# Bias

All your life people have been telling you that’s it’s wrong to have a bias.  A racial bias is racism and it’s bad.  A cognitive bias blinds you to objectivity.  Confirmation bias leads to crackpottery.  From every direction, you are pelted with the message that it is good and just to be neutral, objective, and unbiased and to be otherwise is backwards, ignorant, or evil.  I say f*%k that hippy sh!t.  Without bias, you’ll never get anything done.

As covered in other sections, an input voltage signal on the grid of a tube modulates the flow of electrons from the cathode to the anode.  A load at the anode or cathode then converts this modulating flow of electrons into your new (bigger and/or stronger) voltage signal.  For the grid to do its current flow modulation thing, it must be negative in relation to the cathode.  On the other hand, you could rightly say that the cathode must be positive in relation to the grid.  Creating this difference in voltage between the grid and the cathode is known as biasing and it’s what makes a tube work.

Biasing sets the difference in DC voltage between grid and cathode.  Biasing also therefore sets how much current (flow of electrons) you have through your tube, even when you aren’t giving it a signal.  Yes, the tube is still working if your CD player isn’t plugged in; it’s just amplifying silence (or an infuriating hum) instead of music.  The bias point is also sometimes called the quiescent point.  Quiescent actually means inactive and shit is certainly still active at that point, but whatever.

### Fixed vs Auto

There are two common ways of setting the bias in a tube: fixed bias and cathode bias.  Fixed bias applies a negative voltage to your grid, forcing it below the cathode.  Cathode bias (AKA self or auto bias) sets your cathode voltage above the grid.  The tube doesn’t really give a crap if you want to set the grid negative with respect to the cathode or the cathode positive with respect to the grid, but the choice of how you accomplish it will affect the overall design and performance of your amp.

One important consideration is how your biasing scheme will affect tubes as they age or if replacement tubes will behave identically to the originals.  With a fixed bias, you’ll always apply the same negative bias voltage to the grid (usually through a large resistor); this fixed negative voltage will cause slightly different current flow through different tubes (as well as changing current as tubes age).  This issue is especially important with push-pull output stages because ideally both tubes in a channel should draw equal current to offset the net DC in the output transformer primary.  As a result, some method of balancing the bias voltage for equal current draw in each tube is often necessary. If you are running at the edge of how much current a tube (or your power supply) can handle, fixed bias might shorten the lifespan of your tubes.

Auto-bias is usually achieved by putting a resistor in series with the cathode of a tube and referencing the grid to ground (usually through a large resistor).  As current flows through the tube and cathode resistor, it creates a voltage drop across the resistor and therefore raises the cathode relative to the grid by that voltage.  In this case, the tube is determining its own bias, and a tube that wants to conduct more current would see higher voltage at the cathode, which would make the grid more negative in relation to it and this, in turn, limits the current.  The major downside to cathode bias is that it uses up some of the total voltage available in the amplifier (sometimes a lot in power output stages).

As far as bias is concerned, you can design for higher voltage (more dangerous), lower power (less fun), or a potentially shorter lifespan of your tubes (more costly).  Tradeoffs like this are a recurring theme in tubes (and most other things in life, if you want to philosoraptor).

### Cathode Bias Gain & Bypass

When you auto bias a tube with a cathode resistor you create degenerative cathode feedback.  That sounds like a terrible bone munching affliction and as far as gain is concerned, it kind of is.  Because the cathode resistor drops voltage dependent on the current through it, AC current swings cause the bias to vary with the input signal.  As the input signal goes positive, more current flows through the tube and the bias voltage increases.  As the input signal goes negative, less current flows and the bias voltage decreases.  The anode load, tube, and cathode resistor are also all in series, so the AC voltage developed across the cathode resistor is robbing voltage gain from the anode load.  The gain with an unbypassed cathode resistor is given by:

Gain = Mu * Ra / (Rp + Ra + (Mu +1) * Rk)

Where Ra is the anode load, Rp is the plate resistance, Rk is the cathode resistance, and Mu is Mu. The term “(Mu +1 ) * Rk” is the turd in the punch bowl. From the anode’s perspective, the value of the cathode resistor (Rk) is multiplied by the Mu of the tube plus one, meaning the denominator can grow quickly if the Mu or Rk are large in relation to the load and plate resistance.  But there’s good news: we can nuke the Rk from space (in AC terms).

Capacitors do not pass DC, so our quiescent bias voltage is unchanged.  But capacitors pass AC signals, so this looks like an AC short to the tube, taking the Rk term out of our gain equation.  With a bypassed cathode resistor, the gain looks like:

Gain = Mu * Ra / (Rp + Ra)

Depending on the tube and circuit values, this can be a huge increase in gain, but it’s not without a gotcha. The effect of the capacitor and resistor combination is frequency dependent (key word: reactance) and so we want to chose a capacitor value that will provide an AC short for all audio frequencies of interest.  Too low of a value will roll off low frequencies (bass). To find the minimum capacitor value (in uF) we can use the equation:

C (in uF) = 1,000,000 / (2 * Pi * f * R’)

where R’ = Rk || ((Rp + Ra) / (Mu +1))

Where “f” is the -3db cutoff frequency of the high pass created by the capacitor.  For full-range audio we usually choose a frequency of 5 or 10hz.  This usually results in a hefty value capacitor so electrolytics are the de facto choice.  Electrolytic capacitors usually aren’t ideal for audio (large value variations and non-linear AC behavior) but if you need the gain it’s a necessary evil. If you don’t need the gain or lower output impedance, leaving the cathode resistor un-bypassed is AOK.

There are other ways of biasing your tubes so there’s no need to stop at resistors and capacitors.  For many applications though, it’s a very good compromise of cost, complication, and performance as long as you take your time and do the maths.