can never drive the steady state error to zero. The speed is given by an expression of the form:
ω(t) = ω_setpoint - K.e^-at
The term e^-at gets progressively smaller as t→ ∞, but never vanishes altogether.

The transient performance of the system improves as the gain of the amplifier A₀ is increased.
The transfer function of the motor alone (no tacho) is:

The proportional control loop has a closed loop transfer function:


It can be seen that the system pole has moved from {BR+K_tK_e}/JR_a to {BR+K_tK_e}/JR_a + {HA_oK_t}/JR_a
thus speeding up the system by a term {HA_oK_t}/JR_a
In practice, H(s) is best left a constant. The tacho generator is thus only a converter of units, with no dynamics at all. Any imperfect signal processing performed within the H(s) block results in uncompensatable errors being introduced, so H(s) is left as simple as possible.

It might seem that using a very large gain for A₀ would yield a very good control system, but as A₀ is increased, the appropriate setpoint voltage v_s(t) becomes very small, and electrical noise becomes a disturbance input to the setpoint.

The theoretical limitations of proportional-only control suggest using a better system, rather than trying to work around the limitations.

The feedback configuration will consist of two loops: a velocity loop and a phase loop.
Phase-locked-loops (PLL) typically have a very narrow lock in range, and a velocity loop is required as well to bring the motor into a speed range where the PLL can lock. A multispeed motor with variable pitch would be very tricky to design using ONLY a PLL. A particular type of digital PLL based on a 'state machine' has a very wide lock range, and I will investigate the performance of this circuit too.

Feedback may introduce conditional instability to the system, (oscillation, or runaway). A lead/lag compensator is used to restore a safe margin of stability. Two feedback paths offers greater flexibility in that there is an extra parameter available for adjustment.

The quality and linearity of the tacho pickup is absolutely critical. Any error introduced in the tacho signal directly results in faulty speed regulation - there is no possibility of correcting or compensating such errors. The SP10 motor uses a pickup 'ring', rather than a 'head' (analogous to a tape play head); a pickup ring is less prone to errors of eccentricity. However, direct measurement on the tacho output shows amplitude modulation of the tacho signal at the platter frequency (33,45 or 78 rpm) of about 2% rms. This means that direct use of the tacho amplitude for velocity feedback is not possible. A frequency-to-voltage conversion is required. Such an f/v conversion is used in the original SP10 control sysem - now we know why.
The tacho signal magnitude is around 8, 16, 21, and 40mV_rms for 16, 33, 45, and 78rpm. These are low voltages, and any signal amplification on the tacho must be done with great care since noise picked up here will result in jitter as seen by the frequency discriminator. This is one argument against placing the electronics some distance from the motor in a separate box.
For reference, the tacho frequencies are 52.7, 105.5, 142.5, and 247Hz.

The tacho signal amplifier is shown on page 4, and below is an example of the tacho amplifier output at 33.3 RPM (My circuit, not the original). Timebase: 2mS/div. V_p-p: 17V.

The original SP10 controller does the PLL phase comparison at half the tacho frequency using a linear sample-and-hold phase detector. In 1975 when the SP10 was designed, digital electronics were relatively costly. A linear PLL will now be more costly than a digital PLL.

Two very interesting articles on phasedetectors can be found here:
Kikkert, C.J., Two novel phase trequency detectors. 2006. James Cook University, Queensland Au
and
Gillig, S.F., Linearised three state phase detector. 1990. Motorola, Schaumburg Ill.

An example of the algorithmic state phase frequency detector can be found in the ReVox A710 and B215 cassette decks (below). More elegant than this is hard to find.

The operation of this circuit is easily mimicked by a microprocessor, but the 2 ICs involved are so cheap and universal that it could be advantageous to use CMOS logic for this function.

I'm currently writing a simulation model to evaluate the performance of this ASM PLL. An important aspect is how this type of circuit behaves far from the lock range (low speed, run-up to desired speed); good performance here means no second velocity feedback loop will be required.
The alternative would be a velociy feedback loop and a simpler phase detector - such as a single XOR gate.
The internal goings on of the CD4035 may be slightly daunting when looking at the datasheet, but as wired up above, the logic simplifies to the much more understandable 2 blocks below:
(Pin 7, serial/parallel switches between the two modes)

The shift register records 16 states (0000..1111), but certain states are never entered in practice, leaving only 5 functional states. The state diagram is shown below. The locked condition is characterised by the system alternating between state 3 and state 7. The output then oscillates with a 50% duty cycle, the integrated average being zero. A phase lead results in a transition to state 1 and then state 0, remaining in state 0. The output remains at 0, reducing the motor drive. A phase lag results in a transition to state 15, remaining in state 15, where the output remains at 1, increasing the motor drive.

More control theory, and how the phase detector and tacho fit into the control system appear on page 5.