The Gretsch Chet Atkins preamp has two channels, each with its own bass and treble controls. The channels are combined in a 12AX7-based mixing circuit.
The output is taken from where the plates connect to the 100k plate load resistor RL. The cathode resistor RK is 1.5k. Since the resistors carry the plate current of two triodes, the DC operating point is the same as for a single triode with a 200k plate load resistor and 3k cathode resistor.
AC voltage gain for each channel is less than for a single triode amplifier because of additional negative feedback. Imagine a rising voltage on the left input (channel 1) and no signal on the right. The increasing grid-to-cathode voltage causes an increase in plate current through the left triode, which creates more current through the cathode resistor RK. This raises the voltage across the resistor, causing the cathode voltage to rise and counteract the rising grid-to-cathode voltage induced by the input signal. Classic negative feedback. Moreover, we have common cathode coupling: a rise in cathode voltage for the left triode creates a similar rise in cathode voltage for the right triode. The net result is that the rising input signal causes the total current through the plate load resistor RL to increase while at the same time negative feedback in both triodes causes the total current to decrease.
The Radiotron Designer's Handbook1 calls this a "common plate load mixer." Unfortunately no gain formula is mentioned for the Chet Atkins configuration, where no cathode bypass capacitor is used, so we'll derive a formula here. If you want to use the gain formula but prefer to skip the math, just scroll down to the bottom of the page.
Here is the AC circuit, where DC voltages are considered zero.
The triodes are represented by controlled voltage sources in series with plate resistances rp. When the grid-to-cathode voltage vg1 increases by 1 volt, for example, the voltage across the the controlled voltage source on the left increases by μ volts, where μ is the triode's amplification factor.
We apply Kirchhoff's Voltage Law (KVL) to the left triode plate circuit by taking the sum of the voltages and setting them equal to zero:
Similarly for the right triode we get
Applying KVL to the grid circuits gives us
We substitute these grid voltages into the plate circuit formulas and with a little algebraic re-arrangement the result is
The amplification factor μ for a high-mu triode is much greater than 1 (For the Gretsch's 12AX7, for example, μ = 100.) So these relations are very close:
Next we apply Cramer's Rule and obtain the plate currents i1 and i2 as a function of the input signal voltage vin.
According to Ohm's Law the output voltage is minus the product of the plate resistor value and the sum of the currents.
Substituting the currents that we just computed and simplifying we get the final gain:
The minus sign indicates that this is an inverting amplifier for both channels. The term 2μRK represents negative feedback that is absent if the cathode resistor is fully bypassed by a large capacitor. For a simple triode amplifier with an unbypassed cathode resistor, the feedback term is only (μ+1)RK. So the Gretsch common plate load mixer has approximately twice as much negative feedback.
1F. Langford-Smith, Ed., Radiotron Designer's Handbook, 4nd Ed., (Harrison: RCA, 1953), p. 800.
2Richard Kuehnel, Vacuum-Tube Circuit Design: Guitar Amplifier Preamps, 2nd Ed., (Seattle: Pentode Press, 2009).
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