PG2DTA: Determining the operating point and cathode bias resistor

In an earlier post I discussed how to choose a tube and an appropriate output transformer. The primary value of the transformer is known as the load and is often referred to as RL. The load can be drawn as a straight line on the grid curve graph and is called the loadline. The slope of the loadline can be determined using Ohm's Law again:

I = V / R

Using 5K ohms, the RL we determined for the 6V6 tube, and 100 volts an arbitrary value for V that makes the math easy, I can be calculated with the following equations:

I = 100 / 5000
I = 0.02
I = 20mA

Which means that the slope of the line rises 20mA every 100 volts. Each of the lines on the following graph has a slope of 5K.


The loadline is useful for determining a number of things when designing an amplifier. So, how do you know where to draw the load line? There are many approaches, but the following seems to work. Start by plotting the maximum plate dissipation on the grid curves as was done in my earlier post, Choosing a Power Tube and Transformer. Then draw a loadline that is tangent to the maximum plate dissipation curve. Why tangent? Well, if you go over the maximum plate dissipation curve, then you risk damage to the tube, premature failure or at least distortion. If you go under the curve, then you are wasting power.


If you look closely, you will see that the point at which the loadline is tangent to the curve is around 250V and 48mA. Of course, the easier way to get these values would be to look at the tube data sheet. According to the 6V6 datas heet, the Plate Voltage and Plate Current should be 250V and 49.5 mA - very close to what we came up with above.

These values are known as the quiescence or operating point, sometimes referred to as Iq and Vq. Once you know your operating point, you can calculate the cathode resistor. This is done by looking at the operating point and determining the grid voltage. The operating point does not intersect one of the grid lines already on the graph so you have to guess. In this case the point falls in the center of the -10V and -15V grid lines so -12.5V is a reasonable value.

Once you have the grid voltage, you determine the cathode resistor value using Ohm's law again:

R = V / I
R = 12.5 / 49.5
R = 252.525... Ohms

Once you have calculated the value, then look for a standard resistor value that is close to the calculated value. In this case, 255 Ohms.

The next question is what power rating should the resistor have. This is determined by calculating the power dissipated by the resistor and then doubling the value (at least) for safety. The power dissipation can be calculated using several different but related formulas:

P = I * I * R
P = .0495 * .0495 * 255
P = 0.625 watts

or

P = I * V
P = .0495 * 12.5
P = 0.619 watts

The values are different because the first formula uses the standard resistor value instead of the value originally calculated. Regardless, the values are close enough so that if you double the results you will end up with around 1.2 watts. There aren't a lot of 1.2 watt resistors out there, so round up to the next highest rating - 2 watts should be fine while 3 watts will be extra reliable.

PG2DTA: Choosing a Transformer for the 300B

In my last post of the Practical Guide to Designing Tube Amplifiers I described how to choose an output transformer for a triode-connected 6V6. If you recall, a reasonable load for a triode is twice the Plate Resistance:

RL = 2 * ra

In summary, you calculate ra using the following steps:

• Locate the tube data sheet
• Determine the maximum plate dissipation (Pa-max)
  
• Determine the maximum plate voltage (Va-max)
• Plot Pa-max and Va-max on the Average Plate Characteristics graph
• Draw a line tangent to the grid curve closest to where Pa-max and Va-max intersect
  
• The slope of the tangent is equal to ra

This time I will go over the steps again for a 300B, but with a little less detail.

I located a copy of the 300B data sheet (135K PDF) at the Western Electric website. I figure they are probably a pretty reliable source. The Pa-max and Va-max are reported in the Maximum Ratings section.

The following is the grid curves graph with the Pa-max, Va-max and ra lines plotted.

Examining the tangent line you can see that it rises around 180mA over 125V. We determine the plate resistance using the following formula:

ra = V / mA
ra = 125 / 0.180
ra = 694.444

By doubling the value of ra we have determined that a transformer with a primary of around 1400 Ohms would be appropriate. Suggested values I have seen range from 1250 to 5000 Ohms so again I think the calculations are correct.

PG2DTA: Choosing a Power Tube and Transformer

In my first post of the Practical Guide to Designing Tube Amplifiers (PG2DTA) I defined 10 steps for designing an amplifier. This post will go over Steps 1 and 2:

1. Choose a power tube
   
2. Determine the output transformer to use
   

While I'm sure there are a number of technical reasons that determine why an output tube is suitable for a particular application, I'm not going to attempt to describe them. These days, people aren't building tube amplifiers for technical reasons, they are building them because they like tube amplifiers. The tubes that are suitable for audio output are pretty well known so I'm not going to deviate from the list (tube depot, boi audio works, the tube store).

For the purposes of the guide I am going to use a triode-connected 6V6. Why? Because I like 6V6 tubes in guitar amps and the 6V6 is reasonable inexpensive. In addition, there is a local amplifier competition that I am entering so this is a good exercise.

The simplest way of determining which transformer to use is to look up what other people are using (Hammond, VT4C). However, you will notice that there is not a single value to use for a particular tube. Recommended values for the 6V6 range from 3,500 Ohms to 8,000 Ohms. This value is known as the load and will be used to draw the loadline in step 3 of the PG2DTA. For now, let's try to figure out where these numbers came from.

Generally speaking, a reasonable load for a triode is twice the Anode Resistance or Plate Resistance:

RL = 2 * ra

So, all we need to do is look up the Plate Resistance for the tube and we are done, right? Wrong! Unfortunately the Plate Resistance value changes depending upon the operating point of the tube. This is where things start to get more technical in the books and I get lost, but its really quite simple.

First thing to do is find the data sheet for the tube you want to use. A great source for finding tube data sheets is Duncan Amp's Tube Data Sheet Locator.

Find the maximum plate voltage (Va-max) and the maximum plate dissipation (Pa-max) on the tube datasheet. Unfortunately, there doesn't seem to be a consistent naming convention for these values. Some use the term Anode, some use Plate so you will have to become familiar with all of the terms. I am using General Electric's 6V6 data sheet for this example.

Find the graph that plots the plate current against the plate voltage labeled average plate characteristics. For the 6V6 tube, find those that are for triode connection. Draw a vertical line representing the maximum plate voltage (Va-max). For the 6V6 this will be 315 Volts.

Determine points on the graph representing the maximum plate dissipation (Pa-max). This requires some math, but its simple math. Using Ohm's law P = I * V we can calculate the current required for 9 watts at a number of different voltages with a quick rewrite of the formula. In other words, I = P / V where P = 9 and V = voltages we choose. For example:

9 / 60 = 0.150 or 150mA
9 / 100 = 0.090 or 90mA
9 / 120 = 0.075 or 75mA
9 / 200 = 0.045 or 45mA
9 / 250 = 0.036 or 36mA
9 / 300 = 0.030 or 30mA
9 / 320 = 0.028 or 28mA
9 / 340 = 0.026 or 26mA

You will end up with a graph that looks something like this:


To determine the Plate Resistance, find a grid curve that is closest to the intersection of Va-max and Pa-max and draw a line tangent to the curve. A tangent is the straight line that just touches a curve at a specific point on the curve. Without having the formula that determines the curve, I think the only way to determine the tangent is to do your best drawing it. Here is what I came up with:

Now that we have the Plate Resistance line plotted, calculate the slope of this line by using another of Ohm's laws V = I * R. Because the graphs are not exactly high-resolution, the easiest thing to do is look where the line intersects the grid. In this case, I=0, V=240 and I = 30 and V = 320. Thefore:

I = 30 (30 = 30 - 0)
V = 80 (320 - 240)
R = V / I
R = 80 / 0.030
R = 2,666

As I said above, optimal load for a triode is twice the Plate Resistance:

RL = 2 * ra
5,332 = 2 * 2,666

So, a tranformer with a primary of around 5K is appropriate for a triode-connected 6V6. This is consistent with many designs so I think that we calculated the plate resistance correctly.

A Practical Guide to Designing Tube Amplifiers (PG2DTA)

I've been reading the Radiotron Designers Handbook Version 4 (25MB PDF), the Morgan Jones Valve Amplifiers book, Steve Bench's web pages, and a number of other old books, but there is one thing I cannot seem to find described well - the single-ended amplifier. Maybe its just so simple that nobody goes over it in detail. Maybe its because the authors wanted more power so they focused on pentodes, push pull amps and feedback. Maybe I just missed the forest through the trees because I don't understand all the math and terminology. Whatever the reason, those books don't work for me. While the Boozhound Laboratories site is a great starting point, there are a number of places where I want to know the general idea of why a value was chosen so that I can design my own amplifier.

So here is my Practical Guide to Designing Tube Amplifiers (PG2DTA). I'm going to start with the following circuit - which is about a simple as you can get. There are two stages; a driver stage and a power stage. Personally, I like to refer to the power stage as the output stage, but for the purposes of this guide I will call it the power stage because many of the texts refer to power amplifiers and most websites refer to power tubes.


As you can see, the circuit does not include the values for the different tubes, resistors, capacitors, and voltages. In addition, the power supply is not represented in the diagram except for the magic B+ symbol. I call it magic because it seems to me that B+ is frequently referenced without any details of how to get it.

I have broken down the approach to determine which values to use into the following 10 steps:

1. Choose a power tube
   
2. Determine the output transformer to use
   
3. Determine the operating point
   
4. Calculate the cathode resistor, grid resistor, etc.
   
5. Choose an appropriate driver tube
   
6. Determine the load
7. Determine the operating point
8. Calculate the cathode resistor, grid resistor, coupling capacitor, etc.
9. Choosing a power transformer
10. Modeling your power supply with PSUD

Later I will expand the guide into different areas. For example, going from a single-ended output stage to parallel output tubes or different drivers stages (mu follower, srpp, etc).