2. 6GM8/ECC86 3. 6C45PI 4. 6N1P, 6SN7 5. The Challenge of Ultrapath 6. A solution 7. Conclusions 8. References

.subckt ecc86 1 2 3 ; placca griglia catodo + params: mu=14 ex=1.71 kg1=295 kp=220 kvb=100 rgi=2k + ccg=3p cgp=1.3p ccp=1.8p a2=0.0083 a1=0.022 a0=1.1033 e1 7 0 value= +{v(1,3)/kp*log(1+exp(kp*(1/mu+v(2,3)/sqrt(kvb+v(1,3)* +v(1,3)))))} re1 7 0 1g e2 8 0 value={a2*v(1,3)*v(1,3)+a1*v(1,3)+a0} re2 8 0 1g g1 1 3 value= {(pwr(v(7),v(8))+pwrs(v(7),v(8)))/kg1} rcp 1 3 1g c1 2 3 {ccg} c2 1 2 {cgp} c3 1 3 {ccp} r1 2 5 {rgi} d3 5 3 dx .model dx d(is=1n rs=1 cjo=10pf tt=1n) .ends 
On the plate curves below, grid lines are in .1 V increment, starting from 0. The plot looks odd and artificial because the model is polynomial:
The plot shows that it is difficult to achieve significant voltage swing here, as the max idle plate voltage can not be more than 25V. This remarkably limits grid voltage to 0.4 V in highcurrent region. It's also evident that this tube is model is sensitive to variations in load impedance  because its curves are so complex, which is especially true when plate current is lower than 5 mA. There is a knee in plate curves at around 4..5 mA where plate voltage is < 20V.
First set of distortion data is for 7 mA plate current, which translates to ~ .68V bias:
Input range 10mv..1v. 2:1 winding ratio.

The cathode resistor is 90 ohms which gives the above mentioned 7ma of plate current.The load Rload for this set is 600 Ohm. Distortion curvers for other loads are fairly similar and consistent.
Harmonic ballance shifts in the region of low plate current, though, due to the abovementioned "knee" in plate curves. Below I show some parametric swepps for this region with a disclaimer that Stefano Perugini's plynomial model does look exotic to me. However, I do believe that lowvoltage lowcurrent conditions add some complexity to plate curves that indeed may cause various sonic surprises, and the polynomial model attempts to address that.
This data is for a 300 Ohm load, sweeping the bias resistor value in the range 250...50 Ohm. The first "column" of distortion points (left side) is for 250 Ohm, the last "column" is for 50 Ohm. Input is .1 V.

There is a peculiar local sweet spot around 200 Ohm, where the 2nd harmonic appears to be very low. Apparently, the loadline crosses the "knee", and the shape of the "knee" provides distortion cancellation for the 2nd harmonic. The optimal value is around 190 Ohm. Here is distortion data for varying input voltages with Rk = 190 Ohm:
This significantly differs from Rk=90 data, and such difference would be readily audible. Lower 2nd harmonic and higher 3rd may or may not  depending on your preferences in sonic signatures of SET amps  sound appealing, but the main issue here is that distortion patterns in this region are highly unstable. For example, if we run this configuration with 50K load, the picture is completely different:
What is the value of this analysis? Can Rk=190 Ohm be suggested as the best bias resistor value for this schematic? The answer is no. Even if the model is quite accurate, the sweet spot will position differently for different tubes. On another hand, finetuning the bias resistor may result in interesting but very fragile, sonic surprises, which could be a part of 6gm8/ecc86 reputation. Of course, if 6gm8/ecc86 triodes are parallelled, each must have a separate biasing.
.SUBCKT 6GM8 1 2 3 ; P G K ; + PARAMS: MU=21 EX=1.596 KG1=435 KP=46.5 KVB=24.1 VCT=0.44 RGI=2000 + CCG=3P CGP=1.4P CCP=1.9P E1 7 0 VALUE={V(1,3)/KP*LOG(1+EXP(KP*(1/MU+V(2,3)/SQRT(KVB+V(1,3)*V(1,3)))))} RE1 7 0 1G G1 1 3 VALUE={(PWR(V(7),EX)+PWRS(V(7),EX))/KG1} RCP 1 3 1G ; TO AVOID FLOATING NODES IN MUFOLLOWER C1 2 3 {CCG} ; CATHODEGRID C2 2 1 {CGP} ; GRID=PLATE C3 1 3 {CCP} ; CATHODEPLATE D3 5 3 DX ; FOR GRID CURRENT R1 2 5 {RGI} ; FOR GRID CURRENT .MODEL DX D(IS=1N RS=1 CJO=10PF TT=1N) .ENDSSimulation with this model into a 100K load and a 2:1 transformer does not show a lowdistortion sweetspot but instead displays monotonic increase in 2nd harmonic up to a fairly high value at ~ 4.5 VRMS, and then going down yielding to higher order products (as it always happens when severe limiting/clipping kicks in):
With 4:1 transformer the value of useable VRMS will be halved.
With 4:1 ratio for the output transformer, 6c45pi gives around 320 Ohm of output impedance. No doubling is necessary!. 320 Ohm low enough for most applications, except for lowimpedance headphones.
I added 1 mV of 120 Hz AC to the B+ supply and ran simulation with 6n1p tube's greed to the ground. The choice of 6n1p tube for this test was for its medium amplification factor, not too low (6SN7,) not too high (6c45). The value of C1 was set to 40uF.
The Ultrapath topology developed 8 mV of AC signal on the output. If we disconnect C1 from B+ and connect it to the ground it becomes a classical bypass capacitor. In this configuration, the output AC signal is only ~ 150uV. Finally, with C1 eliminated the output AC signal is ~ 300 uV. In all three tests the AC signal on B+ is the same  1 mV.
In other words, it appears that C1 injects AC ripple into cathode bias and lets the tube amplify it. The ratio 8 mV : 1 mV is close to the gain of the circuit (transformer's ratio is 4:1). One way to understand this is to realize that the tube can be presented works as a groundgrid amplifier here, with input signal (ripple) fed into the cathode.
This is indeed a challenge, as it appears that Ultrapath amplifiers AC ripple of power supply. This will be worse with high mu tubes such as 6c45.
First, SPICE reveals that perfect cancellation happens only if impedance of C1 is much greater than Rk. Otherwise, phase shifts occur, preventing complete cancellation. I suspect what matters here is how much impedances of C1 and the bypass cap lower than Rk. If C1 is 1uF, its impedance at 120Hz is ~ 800 Ohm  more than Rk for 6n1p, 6c45, 6gm8/ecc86. Phase shift is significant, cancellation is not full. 100uF is ~ 8 Ohm at 129 Hz. Can be used for 6n1p with OK success, but still not very good for 6c45 or 6gm8/ecc86. In fact even for 6n1p, where Rk is 500 Ohm, cancellation is not perfect due to slight phase rotation:
on this plot the first curve is for Ck = 3550uF, 2nd  3560,....
Note that in the ultrapath circuit C1 plays the role of a bypass cap, and thus its impedance needs be lower than Rk on the low end of the range, not at 120 Hz, and hence it must be large anyways  as large as "normal bypass". Unfortunately, unlike for normal bypass, C1 must be rated for high voltage.
Second, as the bottom capacitor needs be ~ mu times bigger than C1, with large C1 it can get way too large. Thus for C1 = 100uf and for 6n1p with mu=33 we get 3300uF.
Third, tuning for zero hum is a pain because changing value of a large cap for hum adjustments is a pain. Even after someone assembles the correct value, cancellation will be prefect only for a given mu of a given tube at given point in its lifetime.
Therefore, there is a need to find another method of hum adjustment. What about a solution involving a potentiometer? After some thinking I came up with several approaches. Since I figured out that a proportion of ~ 1/mu of B+ ripple injected into cathode will cancel it out, how about injecting the whole ripple signal into a 1/mu part of the bias resistor? The exact amount is not 1/mu, and John Broskie provides formulas for that, but since I use SPICE, i need to sweep around that value. Which I did with a circuit where Rk was replaced by a parametric pair simulating a potentiometer, with one side of the ultrapath capacitor C1 attached to B+ and another to the tap of the pot. Unfortunately, no cancellation happened  the "load" was too low for C1's impedance. I saw phase shifts suggesting that. Then I switched to circuit where C1 feeds a voltage divider and was able to obtain clean cancellation. Note however, that as the result the circuit lost bypassing and therfore the gain is lower and output impedance is higher.
OK, what about injecting the signal into the grid? This will help to finetune either a classical bypassed stage or a stage with Broskie C:C divider (slightly unbalanced) for lowest hum:
Thus we started with ULTRAPATH, considered Broskie divider and ended up with a topology that provides hum cancellation with or without the ultrapath capacitor C1 present. How about using grid hum injection in the original ULTRAPATH topology? This appeared doable but after closer examination revealed problems, as the grid needs to see even slightly more hum than what reaches the cathode. I'll continue experimenting with this.
The humcancellation topology presented in the last section of this article, which is in between John Broskie cancellation circuits and Jack Elliano ULTRAPATH may be a promising remedy to "tried Ultrapath, but got too much hum" syndrome. A "short audio path" is present here just lik ein the original ultrapath topology, although there are few more parts in the schematic.
[2]. John Broskie Tube Articles.
[3]. John Broskie C:C Hum Cancellation.
[4]. PAENG Design's article on High Gm, High Mu Tubes.
[5]. PAENG Design Tube Models.
[6]. Norman Koren's Improved Tube Models.
[7]. Chris Beck's 6SN7 Ultrapath.
[8] Amperex 6GM8/ECC86 data on Frank's Electron tube Pages.