Nov
25
INTRODUCTION
This article is the second part of my examination of the output impedance (Zout) of triode single ended amplifiers using no negative feedback. Here I intent to examine carefully the effects at low frequencies, which I believe can help you understand the sometimes puzzling bass performance of SE amplifiers. With the help of the Thielle/Small theory, I will show how the high output impedance modifies the the system response.
Although the T/S theory is more familiar to speaker designers, I must use it to arrive at some conclusions relevant to the design of SE amplifiers. I will briefly describe the terms related to the theory as they appear but you may wish to consult the original papers (ref. 1,2,3,4) or a work such as Vance Dickason?s The Loudspeaker Design Cookbook (ref. 5) where these terms are much better presented.
I will include as examples only closed and vented box loudspeakers, because they are by far the most common but you can use the same methods with any other form of correctly modeled bass loading.
T/S PARAMETERS
The low-frequency design of most loudspeakers today is based on the justly famous Thielle/Small theory. At the low frequencies this theory uses an acoustical analogous circuit that allows you to calculate the actual acoustical output of the speaker driver in a box.
If you consider the Zout a simple resistive value, as I did in my first article, it is easy to use the T/S theory to analyze the behavior at low frequencies. Some available formulas take into account the output resistance of the amplifier and the cables. Thielle?s original paper addresses this point, and the work of Small (ref.2) show you very clearly what to do. If you know the basic Thielle/Small parameters of the woofer you?re using in the system (or better yet, if you measure them), you can use the equations (21) and (22) of ref.2 to calculate new parameters for your system. Look at these equations in a slightly different way: it is as if you had a “new” driver with a different Qes and Qts connected to a zero Zout amplifier. It will produce the same results.
Qes’ = Qes (Rg + Re) (1) Re
Qts’ = Qms Qes’ (2)
Qms + Qes’
where,
Rg = output resistance of amplifier + cables
Re = DC resistance of the loudspeaker coil
Qes = Q of driver at resonance due to electrical resistances
Qms = Q of driver at resonance due to nonelectrical resistances
Qts = Q of driver at resonance due to all resistances
Qes’= electrical Q of “new” driver
Qts’= total Q of “new” driver
You should make the calculations for Rg equal to the Zout of the SE amplifier which is typically around 3 to 4 ohms, as I showed in my previous article. If you have significant resistance in the crossover and cables you should add it. You will then have two sets of parameters, each with a different Qts. Using these parameters you can now easily calculate the system response with any loudspeaker design software (or by working through the tables and formulas). Then you can compare the two response curves.
Zout VALUES
All this is not suited for a quick analysis of a loudspeaker low frequency response, because it requires a certain amount of work as well as access to the loudspeaker design parameters which may not be available. But I have done it, looking for any general patterns that might appear.
I used this approach with the loudspeaker referred in the past article which has a KEF woofer with the following parameters: fs = 22.5 Hz Qts = 0.274 Qes = 0.286 Qms = 6.547 Vas = 160 and Re = 6.8 Ohm . The crossover resistance is a high 1.6 Ohm . Table 1 shows the parameters change with different values of Zout, and the final results for a system composed of this driver and a 70l closed box that is lightly stuffed.
TABLE 1
DRIVER PARAMETERS WITH
DIFFERENT VALUES OF Rg
| driver alone | Rg=0 | Rg=crossover resistance |
Rg=crossover resistance +2.7 ohms |
Rg=crossover resistance +3.7 ohms |
| Qes |
0.286 |
0.353 |
0.466 |
0.508 |
| Qms |
6.537 |
6.537 |
6.537 |
6.537 |
| Qts |
0.274 |
0.335 |
0.435 |
0.472 |
| fs (Hz) |
22.5 |
22.5 |
22.5 |
22.5 |
| Vas(l) |
160 |
160 |
160 |
160 |
| driver in box fs (Hz) |
- |
39 |
39 |
39 |
| Q |
- |
0.580 |
0.753 |
0.880 |
Fig. 1 shows the calculated frequency responses and fig.2 shows the actual near field measurements with a very low Zout amplifier and with 3.7 ohms and 2.7 ohms resistors connected between the amplifier and the loudspeaker. Although there were some low pass crossover contributions, the differences between the cases in these measurements are very close to those expected.I have used this method with other speakers and it has worked quite well, but always worried about how to take into account the primary inductance (Lp) of the output transformer. I made the measurements shown in fig.2 with resistors simulating a high Zout. As explained in my former article, the effect of the primary inductance changes the value of the Zout at very low frequencies. Therefore, this use of a fixed resistor value should not be completely accurate.PRIMARY INDUCTANCE
When the finite value of the primary inductance starts to show its effects, the output impedance begins to drop at the low frequencies. Which value of Zout should we use? There is no easy answer. You must consider what happens if you use a low frequency model of the output stage, which includes the primary inductance of the transformer.
Although the original T/S papers include a provision for taking high Zout values in account, they apear to consider only resistive values. I could find nothing about the effect of reactive Zout. The nearest I found was Benson?s work (ref.6) which looks at filters that are directly connected before the loudspeaker but he apparently restricts himself to the case of a zero output resistance amplifier.
Here I will use an acoustical analogous circuit of the whole system, composed of output stage, driver and box. I developed it by connecting the model for low frequencies of the SE output stage (fig.2d of the previous article) with the speaker?s electrical equivalent circuit (see reference1 or 2). The result is circuit a of fig.3.
I have used the T/S assumptions and methods to arrive at an acoustical circuit that permits calculation of the system?s low frequency acoustical response (circuit b in fig.3). In this circuit, the primary inductance appears as an acoustic compliance represented by a capacitor symbol. The relevant new parameters you need are rp, R1, Lp, R2, and n?, which are already defined in my GA 3/97 article and repeated in the figure.
The assumptions needed to make this circuit valid are that the output stage low frequency response is limited by the transformer?s primary inductance, that the iron core losses will be very small, and that the loudspeaker is working as a piston - plus all the other assumptions of the T/S theory.
FURTHER QUESTIONS
In fig.3 also appear the acoustical circuits of the output stage and loudspeaker in a closed box and in a vented box (circuits c and d). Several questions arise. Will the transformer plus loudspeaker be a third order system? If we put the speaker in a closed box will we also have a third order system ? Also will fourth order vented systems became fifth order systems?
The answer to all these questions is yes, but depending mostly on the value of Lp, you can ignore it. For suitably high Lp values the acoustical circuit will, with some use of circuit theory, revert to the original T/S circuits with the values of (rp+R1)/n? and R2 added to form the amplifier output resistance (Rg).
But what are the values of Lp? In attempting to answer this question I wrote a program to calculate the response of the complete system based on the models in fig.3. You can use this program for closed and vented boxes by supplying the driver parameters, the box data and the amplifier values of rp, R1, R2, n? and Lp.
It calculates the system response, computing not only the frequency response of the driver in a box with the variations introduced by the high (and complex) Zout but also accounting for the low frequency response of the output transformer.All the phase responses are accounted for, as well.
TABLE 2
OUTPUT STAGE PARAMETERS
WITH TWO DIFFERENT TRANSFORMERS
AND 300B TUBE
|
transformer 1 |
transformer 2 |
|
| rp(ohms) |
700 |
700 |
| R1(ohms) |
270 |
270 |
| Lp(H) |
14 |
7 |
| R2(ohms) |
0.6 |
0.3 |
| n? |
314 |
350 |
| Zout(ohms) |
3.7 |
2.7 |
| Lp/n?(mH) |
44.5 |
20 |
With the program, I calculated the response of the same loudspeaker I have been using connected to a 300B SE amplifier with two different output transformers.The relevant specifications of the output stage with the two transformers are listed in table 2. The second transformer, specially made, has a low 7H value for Lp. I followed these calculations with the near-field measurements of the loudspeaker connected to the SE amplifier. Fig. 4 and 5 show the results of the simulations with the program and fig. 6 and 7 the near field measurements.SIMULATION OF CLASSICAL ALIGNMENTS
I have tested the model and the program with other speakers and transformers, and they have worked fairly well. With the program, I have simulated countless loudspeakers designs with many different SE output stage parameters. From these simulations, I have selected some that may give a good picture of the overall effect you can expect with a SE amplifier connected to a loudspeaker.
Figures 8 to 15 show these simulations of classical bass alignments. Each figure has four different responses corresponding to four different output stage parameters, as detailed in Table 3.
TABLE 3
OUTPUT STAGE PARAMETERS
USED IN THE SIMULATIONS
|
amp A |
amp B |
amp C |
amp D |
|
| rp(ohms) |
- |
700 |
700 |
700 |
| R1(ohms) |
- |
230 |
230 |
230 |
| Lp(H) |
- |
? |
10 |
3 |
| R2(ohms) |
- |
0.6 |
0.6 |
0.6 |
| n? |
- |
300 |
300 |
300 |
| Zout(ohms) |
0 |
3.7 |
3.7 |
3.7 |
| Lp/n?(mH) |
- |
? |
33.3 |
10 |
In the simulations I have used a fixed relation between the Qes and Qms of the driver such that Qms/Qes = 5. This was a guess at what I believe is a typical driver, and is important because systems with lower relative value of Qms will be less affected by the high Zout. On the other hand, the opposite is true for systems with high Qms. Figure 16 shows one simulation for speakers with the same Qts but with different relations of Qes and Qms. I have used extreme cases so that all normal drivers responses would be within these two cases.For vented box simulations, I have also used a Ql value of seven. Ql represents box losses. Lower values correspond to higher losses. It has na effect similar to Qms in the response.LOW VALUES OF Lp
I believe the first important fact that emerged from all the simulations I have done was the confirmation that the values of Lp/n? found in normal SE transformers has little effect in the system response. The use of Lp/n? permits the comparation of the low frequency performance of transformers designed for different tubes with different impedance.
The specifications listed by some transformers manufactures show that the smallest value for Lp/n? for different type of transformers is usually around 35mH refered to the 8 ohm tap. This sort of value means that only if you have a loudspeaker with very good low frequency extension (something like f3 below 30 Hz) will you need to consider carefully the primary inductance values of traditionally designed SE transformers.
If you are using in your projects off the shelf SE output transformers with a gap, the assumption of resistive behavior for the Zout is sensible. This allows you to use the formulas (1) and (2) from the T/S theory to account for the high Zout, as long as the output transformer holds its specified value of Lp at the bias current you are using.
The second fact is an interesting effect: low values of Lp could help in some cases! The initial effect of a low inductance is to reduce the usual hump, bringing the loudspeaker response a little closer to the intended one. It is not just the reduction you would expect from the falling low frequency response of the amplifier measured into resistive load, because the reduction is greater for higher impedances, as in the resonance peak, where the high Zout introduces the hump.
This is why I have included in the simulations the unusually low Lp value of 3H. A low value of Lp may in some instances bring the speaker?s low frequency response a little closer to that expected by its designer. This is well ilustrated by Fig. 8.
MODIFYING LOW-FREQUENCY PERFORMANCE
High Zout amplifiers always modify the bass performance of the loudspeakers connected to them in a way that depends on the loudspeaker alignment. Although it is very difficult to describe what is a typical loudspeaker load at low frequencies, most of them are very far from a constant 8 ohms resistance, and usually have one or two impedance peaks.
Therefore, it may be possible to optimize the SE amplifier performance at low frequency by balancing the effects of the high Zout with those of the transformer?s primary inductance.For SE amplifiers used with normal loudspeakers there may be a possibility for optimizing the system performance at low frequencies by balancing the effects of the high Zout with the effects of the primary inductance of the transformer. This may work with at least some loudspeakers, and since it pointa to lower values of Lp, you may derive other related benefits such as better high frequency response, better power bandwidth, lower cost and smaller size.
Another possibility is to optimize the design of the whole system when designing both the amplifier and loudspeaker. I am exploring this route right now, and the effect of the variation of the Lp with power levels and frequency remains to be verified more carefully.
Although transformers operating with DC in the primary and with air gap exibhits much less Lp variation than those designed for push pull operation, the sensitivity of the system to these variations may be significant. With high relative values of Lp the difference between its effect and that of an infinite value is very small for frequencies above 20 Hz. But if you depend on its (low) value for an alignment you may need to control it more tightly than usual.
DO SE AMPLIFIERS HAVE POOR L.F. RESPONSE?
It is often taken for granted that tube amplifiers, specially single ended ones of the type I am discussing here, do not have good bass response. One of the main problems seems to be that loudspeakers not designed to work with the high output impedance of these amplifiers will develop a hump in the low frequency response and suffer degradation in their transient response.
But this picture can change if you design the loudspeaker to work with a high Zout. This can be done quite accurately using the same formulas (1) and (2) for the low frequency design and you may even get some additional benefits.
When you design loudspeakers, the efficiency, the box volume and the low frequency response are interrelated in such a way that for any two of these parameters that you fix the third one is already fixed for a given bass alignment. A high output impedance in the amplifier is one way to change this balance. If you model it correctly and use it to your advantage, it can give you more efficiency with the same low frequency response and box volume.
To explain more fully, the speaker efficiency is fixed whenever you select a driver. Choosing the box type and alignment defines the box size and the low frequency extension. With the same box type, alignment and size you can have an efficiency gain for the same low frequency response with amplifiers of high output impedance by choosing another driver.
MAGIC FORMULAS
This result also comes from formulas (1) and (2) as I will show in the following example. If you design a box for a loudspeaker to have a f3 of 40Hz in a vented B4 alignment you will need a woofer with Qts = 0.405. When you use this system with a high Zout amplifier, it will develop the already expected hump.
To correct this you could use another speaker driver with all the same parametersexcept for Qes and Qts. The Qes of the new driver will need to be the Qes of the old driver divided by (Zout+Re)/Re. When this new driver is used with the SE amplifier it will have again a Qts of 0.405.
Also from the T/S theory , the efficiency of a driver is given by equation (33) in ref.2:
KfsіVas , Qes
where _10 K = constant = 9.64 x 10 for Vas in liters fs = resonant frequency of driver Vas = Equivalent Compliance Volume in liters
Since K, fs and Vas can remain unchanged, and the Qes of this new driver will be smaller, its reference efficiency will be greater in dB by 10log((Zout+Re)/Re). For typical values of 3.7 ohms for Zout and 6 ohms for Re you will have a gain of 2.1 dB for any type of bass alignment.
You can see this in fig.17, where I have simulated four different responses. Of course it may be difficult to find exactly this “new” driver but this gain will always exist and may appear translated into more efficiency, a smaller box volume, a lower f3 or any combination of the three.
You need not use single ended tube amplifiers to get high Zout. You could as well use solid state amplifiers with current feedback to get this same characteristic. It may be hard to have the same mids and highs, but this is another debate and as much as the high output impedance is inherent in a single ended amp design, you could use it in your favor. Surprisingly, it is one way to get a better compromise between efficiency, box size and low frequency response for any kind of bass alignment.
CONCLUSIONS
This and my former article show that the high Zout can produce important changes in the loudspeaker frequency response, and I believe that you cannot talk about meaningful listening tests for SE amplifiers without considering it. Probably the only thing that usually mask part of the high Zout effects is the room interaction with the loudspeaker, which may be responsible for even larger variations of the frequency response in the listening position.
I have shown how the bass alignment of typical loudspeakers is altered by a high Zout and also the influence of the finite value of the primary inductance of the output transformer. I did this last with the help of an acoustical model which includes the typical output stage of a SE amplifier. This article deals with the small signal behavior of the system composed of output stage, speaker and box, and although large signal considerations are also needed for the pratical use of the information contained here I believe that it can be used for tree basic purposes:
1. To help in the design of SE amplifiers more suitable to drive an existing normal loudspeaker. This could even include the use of much smaller than usual primary inductance to partially compensate the high Zout effects, and this in turn could lead to several benefits in other areas of transformer performance such as size, low frequency power response, high frequency extension and cost.
2. To help design loudspeakers that not only work correctly with common SE amplifiers, but can achieve an even better bass performance with improved relationship between size, efficiency and low frequency response.
3. To design the amplifier and loudspeaker work with one another as a system optimized to use all the above possibilities. If you can keep the bad effects of the high output impedance under control and design the loudspeaker to take advantage of it, you can actually benefit from it.
Finally, the high output impedance might have even more subtle effects on a system composed of amplifier, cable and loudspeaker which I have not been able to figure out, but I have tryed to detail and understand the more easily measurable effects it produces. Next time I intend to describe an amplifier and loudspeaker that were designed using some of the above ideas.
FIGURES
REFERENCES
ref. 1 - A. N. Thielle - Loudspeakers in vented boxes -Part 1 & 2 - Loudspeaker Anthology Vol.1 pg.181 & pg.192.
ref. 2 - R. H. Small - Direct Radiator Loudspeaker System Analysis - Loudspeaker Anthology Vol.1 pg.271.
ref. 3 - R. H. Small - Closed-Box Loudspeaker Systems - Part 1 & 2 - Loudspeaker Anthology Vol.1 pg.285 & pg.296.
ref. 4 - R. H. Small - Vented-Box Loudspeaker Systems - Part 1,2,3 & 4 - Loudspeaker Anthology Vol.1. pg.316, pg.326, pg.333 & pg.339.
ref. 5 - Vance Dickason - The Loudspeaker Design Cookbook - available from Old Colony Sound Lab, PO Box 243, Peterborough, NH 03458
ref. 6 - J. E. Benson - An Introduction to the Design of Filtered Loudspeaker Systems - Loudspeaker Anthology Vol.1 pg.365.
ACKNOWLEDGMENT
I would like to thank Mr. Manuel M. Pereira for all the discussions about output transformers and other related issues. I’m also indebted for his help and patience with all the changing requirements on the transformers he designed and made for this work.
(published in Glass Audio 6/97)
Eduardo B. E. de Lima
(c) Copyright 1997 Audio Amateur Corp.
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