
The TooleAudio BantamDAC
Frequency Cutoff
As mentioned on the Tweaks page, the output coupling capacitors are the most important factor in the quality of sound for the BantamDAC. In addition, the output coupling caps combine with the connected input impedance of an amplifier to form an RC (ResistanceCapacitance) circuit. This is important, because the physical phenomenon of combining a resistor and capacitor in a circuit results in a varying response to frequency. Specifically, in the case of a series RC circuit, the voltage across the resistor (in this case, the amplifier) is reduced at lower frequencies. In other words, the circuit may form a filter that filters out bass frequencies.
In audio, we try to achieve the full frequency spectrum of human hearing: 20 to 20,000 Hz. Since we can't control the load impedance (well, not usually), we try to select output coupling capacitors that will pass all of the bass frequencies. A tool that's often used to size output capacitors is the cutoff frequency equation, where the "cutoff frequency" is defined as the frequency where the RC circuit's output voltage is reduced by 3dB. This is an important quantity because it also signifies the point of output where power is reduced by onehalf. In human terms, a 3dB reduction in sound output is also that point where the ear can tell the difference in sound reduction.
This equation is expressed as ,where
F is the cutoff frequency and output response is down 3dB,
R is the resistance in ohms,
C is the capacitance in farads, and
2Π signifies a full frequency cycle in radians.
(a little algebra can easily rearrange this and solve for C)
This function works well for many applications. While a 3dB drop at 20Hz may be too much for the audiophile, we often size the cap at two or three more increments in size rating and forget about it. Or, we solve for C at 10Hz or below. The BantamDAC is so small, however, that being this liberal with cap rating gets us into a physical size that's much too big to fit on the board or in the recommended enclosure. We need something that will model the response curve more accurately so that we know exactly what's going on around 20Hz and below. Unfortunately, the cutoff frequency equation above only gives one point on the entire response curve for the circuit. We don't know where the response actually starts to drop, for instance. What's really needed is an equation that models the circuit for all responses, not just the 3dB point.
Back to the actual RC series circuit, depicted here:
Without going into too much detail, the circuit response may be modeled as a differential equation. The reason a differential equation is required is due to the complication of the varying response of R and the varying time factor in charging/discharging C. That's also why we refer to it as an impedance, instead of resistance  because the value of R varies in response to the frequency. Another way of visualizing this is that the voltage, Vo, is a vector whose response deviates in direction according to the frequency (from 1 to +1 over 360°, for instance). This is not explaining it in very much detail, but suffice to say if we were attempting to predict the response of Vo compared to Vin, we could also be referring to the Gain of the circuit. Taking all of this into account, an equation for the entire circuit's response may be written like so:
, where
ω is the angular frequency. (To get frequency in Hertz (Hz), we'll divide both sides by 2Π radians.)
We want this in decibels, so:
At this point, we can actually enter some arbitrary values for omega by entering frequencies from 1 to 20,000 and multiplying each one by 2Π radians to get ω. The result is that we can actually graph these curves for various values of C and R. This gives us the full appearance of the response curve, including a detailed view of the "knee" of the curves between the 3dB cutoff frequency and the actual frequency when the response starts to dip. With this information, we can size capacitors for the BantamDAC in confidence that our selection will not harm the frequency response in any way.
These results are shown below in three sets of graphs. A link to the spreadsheet is also included at the bottom so that you can download it and play with some of these values yourself. Essentially, three sets of curves are shown for each capacitor value. These correspond to the three most common ratings of stereo potentiometers in headphone amps  10K, 50K, and 100K. The volume pot typically sets the input impedance of a connected amp. Furthere, each graph represents an output coupling capacitor value that is in the common film cap sizes that will fit on the BantamDAC. Depending on your goal in frequency response, a 1uf capacitor will assuer full frequency response with all but a 10K input impedance. The 1uf value for C is only down less than 0.25dB at 20Hz for a 50K pot. Regardless, these curves and the spreadsheet will let you determine this on your own. 



RC Frequency Response spreadsheet (*.xls file)
(Rightclick and select "Save As ..." to download.) 
