The JamesBaxandall Passive ToneControl Network
Independent adjustment of bass and treble frequencies in high fidelity audio amplifiers is usually accomplished utilizing specially designed tonecontrol networks. There are versions for these tone controls based only on passive components, such as the groundreferenced James network shown in fig. 1. Among those versions using active devices we must make mention of P.J. Baxandall´s proposal, in which the tone control was devised as a feedback amplifier (refs. 1 and 2).
In this article we shall analyze the James network (also known as the passive Baxandall tone control), obtaining its design equations.
Let´s begin studying the bass control section (fig.2), which has influence over the frequencies below the designed center frequency of the overall James network.
Here, R_{4} offers some isolation between this stage and the treble control section (frequencies above the center frequency are affected by this control). R_{5} represents the input resistance of the amplifier connected to the output of the James network and should be selected such that it imposes no appreciable load on the network. Here we assume that C_{3} and C_{4} are open circuits at the bass frequencies.
With full bass boost (R_{2}´s slider at the upper end), the equivalent circuit for the tone control is as depicted by fig.3.
From this figure and knowing that “s” is Laplace´s variable, we obtain:
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where:
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Then:
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After some algebraic work we arrive to:
_{} ...(1)
The gain for high bass frequencies is:
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The gain at low frequencies is:
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Expression (1) has a zero given by:
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and a pole:
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With the tone control adjusted for maximum bass cut (R_{2}´s slider at the lower end), the equivalent circuit changes to that of fig.4. Again, we neglect the loading effect of R_{4} and R_{5}.
Now the following holds:
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A simple algebraic manipulation takes us to:
_{} ...(2)
The gain at high bass frequencies is now given by:
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For low frequencies the gain is:
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Expression (2) has a zero given by:
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and a pole:
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The gain ratio at low frequencies is:
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For a 40dB control range the following relationship must be satisfied:
_{} ...(3) Then:
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On the other hand, at maximum bass boost, the ratio of the gains for low and high bass frequencies is:
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According to expression (3):
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Then:
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For a 20dB bass boost:
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Solving for R_{1} we arrive to the following relationship:
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We may check that:
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this is, the ratio of the highfrequency and lowfrequency bass gains is also 20dB at maximum bass cut.
For best symmetry in the response curves we must choose s_{01 }= s_{p4}. Therefore:
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Substituting the already obtained relationships between the resistances yields:
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Then, C_{2} = 10C_{1}.
The corresponding Bode plot of the bass response at full boost and full cut is shown below.
Next, we will analyze the treble control. We have already stated that the treble frequencies are those above the center frequency for which the James network is designed. We may consider C_{1} and C_{2} as short circuits at these frequencies. Therefore, R_{1} and R_{3} along with R_{4} are part of the treble network (fig.6).
In order to facilitate calculations, we shall first find the Thevenin equivalent for V_{g}, R_{1} and R_{3}. In fig.7, the equivalent Thevenin voltage source, V_{1}, is given by:
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_{} ...(4)
The equivalent Thevenin resistance, R_{TH,} is:
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With the treble control adjusted for maximum boost (R_{6}´s slider at the upper end), and assuming the current through resistor R_{6} is much smaller than that flowing through R_{4}, we may state the following:
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if R_{5}>>R_{4}.
Substituting the value given by expression (4) for V_{1}:
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Rearranging this equation we get:
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Then:
_{} ...(5)
At sufficiently low treblefrequencies:
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the same as 20.83dB.
At sufficiently high treblefrequencies:
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or 0dB.
Expression (5) has a zero given by:
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and a pole:
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With the treble control adjusted for maximum cut (R_{6}´s slider at the lower end) we may write: _{}
Then:
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From the above equation and bearing in mind (4):
_{} ...(6)
At sufficiently low treblefrequencies:
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or 20.83dB.
At sufficiently high treblefrequencies:
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Expression (6) has a pole given by:
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For best symmetry in the response curves we must choose s_{p7} = s_{05}. Therefore:
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yielding C_{4} = 11C_{3}.
Fig.8 shows the corresponding Bode plot of the treble response at full boost and full cut.
A convenient value for R_{6} must now be computed. We need some knowledge on the constraints the network imposes before we may neglect the current through R_{6}, as compared to that circulating through R_{4} (I_{R6}<<I_{R4}). I_{R4} is given by the following expression when the treble control is adjusted for full cut:
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and I_{R6 }by:
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Therefore, the requirement that must be satisfied is:
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The worst case for the above inequality occurs when s approaches infinity. Translated to our analysis, at sufficiently high treblefrequencies. Then, it is required that:
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or equivalently:
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At sufficiently high frequencies:
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It is clear then that the following must be satisfied:
_{} Then:
_{} ...(7)
If we conduct a similar analysis when the treble control is at full boost, we would find that the condition to be met is R_{6}>>1.1R_{4}. Hence, expression (7) prevails._{}_{ }
We would like to arrive at easytouse design formulae, so, before presenting a design example for the James network we shall try to simplify the expressions for f_{01}, f_{p2}, f_{03}, f_{p4}, f_{05}, f_{p6} and f_{p7}. First, for convenience, the relationships between the components’ values will be repeated. These are:
R_{1} = 10R_{3} _{ } R_{2} = 99R_{3} _{ } C_{2} = 10C_{1}
C_{4} = 11C_{3} _{ } f_{01} is given by:
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_{} ...(8)
f_{p2} is given by:
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_{} ...(9)
f_{03} is given by:
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f_{p4} = f_{01}, required for the desired symmetry on the response curves.
f_{05} is given by:
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_{} ...(10)
f_{p6} is given by:
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f_{p7} = f_{05}, required for the desired symmetry on the response curves.
The center frequency of the James network is taken as the geometric mean of f_{01} and f_{05}, this is:
_{ }_{}_{ }_{ }...(11)
which coincides with the frequency of the minimum of the amplitude response curve when the bass and treble controls are at full boost. Agrees also with the frequency of the maximum of the amplitude response curve when both controls are at full cut. Usually, 1kHz is adopted as the center frequency.
A Design Example
Let us suppose we wish to design a tone control for a transistorized piece of equipment. An adequate value for R_{1} is 10k ohms. R_{3} will then be a 1k ohmresistor and R_{2} a 100k ohmpotentiometer (standard value).
It is convenient that f_{01} and f_{05} be separated one decade in frequency. Then, from expression (11) we may obtain that:
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Being f_{c} = 1kHz, we find that f_{01} must be 316Hz. The value for f_{05} will then be 3.16kHz.
From expression (8) we obtain for C_{2} a value of 503.65nF. Then, C_{1} should have a capacitance of 50.36nF. According to (9), f_{p2} = 31.6Hz.
R_{5} is taken equal to 5 times R_{2} in order to avoid loading effects on the bass network. Therefore, R_{5} = 500k ohms. R_{4} must be chosen such that an affordable value for R_{6} is obtained when using inequality (7). If we make R_{4} = 5k ohms, then R_{6} must satisfy the condition R_{6}>>65k ohms. We may adopt a value of 500k ohms for R_{6}.
From expression (10), with the value chosen for R_{4} we obtain C_{3} = 774.85pF and C_{4} = 8.52nF.
Finally, in order to avoid additional attenuation in the circuit, the output resistance of generator V_{g} (source resistance) must be made some 20 times smaller than R_{1}.
Simulation of the frequency response of the James network
Tone Stack Calculator 1.3 is an excellent software program that can be used for simulation of tonecontrol networks. It may be downloaded from: http://www.duncanamps.com/tsc/
Two simulations have been made using this program. The first simulation uses the above calculated component values and the second one, standard capacitor values for the circuit. No major variation on frequency response has been observed between them. Simulation results are shown below.
Fig.9 Frequency response of the James network with the bass and treble controls at full boost (upper) and full cut (lower).
Fig.10 Frequency response of the James network when standard capacitance values for C_{1}, C_{2}, C_{3} and C_{4} are used.
References
1. Baxandall, P.J. “Negative feedback tone control – independent variation of bass and treble without switches” W.W. 58.10 (Oct. 1952) 402. Correction 58.11 (Nov. 1952) 444.
2. Vargas Patrón, Ramón “Red activa de control de tono” http://www.inictel.gob.pe/publicaciones/rvargas/redactiva.htm
Ramon Vargas Patron LimaPeru, South America July 23rd 2004
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