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In this page I've started to write down the info I gathered from reading several sources. When looking for background information on the web I discovered that there is a wealth of info found such as circuits and DIY projects but there is hardly anything found that describes the basic operating principles nor the formulas and calculations necessary to do the math for your own projects.
Of course I have my SPICE tools and datasheets etc. and these already help a great deal when designing your own amp. But every now and then I need the basic info again and I always seem to have forgotten the details of certain formulas etc.
Therefore on this page some basic tube formula's are found and some more information that will help with the design of an amp.
The formulas used on this page are also available for download in an Excel spreadsheet (see download section) to calculate and create your own amp.
By the way: For the moment I will stick to preamp designs, and the English language.
The following figure describes a generic model for the tube, a common cathode stage, which is the basis for most other stages. By adding or deleting some components in this model we are able to tranform the generic model to one of the more specific models and distill amplification and output impedance from the generic model.
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figure 1.0a: Generic Amplifier |
So we deleted cathode capacitor, load etc. which are not needed to understand the model. By adding additional components, or by taking the output at a different point we can influence gain and output impedance etc. However, the same basic rules and law will still apply.
For the amplification of AC signals the following
formula is very important.
This relationship is one of the fundamental tube
equations, as it describes the tube's amplification in more detail. It tells
us how much the plate current will change as a function of the grid voltage.
S is the factor determining the steepness of the curve and Vp the plate voltage.
In the formula above, S is a constant parameter, but looking at the curves of a typical tube it is clear that S will not be perfectly constant at all. However, for audio applications and when using the tube in normal operating conditions, it is OK to use S this way.
Another important relationship is:
S * r_a = u
By the way, the same is true for the maximum/open loop gain "u". Also this parameter has no constant value but it is perfectly OK to use it's value as a constant in calculations as long as we use the tube within defined and correct boundaries.
How does a tube amplify signals? Since the plate current will not only flow through the tube but also through plate resistor and/or cathode resistor, there will be a voltage drop over these components as a result of this current. This voltage drop allows us to use the tube as an amplifier: A change of voltage at the input will result in a similar change in voltage over either plate or cathode resistor depending on which one we use as the output.
Thie above formula can be used to help calculating all other formulas in this document. You'll probably believe me when I make one calculation, and provide the other formulas for the rest of the circuits without too many calculations (pleez?)
The gain can be calculated using the basic formulas. As we use big capacitors in the powersupply to get rid of ripples in the Vb+ power lines, there is also another effect: For AC, the Vb+ is grounded.
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Figure 1.0b: Triode equivalent Circuit |
Above you find the same generic circuit as g1 where the tube is substituted with its plate resistance Ra and current source. The input V_i is shorted to ground.
The method for calculating the output impedance as used in literature is as follows: Put a AC source on the output, and short the input to ground and then determine the impedance based on the current that starts to flow as a result of that power source.
The common Cathode stage is the most common gain stage to use. In theory the gain that can be achieved by the CCS design is equal to the open gain if the tube (e.g. 100 in case of a 12AX7 tube). However, in real life the gain will be lower due to our choice of components in this stage, about 70% of the theoretical value of mu is typical..
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Figure 1.1: Common Cathode Stage (CCS) |
When we bypass the cathode resistor with a capacitor of sufficiently high value,
the behavior of the tube for AC voltages differs significantly. For bypassed
cathode caps we may substitute Rk=0 in the above formulas.
The formulas above explain why cathodes are bypassed often in phono preamps with passive RIAA filtering. When the output impedance of the tube driving the RIAA network is high compared to the values used in the network (especially the first resistor in series with the tube) tube aging will have an efect in the accuracy of the RIAA network.
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Figure 1.2: The Cathode Follower |
The cathode follower (and all variations) is the second basic circuit used in almost every preamplifier. It is used in the driver stage of the amp. It's amplification factor is at maximum 1, which means that it does not amplify at all but will instead have a (slight) negative amplification.
Why is it then used in amps? Well, there is more than amplification factor
alone: the Cathode Follower delivers more power at the ouput, it has a far better
output impedance than the Common Cathode Stage. It is therefore capable of driving
the power amplifier and the interconnects without loosing in quality of the
audio signal. And, since its distortion is minimal, there is often more too
gain than too loose when using a cathode follower.
As said, the output impedance of the cathode follower is relatively low. The formula below shows that the output impedance is dependent on the plate resistance and the mu factor. Tube with a high plate resistance and/or low mu deliver the best impedance.
The formula also shows that for mu factor much larger than 1, the output impedance
is close to Ro = ra / mu. And that means that R_o will be 1/S. This means that
a tube with low S would be a good candidate for a cathode follower that need
a low output impedance.
The Series Regulated Push Pull (SRPP) stage is a special case. Low distortion, high gain and yet lower output impedance make this stage popular in the 90's.
In this para I will stick to the SRPP design most often used, with the output taken at the cathode of the top tube (V2).
It is possible to take the output also and the plate of the lower tube (V1), and this will result in a better output impedance yet a lower gain. This variant is described in the Aikido section below.
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Figure 1.3: SRPP |
In the figure above, Rk2 is not 0 since we did not bypass it with a cathode capacitor, but Rk1 is 0 because we used a capacitor there.
It is important to remember that the Output Impedance formula is valid for very high load impedances only. Should we use the output to drive a low load (headphone for example) than the gain is also determined by the parallel impedance of: R_load || Z_upper-tube || Z_lower-tube.
The Mu-Follower is a similar circuit to the SRPP stage. It offers even better distortion figures and better output impedance.
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Figure 1.4: Mu-Follower |
The Cascode amp is a special circuit which is not often used anymore in todays designs.
The formula's for gain and output impedance of the Cascode amplifier are found
below.
The output impedance of the cascode circuit is relatively high and as a consequence some tubes (12AX7/ECC83) cannot be used in this circuit. And since we use a plate resistor the power supply voltage must be relatively high in order to assure enough gain.
In many cases this means that we also have to use separate powersupply for the heater of the top tube. The maximum voltage between cathode and heater is in the order of 90V which sometimes means floating powersupply for the heaters.
The Aikido amplifier is documented extensively on the TubeCAD website. The Aikido is a 2-stage preamp, but with minor modifications the circuit could also be used to build a headphone amplifier.
The first stage of the Aikido consists of a Common Cathode Stage with a Tube as plate load. It resembles the SRPP stage a lot, but the difference is that the output is taken at the interconnection point of the plate of the lower tube and the cathode resistor of the top tube (SRPP takes it from the cathode of top tube).
Broskie calls it "a symmetrically grounded-cathode amplifier - pure single-ended operation".
And its architecture is pure symmetrical, which means for example that V_out (DC) will be on half the Vb+ voltage. And as the second stage has a similar symmetrical setup, we will save ourselves a coupling capacitor between the two stages.
Because of the architecture of the first and second stage of the Aikido, distortion is cancelled for a great deal. Both stages are designed without decoupling caps on the cathodes.
The first stage of the Aikido amp is sometime referred to as a SRPP amplifier. However, most sources would say that a SRPP amplifier connects its output to the top of Rk2 resistor (cathode of V2) instead of taking it at the plate of V1.
When building one of my phono amps, I connected the output also this way, and I must say that I like sound quality and noise levels of this setup very much.
The output impedance of the first/SRPP Aikido stage can be computed as follows:
The second stage of the Aikido amp is a cathode follower circuit with a tube as cathode load. Such a stage is called a White Cathode Follower. It means that instead of a cathode resistor a tube is used.
The gain of the output stage can be calculated as follows:
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Page 2: Tube Aging >> |
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