### Low and Midrange Frequency Response

Bass and midrange frequencies are greatly affected by Marshall's substantial adjustment of R1 and somewhat influenced by the 10-percent increase in C2 and C3.

In the paragraphs that follow we will describe how the circuit reacts to bass and middle-range frequencies, computing gain and transition frequencies for maximum bass and maximum midrange settings. Once these key characteristics are known, the effects of Marshall's modifications will be clearly evident. Finally we wrap up our analysis by drawing some overall conclusions on the low and midrange-frequency effects of Marshall's modifications to the 5F6-A tone stack.

## Minimum Bass - Maximum Midrange

The classic way to analyze tone stacks is by using the magnitude plots of Hendrik Wade Bode, which were very popular in the 1950s and 1960s. So in the spirit of the time, I'm going analyze the 5F6-A tone stack and Marshall's modifications in the way they were understood back then.

To begin our analysis, let's look at bass and midrange frequencies when the bass control is at minimum and the midrange control is at maximum. At bass and middle-range frequencies C1 acts as an open circuit.

Because of the high input impedance of the long-tailed-pair phase inverter, there is almost no current flowing through the treble control, thus making its particular setting irrelevant. According to Ohm's Law, the voltage at the top and bottom of the control is essentially the same at midrange and bass frequencies. With the bass control at minimum and the midrange control at maximum, the capacitors C2 and C3 are in parallel. Putting all of this together, the circuit looks like this:

The frequency response as a function of s has a pole at p if the denominator of the response equals zero when s = p. (If the variable "s" looks unfamiliar, please see our tutorial on Laplace notation.) The response has a zero at z if the numerator equals zero when s = z. Poles and zeros represent key transition points in the frequency response. If we can derive an equation for one of the poles as a function of two or three component values, for example, then we know how these part values affect this particular transition point.

With the bass control at minimum and the midrange control at maximum we get a frequency response of

where C2+3 = C2+C3. We use a bit of algebra to separate the pole and the zero of the response so that we can manipulate and graph them individually:

where

We observe that there is a zero at s = -a and a pole at s = 0.

Let's take a look at the extremes of frequency. At DC, where s = 0 we note that H(s) = 0 and there is zero output. At extremely high frequencies, where s approaches infinity, it becomes much larger than a and the response is

Both of these conditions make sense, because at DC C2+3 is an open circuit, producing zero output, and at very high frequencies it acts as a short circuit, producing a simple attenuator formed by the voltage divider of R1 and RM. So we essentially have a simple high-pass filter. Above a certain cutoff frequency the gain is fairly constant. Below it there is increasing attenuation as the frequency decreases. In terms of dB, audio attenuation with the middle-range control set to maximum is

With R1 = 56k and RM = 25k we get a gain of -10dB. The Marshall JMP50 Model 1987 lowers the value of R1 to 33k, which provides -7dB of gain. We conclude that the output signal at middle-range frequencies, middle-range control at maximum, is 3dB higher in the Marshall.

Because of the pole at s = -a the -3dB break frequency is

Using 5F6-A parts values we get f-3dB = 49Hz. In the Marshall, C2+3 is 10-percent higher but R1+RM is 28 percent lower, so the cutoff frequency rises to f-3dB = 62Hz.

Here is the Bode magnitude plot for the 5F6-A.

To plot it we note that the overall gain in dB can be separated into the sum of two parts: one due to the gain constant and the zero and the other due to the pole:

For the term

we plot a line rising at a slope of 20dB per decade passing through the 0dB line at a frequency of

for the 5F6-A. For the Marshall the 0dB crossing frequency is 147Hz because the capacitors are 10-percent larger.

The break frequency for the term

is

so for frequencies below 49Hz we plot a straight line at 0dB. At 49Hz we extend a line that decreases at a rate of 20dB per decade. The approximate gain is the sum of the two components that we have just plotted. Below 49Hz it is a straight line rising at the rate of 20dB per decade until it reaches -10dB at 49Hz. Above 49Hz it remains a constant -10dB. The actual gain is 3dB lower at the break point, as shown in the figure. At frequencies much higher or much lower than 49Hz the approximate gain and the actual gain are almost identical.

For the Marshall we get a higher break frequency and 3dB less midrange attenuation.

## Maximum Bass - Minimum Midrange

With the bass control at maximum and the midrange control at minimum C3 is connected to ground.

The low to middle-range frequency response is then approximately

This approximation holds when RB is much larger than R1, which is the case for both the Fender and Marshall designs. With a little algebra the formula can be rearranged to a form suitable for a Bode plot:

where

There is a zero at s = 0, a pole at s = -a1 and a pole at s = -a2. Let's take a look at the extremes of frequency. At DC, where s = 0, we note that H(s) = 0 and there is zero output. At extremely high frequencies, where s approaches infinity, its magnitude becomes much larger than a1 and a2 and the response approaches

This means H(s) approaches zero at extremely high frequencies, because s is in the denominator and is infinitely large. Both of these conditions make sense, because at DC the capacitors are open circuits, producing zero output, and at very high frequencies C3 short circuits the output directly to ground. So this is a bass bandpass filter. The break points of the two poles are:

For the Marshall, the slightly larger value of C2 lowers the low break frequency by 10 percent. The significantly smaller value of R1 increases the high break frequency to 219Hz.

Here is the Bode magnitude plot for 5F6-A component values.

For the term

we plot a line rising at a slope of 20dB per decade passing through the 0dB line at a frequency of

The break frequency of the term

is exactly the same frequency:

For the Marshall the crossing frequency and break frequency are at 7.2Hz. (The crossing and break frequencies in both cases are identical. From the Bode magnitude plot we see that this means the response is minus 3dB at the crossing frequency and then nearly 0dB in the middle of the passband. Bass frequencies in the middle of the passband have little attenuation, so if middle-range frequencies are significantly attenuated, the effective result is bass boost.)

Using 8Hz for the 5F6-A, we begin by plotting a horizontal line at 0dB. At 8Hz we extend a line that decreases at a rate of 20dB per decade.

We apply the same procedures for

where the break frequency is

The approximate gain is the sum of the three components that we have just plotted and the actual gain is 3dB lower at each of the two break points, as shown in the figure.

Here is the Bode magnitude plot for Marshall's part values.

## Conclusions

The Marshall JMP50 Model 1987 tone stack lowers the value of R1 by 41 percent, from 56k to 33k. This modification is obviously deliberate, and it has substantial effects on bass and low-midrange response. Marshall also raises the values of C2 and C3 by 10 percent, from 0.02uF to 0.022uF, which is likely to be just a parts availability issue. Nevertheless, the capacitor modification does introduce subtle shifts in tone stack response.

With the bass and treble controls at minimum, the midrange control determines the overall attenuation for all guitar frequencies. When the bass control setting is increased, it allows bass to bleed back into the mix, reducing the attenuation for its passband. With the midrange control set to maximum, the Marshall has 3dB less overall attenuation, -7dB gain for the JMP50 versus -10dB for the 5F6-A. We conclude that the Marshall midrange control offers more midrange at the top end of the control setting, but a more accelerated gain variation depending on the setting. The 5F6-A, on the other hand, has less maximum midrange, but finer midrange control.

Marshall's modifications for the guitar raise the low frequency cutoff for the midrange control from 49Hz to 62Hz. For an electric bass this would cause bass cut at low bass control settings, something not seen in the Bassman. We can speculate that Leo Fender's design was driven by the fact that the 5F6-A was designed to be a bass amplifier. The choice of cutoff ceases to carry any significance at 82Hz and above.

The Marshall version clearly shifts more midrange notes into the range of the bass control, allowing more of them to bleed back into the mix. Marshall raises the upper cutoff frequency a whopping 54 percent, from 142Hz to 219Hz. These frequencies don't represent sharply defined shifts in frequeny response, because the transition between the tone stack's bass and midrange response is fairly gradual. At the upper end of the bass passband, even with the bass control set to maximum and the midrange control at minimum, midrange cut increases at the rate of only 20dB per decade, or 6dB per octave. Marshall thus raises this gradual transition region to higher frequencies.

How Marshall's C1 Mod Affects High-Frequency Response