The first stage of the Magnatone MP3 guitar amplifier adds a strange 220kΩ resistor between the 140V plate supply and the cathode.

According to Magnatone's schematic, the measured cathode voltage is 0.5V. Let's see how this matches an average 12AX7 as documented by the data sheet.

According to Ohm's Law, if we assume the cathode voltage is 0.5V then the current through the 470Ω resistor is

0.5V / 470Ω = 1.06mA

The current through the 220kΩ resistor is

(140V - 0.5V) / 220kΩ = 0.63mA

The sum of this current and the plate current passes through the 470Ω resistor, so the plate current is

1.06mA - 0.63mA = 0.43mA

The plate-to-cathode voltage is equal to the plate supply voltage, minus the voltage drop across the 100kΩ plate load resistor, minus the cathode voltage:

140V - (0.43mA)(100kΩ) - 0.5V = 96.5V

Here on the 12AX7 plate characteristic curves, the point for 97V and 0.43mA is marked by the red dot, which indicates the grid voltage should be slightly less than -1V for this DC operating point.

We can repeat these computations for an assumed 0.6V cathode (blue dot, 74V, 0.65mA) and an assumed 0.7V cathode (green dot, 53V, 0.86mA). The blue dot wins: it is in agreement with an imagined -0.6V curve.

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It is interesting to see that the dots form a line.
In fact, the upper left endpoint of the line appears to be at a
plate current of 1.4mA on the Y axis.
The lower right endpoint is at a plate voltage of 140V on the X axis,
so the dots are on a traditional DC load line,^{1} as shown by
the red line of the
12AX7 calculator.

The calculator shows that the same bias can be achieved with a 923Ω cathode resistor for traditional cathode bias. This is not a standard resistor value, but great precision is not the hallmark of guitar amp design. For the discussion that follows, let's assume a circuit designer would select a 1kΩ value to achieve the same DC operating point using traditional cathode bias.

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When the circuit is redrawn, Magnatone's concept becomes clear.

This is not a traditional cathode-biased preamp stage with an extra resistor.
Instead, it is a pair of resistors that form a voltage divider to create
*fixed bias*, as defined by the Radiotron Designer's Handbook.^{2}
Here is a step-by-step procedure to determine the value of R_{2}
for a fixed-bias voltage amplifier based on the plate supply voltage V_{PP}, the
plate load resistor R_{L}, and the desired value of R_{1}:

- Draw a traditional DC load line that
connects the plate supply voltage V
_{PP}with V_{PP}/ R_{L}. - Along the load line, pick a desired DC operating point.
- Note the DC grid voltage V
_{G}and the DC plate current I_{P}at that point. - Solve for R
_{2}using the following formula.

where the cathode voltage is

For a desired R_{1} = 220kΩ and V_{K} = 0.6V,
this procedure yields a result that is very close to Magnatone's value:

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From a DC perspective, does
a pair of 220kΩ and 470Ω resistors
for fixed bias
offer any advantage over a traditional single-resistor design?
Not really, unless you have a warehouse full of them and have 1kΩ
resistors on backorder.
We could also use 150kΩ for R_{1} and 390Ω for R_{2}.
In fact, there are many combinations of values for R_{1} and R_{2} that can be used to create
the same DC operating point.
So perhaps one advantage might be *inventory flexibility*.
Typically, R_{1} has a large value to limit the current that
the voltage divider draws from the high-voltage power supply.

Here is Magnatone's first stage again, this time including the RC ripple filter for the 140V plate supply.

For audio signals, the capacitor acts as a short circuit,^{3} so the
equivalent AC circuit puts the 220kΩ resistor in parallel with the cathode resistor.

220kΩ is huge compared to 470Ω, so
the effective AC cathode resistor value is almost the same as without it:
469Ω.
There is no significant difference in gain when the 220kΩ resistor is included,
so audio performance is the same as for traditional cathode bias **if a 470Ω resistor
is used**.
To create the same DC operating point, however,
we observed that cathode bias requires about **1kΩ**.

If the cathode is bypassed by a capacitor, then there is no negative
feedback from cathode degeneration^{4} and audio performance is the same
for either resistor value.
In this case traditional cathode bias has the advantage that a smaller
capacitor value can achieve full bypass.
Magnatone's resistor is unbypassed, however, so gain is reduced.
A 470Ω value, however, creates less cathode degeneration than a
1kΩ value.
According to the
Preamp Gain and Output Impedance calculator,
the gain is 35.8dB when the cathode is fully bypassed.
When unbypassed, gain is 33.6dB with 470Ω and
31.6dB with 1kΩ.

From a system design^{5} perspective,
greater inventory flexibility and a few less dB attenuation when
unbypassed are not particularly compelling reasons to add a resistor
to the circuit.
It is not surprising, therefore, that Magnatone's approach to establishing a DC
grid bias is extremely rare in guitar amps.
There is, however, a desirable side effect to this circuit: a discharge path for power supply
filter capacitors that is independent of the vacuum tubes.
When the amp is turned off, the voltage across the capacitors slowly bleeds away
via the 220kΩ and 470Ω resistors.

^{1}Richard Kuehnel, **Guitar Amplifier Electronics: Basic Theory**, (Seattle: Amp Books, 2018), p. 57.

^{2}F. Langford-Smith, ed., **Radiotron Designer's Handbook**, 4th ed., (Harrison: RCA, 1953), p. 487.

^{3}Richard Kuehnel, **Guitar Amplifier Electronics: Basic Theory**, (Seattle: Amp Books, 2018), pp. 19-21.

^{4}Richard Kuehnel, **Guitar Amplifier Electronics: Basic Theory**, (Seattle: Amp Books, 2018), pp. 61-65.

^{5}Richard Kuehnel, **Fundamentals of Guitar Amplifier System Design**, (Seattle: Amp Books, 2019).

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