Michael S. McCorquodale ©2005
AbstractA common cathode vacuum tube preamplifier stage is analyzed and simulated. The concept of a linear small-signal AC model for the vacuum tube triode is presented and employed for circuit analysis. Analytical calculations for preamplifier gain and bandwidth are derived and compared to simulation results in the spice3f5 environment where good agreement between the two is shown.
Supplemental MaterialsIn this paper a vacuum tube preamplifier circuit using a 12AX7 triode will be analyzed and simulated. The data sheet for the triode will be used as a reference for determining relevant device performance parameters. The preamplifier circuit will be simulated within the Berkeley spice3f5 environment and a binary of this simulator for the Windows platform is available below. The spice netlist for the preamplifier circuit to be analyzed and simulated is also available below.
12AX7 High-Mu Twin Triode 9-Pin Miniature Type, General Data Sheet, Radio Corporation of America, Electron Tube Division, July 1947. |
Berkeley spice3f5 Binary for Windows platform, courtesy of Prof. Dr.-Ing. Holger Vogt, Universität Duisburg-Essen, Nov. 2005. |
Preamplifier Spice Netlist Spice3f5-compatible netlist for the preamplifier circuit to be analyzed and simulated in this paper. |
Analysis
Prior to performing analysis it is useful to observe that there are two types of electrical signals present in any analog amplifier circuit and the properties of each are not identical. The composite signal present in an amplifier circuit is comprised of time-varying, or alternating current (AC), signals and time-invariant, or direct current (DC), signals. In this work, the DC signals are denoted by capital letters. For example, the DC plate current is represented by I_{P}. The AC signals are denoted by lower case letters and correspondingly the AC plate current is represented by i_{p}. Voltages follow an identical form of notation. Fig.1 illustrates these AC and DC components, using the notation described, for a common cathode preamplifier stage.
A capacitor behaves as a DC block. However, it will pass AC signals in general, but the transfer will be contigent upon signal frequency, type of capacitor, and total capacitance. These differences are attributed to the mechanism of current flow in each case. DC signals move charge by conduction. Electrons from one point in the circuit travel continuously to another point due to an electric field that is created by a potential difference, or voltage. Once the capacitor is charged fully by the conduction of charge to each plate, the capacitor appears as an open circuit to DC signals. In contrast, AC signals propagate through capacitors by the mechanism of displacement. Small amounts of charge are moved to and from the plates of the capacitor and the charge is never conducted through the capacitor. Nevertheless, this displacement of charge is the means by which AC signals propagate through capacitors. The AC signal is superimposed upon the DC signal, thus the composite signal can be decomposed and each component can be treated separately by superposition. For completeness, it is worth noting that this is only true to a first order since nonlinearities in the circuit will cause AC signal components to give rise to DC signal components, but this is beyond the scope of this paper. Lastly it will be shown that the DC bias of the amplifier affects certain AC properties of the circuit, such as the gain.
Begin the analysis by recalling the common cathode configuration, shown in Fig. 2.
The grid bias resistor, R_{GB,} provides a high resistance DC path to ground, thus it biases the grid to zero volts DC. Moreover, it sets the input resistance into the preamplifier. It is generally desirable for the input resistance of an amplifier to be as high as possible so the input signal is not attenuated by the voltage divider set up by the amplifier input resistance and output impedance of the signal source. The bypass capacitor, C_{K}, provides an AC path from the cathode to ground. Therefore, the cathode resistor, R_{K}, does not influence the AC signal since the cathode is effectively an AC ground. The grid resistor, R_{G}, limits the frequency response of the preamplifier by interacting with the capacitance between the grid and plate to create an RC circuit with a -3dB point, or half power point, of approximately 20kHz as will be shown subsequently. This resistor limits the frequency response and damps any oscillations that may occur at higher frequencies beyond the audio bandwidth. The plate and cathode resistors simply limit the DC current in the tube, or more specifically, they are what set the bias, or quiescent, point.
Next, it is instructive to define certain AC parameters of the tube so an AC model, more commonly referred to as the small signal model, may be derived. Be aware that the small signal model is exactly that: a model. Models are intended to approximate the real physical operation of a device and the most useful models are typically linear. The operation of tubes, or any device for that matter, is nonlinear in general. However, if one can represent the operation of the device by familiar and simple linear circuit elements, such as resistors and capacitors, then the model facilitates manual analysis. The linear small signal model parameters are as follows:
Here r_{p} is the dynamic plate resistance, g_{m} is the mutual conductance (or transconductance), and µ is the amplifier gain. µ is not the gain of the amplifier stage, rather this parameter quantifies the open loop, or unloaded, gain of the amplifier. Thus µ is actually the maximum possible gain for the stage.
If the reader is unfamiliar with calculus, the ? operator represents a partial derivative of a multivariable function with respect to the variable in the denominator. This can be thought of as a differential change, or a very small change, of one parameter with respect to the other. The straight bar means "while holding constant." Thus in words, the dynamic plate resistance is a small change in the plate voltage with respect to a small change in the plate current while holding the control grid voltage constant.
Similarly the mutual conductance is the change in plate current with respect to the grid voltage while holding the plate voltage constant. Thus if the grid voltage changes slightly, then a small amount of change in the plate current is induced. This ratio is the mutual conductance which is fundamental to tube operation. The control grid controls the current from the plate to cathode. Here that fact has been quantified.
With these parameters, a linear "small signal model" for the device can be derived. Although there is no absolute definition of what a small signal is, it is safe to assume that the audio signal is always small in preamp circuitry. A more rigorous definition that holds generally is that a small signal must incur no more than 5% total harmonic distortion (THD). The idea behind such a metric is that at 5% THD nonlinearity is substantial, thus the linear model becomes insufficiently accurate. Power amplifier stages cannot be modeled by a small signal model because the signal is large, THD is high, and the linear differential parameters do not apply.
The linear small signal models for the triode are shown in Fig. 3. The triangular voltage and current sources in Fig. 3 represent dependent sources meaning that the voltage or the current depend on another voltage or current elsewhere in the circuit. There are no capacitors in the models in Fig. 3, thus there are no frequency dependent devices in the model. Although this is a very simple model, it is accurate for calculating the midband gain, or the gain in the middle of the audio bandwidth.
(A)
(B)
In Fig. 3A the small signal model makes use of a dependent voltage source. Thus, when a grid-to-cathode voltage is applied, a corresponding plate-to-cathode voltage is induced that is multiplied by the parameter µ. The plate resistance is then in series with this source. A dual of this circuit can be derived by replacing the voltage source with an equivalent current source in parallel with the plate resistance, a transformation known as Norton equivalency. The two models are electrically identical at the terminals. For example, using the dependent voltage source model in Fig. 3A will yield a plate voltage of -µv_{gk}. In comparison, a plate voltage of -g_{m}v_{gk}r_{p} can be determined using the dependent current source model shown in Fig. 3B. However, the product g_{m}r_{p} is equal to µ as shown in Eq. 1, thus the two models are equivalent at the plate terminal.
Parasitic capacitors exist between each terminal of the tube and affect the frequency response of the preamplifier. These capacitors are illustrated in Fig. 4. They arise from the physical construction of the device.
These frequency-dependent components can be incorporated into the midband small signal model. The modifications for the dependent current source model are shown in Fig. 5.
The reader may wonder why in this model the capacitors are not short circuits to the AC signal. This is due to the fact that these capacitors do effect the AC signal. The capacitors that were shorted circuited previously were bypass capacitors and are typically very large so the AC impedence is very small. Consequently, these capacitors can be modelled accurately as a short-circuit. The parasitic capacitors in the model above are are small by comparison, thus the associated impedance is not negligible. These devices, in fact, dominate the frequency response of the preamplifier stage as will be shown.
With the new linear small signal model for the triode a small signal equivalent circuit for the common cathode preamplifier stage can be derived. There exist some simple rules for determining the small signal equivalent circuit for the preamplifier stage. This is to be done for analysis only, thus do not actually do this to a circuit. The rules are as follows:
- Short all coupling and bypass capacitors by virtue of presenting a low impedance to the AC signal.
- Short the power supply to ground by virtue of it being a fixed voltage.
- Replace the active devices (tubes) with the appropriate linear small signal model.
- Reduce parallel and series passive devices the from new circuit as appropriate.
- Perform the desired analysis and compute the desired performance parameters.
Fig. 6A shows the midband small signal model for the preamplifer circuit shown in Fig. 2 where the aforementioned rules have been applied. The midband gain can be determined from this schematic, while Fig. 6B shows the frequency-dependent model including the parasitic capacitors which can be used to determine the bandwidth, or -3dB frequency, of the preamplifer. In both figures, the dependent current source model has been selected because it is simpler to analyze as will be shown. However, the dependent voltage source model could also be employed and results would be identical.
(A)
(B)
Referring to Fig. 6A, the voltage at the plate, or output, will be the product of the current to the output and the parallel combination of r_{p} and R_{A}. The current at the output is g_{m}v_{gk}, v_{gk} = v_{in}, thus the gain is v_{out}/v_{gk}. The derivation of the gain expression is merely a matter of algebra and is shown below where A_{v} denotes the voltage gain.
In the expressions above the operator || represents the parallel combination of resistances, or specifically the inverse of the sum of the inverses for each resistor.
The amplification factor (µ), the mutual conductance (g_{m}), and the dynamic plate resistance (r_{p}) are functions of the bias point as can can be seen in the data sheet for the device. This is exactly how the DC and AC signals are interdependent. It should be apparent that R_{A} is incorporated into the final gain equation. Clearly changing R_{A} will change the bias point which will also change the AC model parameters and hence the gain. Prior to performing a SPICE simulation, a numeric example is presented. Examine the circuit in Fig. 7.
The current source AC midband model for this stage was shown in Fig. 6A for this circuit. In Fig. 6B the frequency-dependent model is presented. It is well known that in the common cathode configuration, the input-to-output capacitance dominates the frequency response and all other parasitic capacitors can be ignored. The parasitic capacitor that is coupled from the input to the output is C_{GA}. Using the well known Miller's Theorem, this feed-forward capacitor can be modeled by an equivalent input capacitor that is multiplied by the gain of the stage and is in shunt with the input signal. This concept can be investigated further in any elementary circuits text. The Miller transformation enables the schematic in Fig. 6B to be reduced to the circuit shown in Fig. 8.
For the cathode and plate resistor values shown in Fig. 7, the DC grid voltage is 0VDC while the plate voltage will be approximately 250VDC. Referring to the datasheet for the 12AX7, where the relevant graph is pictured in Fig. 9, the plate resistance and mutual conductance (or transconductance) can be determined. Interpoltaed lines have been drawn on the graph manually between the curves for plate voltages of 300VDC and 200VDC. The mutual conductance and dynamic plate resistance are found by locating the points at which these curves intersect the line where the grid voltage = 0VDC. As shown, the dynamic plate resistance and mutual conductance are 36kΩ and 2.8mmhos respectively.
The gain is then calculated as follows:
In the last calculation in Eq. 3 the linear gain is converted to decibels, or dB, which is a common manner in which to represent the gain of amplifier stages.
The bandwidth, or -3dB frequency, can be calculated by noting that the RC time constant associated with the input grid resistance and Miller capacitor dominates the frequency response. All other parasitic capacitors can be ignored. Thus the RC time constant associated with this devices can be calculated and the -3dB frequency can be determined by taking the reciprocal of this time constant and converting from radians to Hertz. The value of C_{GA} can be obtained from the 12AX7 data sheet and is 1.7pF as shown in Fig. 10.
The bandwidth is then calculated as follows:
These calculations are compared to simulation next.
SPICE SimulationAlthough the previous analysis was trivial, using a more complex model would unneccessarily increase the complexity of the analysis. Doing so is not insightful as the purpose of manual calculations is to simply determine the parameters that most significantly affect the preamplifier performance and then make adjustments to the design accordingly. Therefore, utilizing a simulation program is an indispensible tool since the designer can increase the complexity and accuracy of the model without increasing the design time.
The Berkeley spice3f5 package was utilized for the simulations that follow. This paper will not address SPICE netlist format or the use of SPICE, but there exist numerous references on the subject that can aid the reader. The SPICE netlist for the preamplifier stage can be downloaded at the beginning of this paper. Additionally, the commands required to generate the results shown will be presented. The reader will not require any additional information in order to reproduce identical results.
Fig. 11 presents the schematic of the circuit to be simulated. The circled numbers refer to the nodes in the netlist at the top of the hierarchy. The uncircled numbers refer to
the nodes within the preamplifier subcircuit model. As shown, v(1)
corresponds to the input voltage, v(2)
corresponds to the output voltage, and
v(3)
corresponds to the power supply voltage.
Three different simulations were performed: OP, TRANS, and AC. The OP analysis provides the DC operating currents and voltages. Transient analysis determines the response in the time domain and AC analysis shows the frequency response, or the gain against frequency. Results of the OP analysis are shown in Table 1.
i(va) = 1.104340e-03
i(vin) = 0.000000e+00
v(1) = 0.000000e+00
v(2) = 2.395655e+02
v(3) = 3.500000e+02
To generate these results, comment the TRANS and AC analyses in the netlist, uncomment the OP analysis, and execute the commands shown below.
--
Program: Spice, version: 3f5 Build 2.5
Compiled with MINGW32 GCC-3.2.1 within MSYS
Date built: Sun Feb 29 13:18:47 MEZMZ 2004
Spice 1 -> cd netlist directory
current directory: netlist directory
Spice 2 -> preamplifier.sp
Circuit: PREAMPLIFIER
Spice 3 -> run
Spice 4 -> print i(va) i(vin) v(1) v(2) v(3)
--
Here, i(va)
is the current drawn from the supply. Thus, the bias current is approximately 1.1mA as predicted. v(1)
is the grid voltage and it is, of
course, biased to ground via the 1MΩ resistor. There is no grid current, as shown by i(vin)
since vin is connected directly to the grid. The plate voltage,
v(2)
, is at approximately 240V while the supply voltage, v(3)
, is at 350V.
The transient simulation provides the time domain response for a given input. The input signal, v(1)
, was slected to be 100mV and 1kHz as shown in Fig. 12A. The
output signal, v(2)
, shows significant amplification at the plate as shown in Fig. 12B. By inspection, the gain is approximately 60 since the output is swinging just
approximately 6V zero-to-peak for a 100mV zero-to-peak input.
(A)
(B)
To generate these results, comment the OP and AC analyses in the netlist, uncomment the TRANS analysis and execute the following commands.
--
Program: Spice, version: 3f5 Build 2.5
Compiled with MINGW32 GCC-3.2.1 within MSYS
Date built: Sun Feb 29 13:18:47 MEZMZ 2004
Spice 1 -> cd netlist directory
current directory: netlist directory
Spice 2 -> preamplifier.sp
Circuit: PREAMPLIFIER
Spice 3 -> run
Spice 4 -> plot v(1)
Spice 5 -> plot v(2)
--
Similarly, by performing an AC analysis, the frequency response of the preamplifier is determined. The results are shown in Fig 13.
(A)
(B)
To generate these results, comment the OP and TRANS analyses in the netlist, uncomment the AC analysis and run the following commands.
--
Program: Spice, version: 3f5 Build 2.5
Compiled with MINGW32 GCC-3.2.1 within MSYS
Date built: Sun Feb 29 13:18:47 MEZMZ 2004
Spice 1 -> cd netlist directory
current directory: netlist directory
Spice 2 -> preamplifier.sp
Circuit: PREAMPLIFIER
Spice 3 -> run
Spice 4 -> plot vdb(2) xlimit 100 1e5
Spice 4 -> plot vdb(2) xlimit 1e3 1e5
--
The gain is very flat over the frequencies of interest for audio amplification. Moreover, it can be seen that the midband gain is approximately 35dB which is very close to the analytical prediction of 37.4dB. In Fig. 13B, the frequency response is zoomed and the frequency at which the gain decreases by 3dB is approximately 22kHz. This result also corresponds well with the analytical prediction of 18.6kHz. The analytical calculations for the midband gain and -3dB frequency are in error by +6.86% and -15.5% respectively when compared to simulation results.
Design LessonsThough analysis yielded fairly accurate results in this example, the purpose of analysis is not to obtain the most accurate result possible, rather it is to understand the circuit parameters that contribute to the performance parameters of interest, such as the gain. With this in mind, the designer can easily adjust circuit parameters and use a simulator to determine critical performance parameters accurately.
With this in mind, an obvious question is, how can the gain be increased? As shown in Eq. 2, increasing R_{A} will achieve that goal. However, it does not come without a penalty. Consider Eq. 3 where it can be seen that as the gain increases, the bandwidth decreases. Confirming this, a simulation was performed where R_{A} was increased to 200kΩ. Results are shown in Fig. 14.
(A)
(B)
In Fig. 14A it can be seen that the gain has increased to nearly 37dB which corresponds to a linear gain of nearly 71. However, the bandwidth is now less than 20kHz as shown in Fig. 14B. This simple example should equip the reader with both analytical and design skills for the development of vacuum tube audio amplifier stages.
ConclusionThe concept of a linear small-signal equivalent model for both the triode vacuum tube was presented in this work and applied to the analysis of a preamplifier. Analytical predictions were compared to simulation where good agreement was found. This work prepares the reader for more advanced analysis and simulation, which are required for advanced audio system design and analysis.
Related ReferencesLouis N. Ridenour et al., Vacuum Tube Amplifiers, 1st ed., Massachusetts Institute of Technology Radiation Laboratory Series, New York: McGraw-Hill Book Company, 1948.
Alfred J. Cote Jr. and J. Barry Oakes, Linear Vacuum-Tube and Transistor Circuits: A Unified Treatment of Linear Active Circuits, New York: McGraw-Hill Book Company, 1961.
Karl T. Compton et al., Magnetic Circuits and Transformers, New York: John Wiley & Sons, 1950.
K. O'Connor, The Ultimate Tone, Canada: London Power Press, 1995.
F. Langford-Smith et al., The Radiotron Designer's Handbook, 4th ed., Sydney: Wireless Press, 1953.
Paul R. Gray and Robert G. Meyer, Analysis and Design of Analog Integrated Circuits, New York: John Wiley & Sons, 1993.
Ron M. Kielkowski, SPICE: Practical Device Modeling, New York: McGraw-Hill Book Company, 1995.