Analysis of the voltage gain of Millenium Amplifier
This article shows how the voltage gain of each stage is determined. This is used to compute the system open loop and close loop gain with feedback.
Stages A1 to A5
The diagram shown the 5 gain stages of the Millenium amplifier which is briefly described here.
A1 - Front-end common cathode voltage gain stage
A2 - Phase splitter which splits the to 2 complementary opposing signal phases
A3 - Pre-amplifier common-cathode stage that amplifies the complementary signal phases
A4 - Power amplifier stage to convert the voltage signal to drive the reflected load from the speaker
A5 - Impedance matching stage which is a transformer chosen to match the speaker impedance for maximum power transfer. This stage does not provide any gain, but as result of matching the impedance the output voltage is reduced by the turns ratio of the transformer.
Going into greater detail, each stages A1 to A5 are examined closely for its voltage gain.
With A1,A2,A3,A4 and A5 determined, compute Open Loop gain, AOL = A1*A2*A3*A4*A5
With feedback determined, compute the Close Loop gain, ACL = AOL/(1+B*AOL)
Stage A1
A1 - Pre amplifier voltage gain stage. This is the first stage to provide the initial voltage gain, but also to raise the anode voltage sufficiently high to 105V so that it is directly coupled to the phase splitter without capacitor. This is crafted so that the anode voltage of U1A is slight lower than the cathode voltage of U1B to provide a negative bias for the grid of UIB. U1A cathode voltage is bypassed with capacitor C2 so the effective cathode resistor is bypassed for AC signal. The effective gain of the A1 stage is similar to the common cathode amplifier configuration with bypass resistor. We will ignore the feedback network for now and discuss this in the feedback network setting. So the feedback network is excluded in the analysis for now.
For common cathode configuration, Av is the voltage gain equation
Referring to the datasheet for 6189 which is similar to 12AU7 operating at 100V, 2.2mA
Plate resistance Rp=13K, Amplification factor U=15, Transconductance gm=1200 uA/V
with RL = R3 = 100K
A1 = 15*100/(100 + 13) = 13.3
Stage A2
A2 - A2 stage is designed such that the anode and cathode resistance are equal (22k) and the cathode is not bypassed with capacitor. When grid-cathode voltage decreases, anode current increases and the cathode voltage increases and the anode voltage decreases because of increased voltage drop at across R7 and R8 respectively. When grid-cathode voltage increases, anode current decreases and the cathode voltage decreases and the anode voltage increases because of decreased voltage drop at across R7 and R8 respectively. This create signals of opposing phase at the anode and cathode respectively. The design is also biased so that the cathode (@108V) is about 1/3 of the supply voltage (310V) and the anode (@210V) at about 2/3 of the supply voltage to allow for the full translation of the voltage swing. This design is also called the Cathodyne phase splitter with the voltage gain equation below.
Bias of U1B is at 102V, 4.4mA and from the datasheet, U=17.5, Rp (ra) = 13K,
Referring to the Millenium circuit Rk = Ra = 22K
A2 = 17.5 * 22/(13+22+(17.5+1)*22) = 0.917
Stage A3
A3 - This is a push-pull front end voltage amplifier with a shared cathode resistor R13 between U2A and U2B. Bypass cathode capacitor is not used because the signal at U2A and U2B are opposing phase, the cathode current flowing pass R13 is effectively constant creating a stable DC at the cathode. The shared cathode resistor though allows for a stable cathode DC bias but will reduce the AC gain, since the cathode resistance will be present as a localized negative feedback to AC signal. The voltage gain for the common cathode push pull front end is as follows;
For the Millenium design, the cathode voltage is at 7.55V, the cathode current in each leg of the push pull is half that in the cathode resistor (1K).
Cathode current = 0.5*7.55/1K = 3.9mA
Bias of U2A and U2B are at 212V, 3.9mA
From the datasheet for 6SN7, Plate resistance = 11K, Transconductance=1900 uA/V, U = 19
Using the unbypassed equation, with RL=47K, Rk=1k
A3 = 19* 47/(11+47+(19+1)*1) = 11.45
However, if the cathode resistor is bypassed, the gain A3 will be higher
A3 = 19*47/(11+47+0) = 15.4
Stage A4
A4 - This is the power output stage of U3A and U4A operating in class AB ultra linear mode. The amplifier in low signal condition operates in class A and transits to class B when the signal level increases. The impedance presented to the U3A and U4A must be understood for each half of the positive and negative half cycle. In class A operation, where both U3A and U4A are active in the full cycle, the reflected impedance to U3A is presented by the upper half of the split primary coil and the reflected impedance is presented to U4A by the lower half of the split primary coil . However in class B, only one of the 2 tube in U3A and U4A are active with the other tube in cutoff, the effective impedance is only present to the active tube which means for at most half of the full cycle. Output push pull can be configured as a pentode or triode configuration. Pentode configuration offers higher power output and efficiency but with higher distortion and triode offers lower power output but with lower distortion. The middle ground is the ultra-linear configuration which is a pentode configuration with screen feedback tied to the output transformer tapped at about 40~43% of applied anode voltage.
With the ultra linear configuration it is technically a pentode configuration with local feedback, thus taking advantage of the power efficiency of pentode design with reduction to the distortion with feedback. The effective Rp (plate resistance) and U (amplification factor) has effectively changed from open loop to close loop plate resistance and amplification factor with a [1+BAOL] factor where B is the transformer tap ratio and AOL is the Ug2g1 inner gain between screen and control grid.
The equation for general ultra linear configuration is shown below
A4 computation requires Rp_UL and UUL
From the datasheet from Mullard EL34 (only available for 250V)
For ultra linear configuration, with K = 0.4 for Hammond 1645,
Up = 11*15 = 165 (amplification factor in pentode mode)
UUL = 165/(1+0.4*11) = 30.5
Observe that the effective amplification factor is between triode mode and pentode mode.
Rp_UL = 15/(1+0.4*11) = 2.78K
Observe that the effective anode resistance is between triode mode (0.91K) and pentode mode (15K).
Reflected Impedance of the Load to Anode (RL) with a transformer of input impedance RL
For class AB configuration, the amplifier could be in class A or class B. For both class A, the reflected load per tube is half of the input impedance (ZL/2). For class B operation, this is further reduced because only one tube is driven while the other is cutoff. The driven tube is operating only for 1/2 cycle of the time.
Class A operation - RL(eff) = RL/2
Class B operation - RL(eff) = RL/4
When considered for single ended from either the push or pull side, the reflected impedance is reduced
For Hammond 1645, Primary impedance = 5K
RL(eff) = 5K/2 = 2.5K (class A)
RL(eff) = 5K/4 = 1.25K (class B)
Using the classic common cathode equation with the effective amplification factor and effective plate resistance for ultra linear configuration,
A4 = 30.5*2.5/(2.5+2.78) = 14.44 (class A)
A4 = 30.5*1.25/(1.25+2.78) = 9.46 (class B)
Observe than the voltage gain drop when transiting from class A to class B operation. However, it will be mitigated with overall feedback to stabilize the close loop gain.
Since the analysis assumes a single ended push or pull stage, for push pull the net voltage gain at the input of the transformer is doubled.
A4pp = 30.5*2.5/(2.5+2.78) = 28.88 (class A)
A4pp = 30.5*1.25/(1.25+2.78) = 18.92 (class B)
Stage A5 is not an amplifying stage but a transformer whose main purpose is to match the impedance of the output tube to the speaker. To match to the output tube the load has to be about the same impedance of the plate which computed from A4 is Rp_
UL = 2.78K. However, speaker does not have such high impedance, so a impedance converter is needed. This converter comes in the form of a wide audio frequency transformer that can convert the low speaker impedance to match the required plate impedance. A matching transformer should be about 2 times the Rp_
UL. Therefore, a matching transformer of between 4.5K to 7K is normally chosen. In this case, for the Hammond 1645 the transformer input impedance is 5K. A5 gain stage is related to the transformer turns ratio which is related to the impedance by this equation
Since A5 = Vout/Vin = Ns/Np
and Zp = 5000 ohms, Zs = 8 ohms
Np/Ns = SQRT(5000/8) = 25
Note: The transformer turns ratio is Np/Ns = 25
Since the push pull is treated as 2 separate single ended windings driving the common load
Effective winding ratio = 0.5*25 = 12.5
A5 per single ended = 1/(12.5) = 0.08
Feedback Network (B)
The feedback network feeds a small portion of signal from the output to oppose the input at the cathode U1A thus forming the negative feedback loop.
Note that the for mid band analysis C3 can be ignored. C3 is intended for high frequency roll by forcing a more feedback.
feedback nominal gain B = R5/(R5+R6)
B = 68/(68+2670)
B = 0.02484
Open Loop Gain AOL
AOL is the open loop gain before negative feedback is applied.
Open Loop gain, AOL = A1*A2*A3*A4*A5
For Class A operation,
AOL = 13.3*0.917*11.45*28.88*0.08 = 322.64
For Class B operation,
AOL = 13.3*0.917*11.45*9.46*0.08 = 211.36
Close Loop Gain ACL
ACL is the close loop gain after global negative feedback is applied.
Close Loop gain, ACL = AOL/(1+B*AOL)
For Class A operation,
ACL = 322.64/(1+0.02484*322.64) = 35.79
For Class B operation,
ACL = 211.36/(1+0.02484*211.36) = 33.82
This shows that the close loop gain is between 33.82 to 35.79 from full class A to class B. In actual the close loop gain will move between these 2 limits. Of course, the fluctuation will introduce unwanted distortion. This could further be reduced by increasing the open loop gain especially by increasing A1 or A3. With higher open loop gain, the close loop gain variation will be reduced. For example increasing the open loop gain by 5 time.
For Class A operation,
ACL = 322.64*5/(1+0.02484*322.64*5) = 39.28
For Class B operation,
ACL = 211.36*5/(1+0.02484*211.36*5) = 38.78
Now the difference between class A and class B is significantly narrowed.
Validation on Millenium
The actual system was tested with input of 44mV and the output delivered is 1.6V. The close loop gain is 36.3 or 31dB. Since this signal is at a relatively low level and it can be expected that the amplifier is likely in class A operation.
AOL = 36.3/(1-0.02484*36.3)
AOL = 369 (51dB)
The actual open loop gain turns out to be slightly higher which is good because the distortion between class A to class B transition will be lesser than the analysis.
Summary of observation from the analysis
The open loop gain of the system is not very high from a moderate voltage gain in each stage, resulting is a possible distortion from gain changes when transiting from class A to class B operation. To improve the system gain stability, the open loop gain needs to be increased to result the gain change from this transition. One possible idea is to have the 12AU7 (6189) changed to 12AX7 which should give a 5 fold increase in open loop gain.
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