1.3.1 Geometry

The majority of practical vacuum tubes have geometries which are cylindrically symmetrical, or close approximations to it. The flow of the DC current is radial or axial, and driven by a static electric field in the same direction (see Chapters 5, 8, and 9).

1.3.2 Electron Dynamics

The voltages employed in vacuum tubes are high enough, in many cases, for the velocities of the electrons to be at least mildly relativistic. The kinetic energy of a relativistic electron is [Reference Rosser20]:

$T=m{c}^2-m_0{c}^2,$(1.1)

where $m_0{c}^2$ is the rest energy of the electron and the relativistic mass is

$m=\frac{m_0}{\sqrt{1-u^2/c^2}}=\gamma m_0,$(1.2)

where u is the velocity of the electron and c the velocity of light. If an electron starts from rest at the cathode then its velocity at a point where the potential is V, relative to the cathode,Footnote ⁴ is found by using the principle of conservation of energy:

$eV=m_0{c}^2\left(\frac{1}{\sqrt{1-u^2/c^2}}-1\right).$(1.3)

This equation can be rearranged as

$u=c{\left[1-\frac{1}{{\left[1+\left(V/V_R\right)\right]}^2}\right]}^{\frac{1}{2}},$(1.4)

where $V_R=m_0{c}^2/e=511\text{kV}$ is the rest energy of the electron expressed in electron volts. When $V/V_R\ll 1$ the fraction can be expanded by the Binomial Theorem to give the approximate expression:

$u=c\sqrt{\frac{2V}{V_R}}.$(1.5)

Figure 1.3 shows a comparison between the velocities calculated using the exact and approximate equations. The error in the approximate velocity is 1% at 7 kV so that the exact formula should be used for voltages higher than this.

Figure 1.3: Comparison between exact and approximate electron velocities as a function of the accelerating potential.

The force acting on an electron is equal to the rate of change of momentum

$F=\frac{d}{dt}\left(mu\right)=m\frac{du}{dt}+u\frac{dm}{dt}.$(1.6)

If the force acts in the direction of the motion of the electrons then

$F=\frac{m_0}{{\left(1-u^2/c^2\right)}^{3/2}}\cdot a={\gamma }^3{m}_{0}a,$(1.7)

where a is the acceleration and ${\gamma }^3{m}_0$ is sometimes called the longitudinal mass. When the force acts at right angles to the direction of motion then

$F=\frac{m_0}{{\left(1-u^2/c^2\right)}^{1/2}}\cdot a=\gamma m_{0}a,$(1.8)

where $\gamma m_0$ is the transverse mass. If the longitudinal velocity is approximately constant and much greater than the transverse velocity then it is possible to use the classical equations of motion with relativistic corrections to the mass of the electron.

The DC electron current in a tube may be collimated by a magnetic field which is either parallel to the direction of the DC current or perpendicular to it (see Chapters 7 and 8). Tubes in which the field is parallel to the current are described as ‘Type O’ and those in which it is perpendicular as ‘Type M’. The current may also be collimated by a static electric field but this is rather rare.

1.3.3 Modulation of the Electron Current

For the present we will assume that the input RF voltage is purely sinusoidal with amplitude $V_1$ and frequency $\omega$. Practical input signals are discussed in Section 1.6. It can be shown that the three possible methods of modulation of the current by the RF input voltage are [Reference Gabor21]:

• Emission density modulation in which the current emitted from the source is varied.

• Deflection and concentration modulation in which the electrons are deflected sideways.

• Transit-time modulation in which the electron velocity is varied.

The effect of any of these methods of modulation, or combinations of them, is to produce a bunched stream of electrons whose current varies with time at the frequency of the input signal. Because the process is non-linear the time variation of the current is not sinusoidal but can be represented by a Fourier series. The amplitudes and phases of the harmonic components depend upon the amplitude of the input voltage, and on the distance from the source. The modulation may be either in the same direction as the DC current flow, or normal to it. Thus the current at some point $z$ can be written

$I\left(z,t\right)=I_0+{\displaystyle \sum _{n=1}^{\infty }{I}_n\left(V_{in},z\right)}\exp \left(jn\omega t\right),$(1.9)

where $I_0$ is the current in the unmodulated stream, $I_n\left(V_{in},z\right)$ are complex amplitudes, and modulation in the axial direction has been assumed. The DC and time-varying parts of the current may be in different directions in space. We note that the real current cannot be negative. Its maximum value, relative to $I_0$, is determined by the process of modulation, or by the maximum current which can be drawn from the source. The ratio I0/I0 cannot exceed 2.0 (see sections 11.8.4 and 13.3.4).

1.3.4 Amplification, Gain, and Linearity

The RF output voltage is obtained by passing the modulated stream through a region at an effective position $z_2$ where energy is removed from the electron bunches by an RF electric field. This can be represented by an impedance $Z_n\left(V_{in}\right)$ which depends on frequency and on the magnitude of the input signal. Thus the output voltage is

$V_{out}\left(t\right)=\mathrm{Re}\left\{{\displaystyle \sum _{n=1}^{\infty }{I}_n\left(V_{in},z_2\right)}{Z}_n\left(V_{in}\right)\exp \left(jn\omega t\right)\right\},$(1.10)

where, for simplicity we assume that the characteristic impedances of the input and output waveguides are the same. The impedance $Z_n$ is effectively zero above some value of the harmonic number n determined by the nature of the output section. Thus the number of harmonics at which there is appreciable output power is small and, in some cases, limited to the fundamental. If the output power is to be radiated it may be necessary to filter out the harmonics to comply with the regulations determining the bandwidth available to the system, as specified by international agreement and national regulations [22, 23]. The transfer characteristic of the amplifier at the frequency $\omega$ is then

$V_{out}\left(\omega t\right)=A\left(V_{in}\right)\cos \left(\omega t+\Phi \left(V_{in}\right)\right),$(1.11)

where $A\left(V_{in}\right)$ is the AM/AM (amplitude modulation) characteristic which is usually plotted on decibel scales and $\Phi \left(V_{in}\right)$ is the AM/PM (phase modulation) characteristic of the amplifier.

The details of the transfer characteristics depend upon the type of amplifier but many of them show the same features. Figure 1.4 shows, as an example, a typical AM/AM curve of a travelling-wave tube (TWT). The output power is proportional to the input power at low drive levels (slope 1 dB/ dB) but reaches saturation as the input is increased. In the figure the input and output powers have been normalised to their values at saturation. The gain of the amplifier in decibels is given by

$G_{dB}=10\log \left|\frac{P_{out}}{P_{in}}\right|,$(1.12)

where $P_{in}$ and $P_{out}$ are, respectively, the input and output power. Since we have assumed that the input and output waveguides have the same characteristic impedances we can write

$G_{dB}=20\log \left(\frac{A\left(V_{in}\right)}{V_{in}}\right).$(1.13)

The difference between the linear (small-signal) and the saturated gain is the gain compression which may be used as a measure of non-linearity. It is common for a tube to be described in terms of its saturated output power, but the output power available under normal operating conditions may be less than this. For example the tubes used in particle accelerators are normally operated ‘backed-off’ from saturation to provide a control margin for the operation of the accelerator. In a TWT there is normally some output at second, and higher, harmonic frequencies. This also depends on the input drive level as shown in Figure 1.4. At low drive levels the second harmonic output is proportional to the square of the input power, giving a slope of 2 dB/dB. The maximum second harmonic power does not necessarily occur at the drive level that saturates the fundamental. The intersection between the projections of the linear parts of the fundamental and second harmonic curves, known as the second-order intercept point, is a measure of the second-order distortion of the amplifier.

Figure 1.4: Typical curves of fundamental and second harmonic output power of an RF amplifier plotted against the input power, normalised to saturation.

The corresponding AM/PM curve for a TWT is shown in Figure 1.5. The phase is plotted relative to the phase at low drive levels, and the input power is normalised to the input power at saturation. The phase of the output signal relative to the input is constant at low drive levels but changes as the drive level is increased. Both the amplitude and phase of the output depend on frequency, and on the operating conditions of the amplifier, including the voltages applied and the external RF matches. Thus any ripple in the voltages applied may result in amplitude and phase modulation of the output signal. In an oscillator, voltage ripple may also produce frequency modulation.

Figure 1.5: Typical curve of the phase of the output voltage of an RF amplifier against input power normalised to saturation.

1.3.5 Power Output and Efficiency

An important consideration for many applications of tubes is the efficiency with which the DC input power is converted into useful RF power. The principle of conservation of energy requires that, in the steady state, the total input and output powers must balance, that is

$P_{RFin}+P_{DCin}=P_{RFout}+P_{Heat}.$(1.14)

The DC input power includes the power in the electron stream, the cathode heater, and electromagnets. If the tube has a depressed collector (see Section 10.3) the DC power is reduced by the power recovered by the collector. When a comparison is made between alternative tube types, or between vacuum tube and solid-state amplifiers, then the DC input power should be specified at the input to the power supply to allow for losses in it. Any power required for cooling fans or pumps must also be included. The RF output power comprises power at the fundamental frequency and its harmonics. Heat is generated by the impact of electrons on the collector and the tube body and by RF losses in the tube body, connecting waveguides and windows.

A number of different definitions of efficiency are in use and it is important to distinguish between them. The overall efficiency is defined here as the ratio of the fundamental RF output power $\left(P_2\right)$ to the total input power to the tube

${\eta }_O=\frac{P_2}{P_{DC}+P_{RFin}}.$(1.15)

If the gain of the amplifier is high, the RF input power is much smaller than the DC input power so that approximately

${\eta }_O\approx \frac{P_2}{P_{DC}}.$(1.16)

The heater and electromagnet powers are typically much smaller than the stream power in continuous wave (CW) tubes but they can be comparable with the stream power in pulsed tubes. The power added efficiency, sometimes used when the gain is small, is

${\eta }_A=\frac{P_2-P_{RFin}}{P_{DC}}.$(1.17)

This efficiency is effectively identical to that in (1.16) if the gain is 20 dB or more. The efficiency of tubes with depressed collectors is discussed in Section 10.3. The RF efficiency is defined here as the ratio of the useful RF output power to the power input to the electron stream (less any recovered by the collector) plus the RF input power.

${\eta }_{rf}=\frac{P_2}{P_{stream}-P_{recovered}+P_{RFin}}.$(1.18)

The electronic efficiency is the efficiency with which power is transferred from the electron stream to the RF electric field of the output circuit.

${\eta }_e=\frac{P_{RFout}+P_{loss}}{P_{stream}},$(1.19)

where $P_{loss}$ is the total RF power loss in the output circuit. If the output circuit is resonant so that only power at the fundamental frequency is included

${\eta }_e=\frac{P_2+P_{2,loss}}{P_{stream}},$(1.20)

where P_2,loss is the RF loss at the fundamental frequency. Then the circuit efficiency can be defined by

${\eta }_c=\frac{P_2}{P_2+P_{2,loss}}.$(1.21)

If, in addition, the RF input power is small enough to be neglected, and the tube does not have a depressed collector then the RF efficiency is

${\eta }_{rf}={\eta }_e{\eta }_c.$(1.22)

1.3.6 Bandwidth

The AM/AM, and AM/PM, curves depend upon the frequency of the input signal. Then the transfer function of the amplifier may be written $H\left(\omega ,V_{in}\right)$. Figure 1.6 shows a typical graph of the RF output power of an amplifier as a function of frequency. This graph may be plotted at constant input power, or showing the saturated output power at each frequency. The bandwidth is defined in terms of the frequencies at which the power falls below the maximum by a specified amount (e.g. 1 dB bandwidth). This may be expressed in absolute terms or as a percentage of the centre frequency. The figure also shows that the output power may vary within the band and this can affect the system in which the tube is employed. These ripples are normally greatest when the tube is not saturated and are reduced at saturation by the effects of gain compression. The custom of using a decibel scale for this graph can be misleading. It is important to remember that a change of −1 dB represents a reduction in power by 20% and a change of −3 dB a reduction of 50%. Similar graphs of efficiency, small-signal gain, and saturated gain against frequency can also be plotted. The bandwidth specified for a tube may be to permit rapid changes in frequency (as in frequency-agile radar), or to enable it to be used with multiple and modulated signals (see Section 1.6).

Figure 1.6: Typical graph of the output power of an RF amplifier plotted against frequency.

In addition to power at harmonics of the input frequency the output of a tube may also include non-harmonic power under some operating conditions (for example during the rise and fall of the pulse during pulsed operation). These out-of-band emissions are undesirable, and limits are usually specified to avoid interference with other systems. For example it has been known for out-of-band emissions from marine radar, and from industrial microwave ovens, to interfere with microwave communication links. The tube also contributes to the noise in the system as discussed in Section 1.6.1.

1.3.7 The Electromagnetic Structure

The electromagnetic structures used to modulate the electron stream, and to extract power from it, can be divided into three groups:

• Fast-wave structures in which the phase velocity of travelling waves is greater than the velocity of light (Chapter 2).

• Resonant (standing wave) structures in which the RF electric field profile is time-varying but fixed in space (Chapter 3).

• Slow-wave structures which employ the field of a travelling electromagnetic wave whose phase velocity is less than the velocity of light (Chapter 4).

The bandwidth of a tube using a resonant structure is typically restricted to a few percent by the loaded $Q$ of the resonance (see Section 3.2.2). If the higher-order modes of the structure do not coincide with harmonics of the input frequency then the harmonic output of the tube is small.

The bandwidths of tubes using travelling-wave structures can be broad because the phase velocity and the impedance presented to the electron current by the structure vary slowly with frequency. The interaction is distributed so that the bunching section and the output section of the tube are effectively combined. For useful interaction to take place the mean electron velocity must be approximately synchronous with the phase velocity of the wave. There may be interaction at harmonics of the input frequency leading to appreciable output power at the second and higher harmonics as shown in Figure 1.4.

1.3.8 Coupled-Mode Theory

Useful insights into the properties of travelling-wave structures and their interactions with electron streams under small-signal (linear) conditions are given by coupled-mode theory [Reference Pierce24–Reference Pierce28]. For this purpose the small-signal modulation of the electron stream is described by a pair of normal modes whose impedances have opposite signs. The properties of space-charge waves on electron beams are discussed in detail in Chapter 11.

The properties of a wave propagating as $\exp j\left(\omega t-\beta z\right)$ are the solutions to the dispersion equation

$D\left(\omega ,\beta \right)=0,$(1.23)

where the frequency (w) and the propagation constant (β) are real if there is no attenuation.Footnote ⁵ The solutions of the dispersion equation can be plotted on a dispersion $\left(\omega ,\beta \right)$ diagram. In this diagram the phase velocity of the wave is

$v_p=\frac{\omega }{\beta }$(1.24)

and the group velocity, which is taken to be the velocity of propagation of energy, is

$v_g=\frac{d\omega }{d\beta }.$(1.25)

The phase velocity is positive in the first quadrant of the $\left(\omega ,\beta \right)$ diagram and negative in the second quadrant. The group velocity can be either positive or negative depending upon the properties of the mode. We shall see in Chapter 4 that waves having positive phase velocities and negative group velocities occur in periodic structures.

Where two waves propagate in the same region of space they may interact to produce coupled modes if their dispersion curves intersect as shown in Figure 1.7. The dispersion equation for the coupled modes can then be written in the form

$D_1\left(\omega ,\beta \right)D_2\left(\omega ,\beta \right)=\kappa \left(\omega ,\beta \right),$(1.26)

where $\kappa \left(\omega ,\beta \right)$ represents the coupling between the modes. The coupled modes are the solutions to this equation. In order to investigate the properties of coupling under different conditions we will redefine $\omega$ and $\beta$ so that the two uncoupled modes intersect at the origin. This transformation does not change the group velocities of the modes.

Figure 1.7: Dispersion diagram for two modes of propagation.

The properties of the different possible kinds of mode coupling can be illustrated by assuming that the uncoupled dispersion curves are straight lines so that

$D\left(\omega ,\beta \right)=\left(\omega -\beta v_g\right)=0.$(1.27)

The dispersion equation for the coupled modes is then

$\left(\omega -\beta v_{g1}\right)\left(\omega -\beta v_{g2}\right)={\kappa }_0,$(1.28)

where ${\kappa }_0$ is regarded as a constant close to the origin. This equation can be expanded to give

${\omega }^2-\left(v_{g1}+v_{g2}\right)\omega \beta +v_{g1}{v}_{g2}{\beta }^2-{\kappa }_0=0,$(1.29)

which is quadratic in both $\omega$ and $\beta$. Solving for $\omega$ gives

$\omega =\frac{1}{2}\left\{\left(v_{g1}+v_{g2}\right)\beta \pm \sqrt{{\left(v_{g1}-v_{g2}\right)}^2{\beta }^2+4{\kappa }_0}\right\}$(1.30)

and for $\beta$ gives

$\beta =\frac{1}{2v_{g1}{v}_{g2}}\left\{\left(v_{g1}+v_{g2}\right)\omega \pm \sqrt{{\left(v_{g1}-v_{g2}\right)}^2{\omega }^2+4v_{g1}{v}_{g2}{\kappa }_0}\right\}.$(1.31)

The constant ${\kappa }_0$ is positive if the signs of the impedances of the uncoupled waves are the same, and negative if they are opposite. Likewise the signs of the group velocities of the waves may be the same, or opposite. Hence all possible combinations are covered by four cases, which can be explored using Worksheet 1.1. It can be shown that the properties of the coupled modes can be revealed by considering the solutions of (1.30) when $\beta$ is real and the solutions of (1.31) when $\omega$ is real as follows [Reference Briggs26, Reference Sudan29]:

Case A: The group velocities and the impedances have the same sign $\left(v_{g1}{v}_{g2}>0,{\kappa }_0>0\right)$.

When $\beta$ is real the solutions of (1.30) are always two real values of $\omega$ and, when $\omega$ is real, the solutions of (1.31) are two real values of $\beta$. These solutions represent travelling waves with the modified dispersion diagram shown in Figure 1.8. In Figures 1.8 to 1.11 the uncoupled modes are shown by dotted lines, the real parts of the solutions by solid lines and the imaginary parts by dashed lines.

Figure 1.8: Coupling between two forward waves with impedances having the same sign: (a) $\omega$ for real $\beta$, and (b) $\beta$ for real $\omega$.

Case B: The group velocities have opposite signs and the impedances have the same sign $\left(v_{g1}{v}_{g2}<0,{\kappa }_0>0\right)$.

When $\beta$ is real the solutions of (1.30) are always two real values of $\omega$ but, when $\omega$ is real and close to the origin, the solutions of (1.31) are complex conjugate pairs of values of $\beta$, as shown in Figure 1.9(b). Physically, the complex values of $\beta$ represent travelling waves whose amplitudes decay exponentially with distance. For example see Figure 4.28(a) where the forward and backward waves in a waveguide are coupled to one another by periodic discontinuities to produce frequency stop-bands in which there are decaying (evanescent) waves.

Figure 1.9: Coupling between a forward wave and a backward wave with impedances having the same sign: (a) $\omega$ for real $\beta$, and (b) $\beta$ for real $\omega$.

Case C: The group velocities have the same sign and the impedances have opposite signs $\left(v_{g1}{v}_{g2}>0,{\kappa }_0<0\right)$.

The solutions for both $\omega$ and $\beta$, close to the origin, are complex conjugate pairs, as shown in Figure 1.10. It can be shown that the complex solution for $\beta$ corresponds to a convective instability in which the waves grow and decay exponentially in space (see Figure 1.10(b)) [Reference Briggs26, Reference Sudan29]. An example of this type of coupling is the growth of waves in a travelling-wave tube as shown in Figure 11.19.

Figure 1.10: Coupling between two forward waves with impedances having opposite signs: (a) $\omega$ for real $\beta$, and (b) $\beta$ for real $\omega$.

Case D: The group velocities and the impedances have opposite signs $\left(v_{g1}{v}_{g2}<0,{\kappa }_0<0\right)$.

When $\beta$ is real and close to the origin the solutions of (1.30) are a complex conjugate pair of values of $\omega$ and, when $\omega$ is real, the solutions of (1.31) are always two real values of $\beta$, as shown in Figure 1.11. The solution with complex $\omega$ is a non-convective, or absolute, instability in which the two solutions grow and decay exponentially in time. The condition for this instability to exist is that there is at least one value $\omega ={\omega }_s$ in the upper half of the complex plane for which there are coincident roots in ${\beta }_s$ [Reference Sudan29]. For the case considered here the condition for the existence of coincident roots is found from (1.31)

${\omega }_s=\pm 2j\frac{\sqrt{\left|v_{g1}{v}_{g2}\right|\left|{\kappa }_0\right|}}{\left|v_{g1}\right|+\left|v_{g2}\right|}.$(1.32)

The condition is satisfied in this case when ${\kappa }_0=0$ and $\omega =0$ so that an absolute instability exists for all negative values of ${\kappa }_0$. The corresponding values of $\beta$ can be found from (1.31):

${\beta }_s=\frac{\left|v_{g1}\right|-\left|v_{g2}\right|}{2\left|v_{g1}{v}_{g2}\right|}{\omega }_s.$(1.33)

Thus ${\beta }_s$ is complex when ${\omega }_s$ is complex. In general the start-oscillation condition can be found by finding the least magnitude of the coupling for which coincident roots of $\beta$ exist when $\omega$ is real (see Figures 17.5 and 17.6) [Reference Chu and Lin30]. If the coupling is increased beyond this point both the frequency and the propagation constant are complex (see Section 11.7). Examples of this kind of instability are backward-wave oscillations in a travelling-wave tube (Section 11.7) and gyrotron oscillators (Section 17.3).

Figure 1.11: Coupling between forward and backward waves with impedances having opposite signs: (a) $\omega$ for real $\beta$, and (b) $\beta$ for real $\omega$.

The properties of coupled modes remain the same in more general cases where the uncoupled dispersion diagrams are not straight lines, $\kappa$ is a function of $\omega$ and $\beta$, and more than two modes are involved in the coupling. Thus a qualitative description of the properties of the coupled system can be derived from an examination of the dispersion diagram of the uncoupled modes and the form of the dispersion equation for the coupled system [Reference Sudan29].

1.3.9 Classification of Vacuum Tubes

The preceding sections have reviewed the different ways in which the functional blocks in Figure 1.2 can be realised, and that can be used as a basis for classifying tubes. The discussion in this book is restricted to the types of tube which are of continuing practical importance. A summary of their principal features is given in Table 1.1. A few variations on the main types of tube exist, for example: triodes and tetrodes with DC and RF current flow in the axial direction, and hybrid tubes in which the bunching section resembles a klystron and the output section a travelling-wave tube. It is possible to think of the inductive output tube as a hybrid tube with a triode bunching section and a klystron output section. Many other possible types of tube can be envisaged [Reference Feinstein and Felch31–Reference Harvey33].

1.6.1 Noise

Any communications system adds white noise to the input signal so that the signal received is corrupted to some extent [Reference Dunlop and Smith39]. The thermal noise power in watts caused by the random motion of charge carriers in resistive materials that is delivered to a matched load is

$N_p=kTB,$(1.35)

where k is Boltzmann’s constant $\left(k=1.380\times {10}^{-23}\text{J}{\text{K}}^{-\text{1}}\right)$, T is the absolute temperature and $B$ is the bandwidth of the system in Hz. The effectiveness of an analogue communication system is measured by the signal to noise ratio $\left(S_p/N_p\right)$ where $S_p$ is the average signal power. This is commonly expressed in decibels as

$SNR=10{\log }_{10}\left(\frac{S_p}{N_p}\right).$(1.36)

The minimum acceptable SNR for reliable communication is generally taken to be about 10 dB.

For digital communications the corruption of the signal may result in some of the bits being received incorrectly. Shannon’s theorem states that a noisy channel will theoretically support error-free data transmission at a channel capacity given by

$C=B{\log }_2\left(1+\frac{S_p}{N_p}\right).$(1.37)

This can be expressed alternatively as

$C=B{\log }_2\left(1+\frac{E_b}{N_0}\cdot \frac{R}{B}\right),$(1.38)

where $E_b$ is the energy per bit, $R$ is the mean bit rate, and $N_0$ is the spectral energy density of the noise. Shannon’s law represents an ideal which cannot be achieved in practice, but it remains useful for comparison with practical systems. In a digital transmission system the energy per bit, which is related to the bit error rate, is commonly used in place of the signal to noise ratio [Reference Ziemer and Meyers40]. A second performance measure is the bandwidth efficiency $\left(R/B\right)$ in bits s⁻¹ Hz⁻¹.

Each part of a communications channel contributes some noise to the system. Here we are concerned specifically with the contribution from the final power amplifier. The noise generated by an amplifier can be described by its noise figure, which is the ratio of the signal to noise ratio at the input to that at the output

$F=\frac{SN{R}_i}{SN{R}_o}.$(1.39)

If an amplifier has a power gain $A_p$ then the signal to noise ratio at the output is

$SN{R}_o=\frac{A_p{S}_i}{A_p{N}_i+N_a},$(1.40)

where $S_i$ and $N_i$ are the signal and noise powers at the input and $N_a$ is the noise power added by the amplifier. Thus

$F=\frac{A_p{N}_i+N_a}{A_p{N}_i}.$(1.41)

The noise figure is standardised by fixing the input noise power as that given by (1.35) when $T=290\text{K}\text{.}$ The noise added by the amplifier can be described by an equivalent noise source at its input such that

$N_a=k{T}_{e}B{A}_p,$(1.42)

where $T_e$ is the effective noise temperature of the amplifier. The noise performance can also be described by the total noise power at the output, or by the noise power density (noise power per Hz).

1.6.2 Analogue Modulation

In order to transmit the baseband signal over a wireless channel it must be superimposed upon an RF carrier wave by a modulator. The properties of the carrier wave which can be varied by the modulator are its amplitude, frequency, and phase, or some combination of them. In general the modulated signal at the input to the power amplifier can be described by a sinusoidal carrier whose amplitude and phase are functions of time [Reference Berman and Mahle42, Reference Saleh43]. Thus

$V_1\left(t\right)=r\left(t\right)\cos \left({\omega }_{0}t+\psi \left(t\right)\right),$(1.43)

where ${\omega }_0$ is the carrier frequency. The rates of change with time of the amplitude $\left(r\left(t\right)\right)$ and phase $\left(\psi \left(t\right)\right)$ are normally much slower than that of the carrier wave so that it is permissible to regard the instantaneous response of the amplifier as being that for a single unmodulated carrier wave. The response to more complex signals can therefore be deduced from the steady-state, single-carrier, transfer characteristics. Figure 1.12 shows examples of simple analogue modulation schemes. For purposes of illustration the modulation frequency has been taken to be one eighth of the carrier frequency although the ratio is normally much smaller than that.

Figure 1.12: Time and frequency domain representations of analogue modulated carriers: (a) and (b) double sideband amplitude modulation; (c) and (d) double sideband suppressed carrier amplitude modulation; (e) and (f) phase modulation.

The simplest example to understand is the amplitude modulation shown in Figure 1.12(a). The envelope of the carrier carries information such as an audio waveform. In the time domain the modulated waveform is given by

$f\left(t\right)=\left[K+S\left(t\right)\right]\cos \left({\omega }_{0}t\right),$(1.44)

where K is the amplitude of the unmodulated carrier and $S\left(t\right)$ is the signal waveform. When the signal is sinusoidal with frequency ${\omega }_m$ the Fourier transform of $f\left(t\right)$ yields the spectrum in the frequency domain shown in Figure 1.12(b). This representation shows the amplitudes, but not the phases, of the components of the signal. The central line at the carrier frequency is flanked by two sidebands at frequencies $\left({\omega }_0\pm {\omega }_m\right)$. This is known as double sideband amplitude modulation. For a more general modulating waveform the sidebands are the frequency-shifted Fourier transforms of the baseband waveform. Faithful transmission of the modulated waveform requires the bandwidth of the amplifier to be at least $2B$ (the Nyquist limit) where B is the baseband bandwidth. The instantaneous power in the modulated signal varies with time, as can be seen from Figure 1.12(a). Faithful transmission of the signal requires the amplifier to be linear up to the highest instantaneous power. Any non-linearity of the amplifier produces a distortion of the signal in the time domain which is reflected in changes in its spectrum (see Section 1.6.4).

Figures 1.12(c) and (d) show double sideband suppressed carrier modulation. The modulated waveform is

$f\left(t\right)=S\left(t\right)\cos \left({\omega }_{0}t\right).$(1.45)

The spectrum differs from that in Figure 1.12(b) in the absence of the carrier frequency. The bandwidth requirements for the amplifier are the same as in the previous case and the dynamic range is greater. However, this modulation scheme is less susceptible to interference from noise than the previous case because all the power is in the sidebands.

Figures 1.12(e) and (f) show phase modulation. The amplitude envelope of the signal is constant in the time domain so that the instantaneous power is constant. This has the advantage that the working point of the amplifier is virtually constant, so that the output signal is only affected by the frequency dependence of the gain and phase of the amplifier. The modulated signal, assuming unity amplitude, is

$f\left(t\right)=\cos \left({\omega }_{0}t+S\left(t\right)\right),$(1.46)

so that the dependence of the modulated signal on the modulation is non-linear. For sinusoidal modulation at frequency ${\omega }_m$ (1.46) can be written

$f\left(t\right)=\cos \left({\omega }_{0}t+\beta \sin {\omega }_{m}t\right),$(1.47)

where $\beta$ is the modulation index. The spectrum for this waveform, shown in Figure 1.12(f) for $\beta =1$, has additional lines at frequencies $\left({\omega }_0\pm n{\omega }_m\right)$ where $n=1,2,3,\dots$. Thus the bandwidth required for faithful amplification is greater than for amplitude modulation schemes. A rule of thumb for the transmission bandwidth is Carson’s rule [Reference Ziemer and Meyers40]

$B_T=2\left(\beta +1\right){\omega }_m.$(1.48)

Frequency modulation can also be represented by (1.47) so that its properties are similar to those of phase modulation. The properties of analogue modulation schemes can be explored using Worksheet 1.2.

1.6.3 Digital Modulation

The simplest digital modulation schemes are amplitude shift keyed (ASK), phase shift keyed (PSK), and frequency shift keyed (FSK) modulation [Reference Dunlop and Smith39]. Figure 1.13 shows time and frequency domain representations of these methods of modulation, using the same carrier and modulation frequencies as before. It may be observed that the bandwidths of these signals are greater than those for the equivalent analogue modulation, with the exception of binary FSK modulation, which may be considered a special case of frequency modulation. The properties of binary digital modulation schemes can be explored using Worksheet 1.2.

Figure 1.13: Time and frequency domain representations of simple digitally-modulated carriers: (a) and (b) binary ASK; (c) and (d) binary PSK; (e) and (f) binary FSK.

We saw in Section 1.6.1 that the bit rate in a digital system having fixed bandwidth can be increased by using a greater number of symbols. Two common implementations of this principle are illustrated in Figure 1.14 which shows the positions of the symbols on phasor diagrams known as constellation plots. Figure 1.14(a) shows binary phase-shift keyed (BPSK) modulation with a phase shift of 180° corresponding to Figure 1.13(c) and (d). Quadrature phase-shift keyed (QPSK) modulation is shown in Figure 1.14(b), and quadrature amplitude modulation with 16 symbols (16-QAM) is shown in Figure 1.14(c). The effect of noise in the transmission channel is to produce a scatter of points around the desired constellation points at the receiver. If the scatter is too great then the receiver will fail to identify the symbols correctly leading to errors in transmission. Table 1.6 shows the bandwidth efficiencies and $E_b/N_0$ figures for these three channels at a bit error rate of ${10}^{-6}$. These figures assume coherent modulation so that the clock signals at the transmitter and the receiver are locked to one another. It can be seen that the bandwidth efficiency increases with the number of symbols according to (1.34) as expected.

Figure 1.14: Examples of constellation diagrams for digitally-modulated carriers: (a) binary phase-shift keyed (BPSK), (b) quadrature phase-shift keyed (QPSK), and (c) quadrature amplitude modulation with 16 symbols (16-QAM).

Table 1.6: Comparison between examples of coherent digital modulation schemes [Reference Ziemer and Meyers40]

Modulation type	Bandwidth efficiency (bits s−1 Hz−1)	$E_b/N_0$
BPSK	0.5	10.6 dB
QPSK	1.0	10.6 dB
16-QAM	2.0	14.3 dB

From the point of view of the design of the power amplifier it is important to note that BPSK and QPSK are constant envelope schemes which allow the efficiency of the power amplifier to be high, subject only to the need to maintain adequate linearity. On the other hand the amplitude variation in 16-QAM means that the amplifier is only working at the highest power conversion efficiency at four of the constellation points. In addition, the overall transmitter power must be high enough to achieve an acceptable bit error rate for the innermost constellation points where the transmitted power is least. If all constellation points are equally probable then the average efficiency is about half of that at the outermost points. This makes 16-QAM unsuitable for applications such as satellite downlinks where high efficiency is important.

1.6.4 Multiplexing

It is commonly the case that one telecommunications channel carries data from multiple sources. The interleaving of the different data streams, known as multiplexing, can be achieved in many different ways [Reference Ziemer and Meyers44]. Of these, the method which has implications for the design of a vacuum tube amplifier is frequency domain multiplexing (FDM) in which the input to the tube comprises a number of carriers each of which is modulated by a separate data stream [Reference Berman and Mahle42]. In a typical scheme the available bandwidth is occupied by N carriers at frequency intervals $2\Delta \omega$. If the centre frequency is ${\omega }_0$ then the channel frequencies are

${\omega }_n={\omega }_0+n\Delta \omega \text{for }n=\pm 1,\pm 3,\pm 5\dots ,$(1.49)

where each of these sub-carriers is frequency-modulated. The non-linearity of the final power amplifier produces mixing between the carriers leading to co-channel interference. It is therefore necessary to have some way of calculating this interference and specifying the maximum acceptable level.

The simplest approach is to consider the central pair of carriers at frequencies $\omega ={\omega }_0\pm \Delta \omega$ with equal amplitudes. The two carriers each contribute half of the RF input power so that their amplitudes are related to that of a single carrier having the same total power by $V_c=V_1/\sqrt{2}$. The input to the final power amplifier can be written in the form given in (1.43)

$V_{in}\left(t\right)=2V_c\cos \left(\Delta \omega t\right)\cos \left({\omega }_{0}t\right)=r\left(t\right)\cos \left({\omega }_{0}t\right),$(1.50)

where $r\left(t\right)$ is the time-varying amplitude. Then, from (1.11), the output of the amplifier is

$V_2\left(t\right)=A\left[r\left(t\right)\right]\cos \left({\omega }_{0}t+\Phi \left[r\left(t\right)\right]\right).$(1.51)

The non-linearity of the amplifier produces signals at frequencies given by

${\omega }_{m,n}=m\left({\omega }_0+\Delta \omega \right)\pm n\left({\omega }_0-\Delta \omega \right),$(1.52)

where $m,n=0,1,2\dots$. The amplitudes of the signals at these frequencies can be found by Fourier analysis [Reference Stette45]. Thus, setting $m=1$ and $n=0$ gives the carrier frequency ${\omega }_{1,0}={\omega }_0+\Delta \omega$ at which the amplitude is

$V_{1,0}\left(V_1\right)=\frac{1}{\pi }{\displaystyle \underset{0}{\overset{\pi }{\int }}A\left[r\left(\varphi \right)\right]}\cos \left\{\Phi \left[r\left(\varphi \right)\right]\right\}\sin \varphi d\varphi ,$(1.53)

where $\varphi =\Delta \omega t$. The analysis is conveniently carried out in the base band. Since the AM/AM and AM/PM characteristics of the amplifier normally vary only slowly with frequency it is possible to assume that they are constant, so that the amplitude of the other carrier $\left(V_{0,1}\right)$ is the same. Hence the total RF output power at the frequencies of the two carriers is given by replacing $V_c$ in (1.50) by the equivalent single carrier amplitude $V_1$. This power can be plotted against the RF input power for comparison with the single-carrier AM/AM characteristic as shown in Figure 1.15. At small input powers the amplifier is linear and the power gain is identical to that with one carrier. However, as the input power approaches the single-carrier saturation level the output power is less with two carriers than with one. The reason for this is illustrated in Figure 1.16 which shows typical two-tone input and output waveforms for an amplifier, normalised to the saturated output voltage of the amplifier. For clarity of explanation the carrier frequency shown is much lower relative to the frequency of the envelope than would normally be the case. The output voltage is reduced when the instantaneous amplitude of the signal exceeds the saturation level of the amplifier. Thus the saturated output power under multi-carrier operation is less than that for a single carrier at the same input power. Figure 1.15 also compares the effects of AM/AM conversion alone (typical of klystrons) with those in which AM/PM conversion is also included (typical of TWTs) [Reference Stette45].

Figure 1.15: Typical AM/AM transfer characteristics for two-carrier operation showing third-order (IM3) and fifth-order (IM5) intermodulation products.

Figure 1.16: Typical waveforms for two-carrier operation of a non-linear amplifier: (a) input, and (b) output.

When we set $m=2$ and $n=1$ in (1.52) we obtain a third-order intermodulation product at frequency ${\omega }_{2,1}={\omega }_0+3\Delta \omega$ whose amplitude is

$V_{2,1}\left(V_1\right)=\frac{1}{\pi }{\displaystyle \underset{0}{\overset{\pi }{\int }}A\left(2V_c\cos \varphi \right)}\cos \left[\Phi \left(2V_c\cos \varphi \right)\right]\sin 3\varphi d\varphi .$(1.54)

In the same way, setting $m=3$ and $n=2$ gives a fifth-order intermodulation product at ${\omega }_{3,2}={\omega }_0+5\Delta \omega$ whose amplitude is

$V_{3,2}\left(V_1\right)=\frac{1}{\pi }{\displaystyle \underset{0}{\overset{\pi }{\int }}A\left(2V_c\cos \varphi \right)}\cos \left[\Phi \left(2V_c\cos \varphi \right)\right]\sin 5\varphi d\varphi .$(1.55)

The amplitudes of the other intermodulation products at ${\omega }_{1,2}$ and ${\omega }_{2,3}$ are the same. Then the total third-order (IM3) and fifth-order (IM5) intermodulation powers can be plotted as shown in Figure 1.15. The frequencies of these signals coincide with those of possible adjacent carriers resulting in co-channel interference. The acceptable level of interference is commonly expressed as the ratio $\left(C/I3\right)$ of the fundamental power to that of the third-order intermodulation product, expressed in decibels. At low drive levels the slope of the graph of the third-order product is 3 dB/dB and that of the fifth-order product is 5 dB/dB. Thus the ratio $C/I3$ can be increased by operating the amplifier at a reduced drive level. This is commonly expressed in terms of the output back-off from saturation (in dB) necessary to achieve an acceptable value of $C/I3$. The linear parts of the curves for the intermodulation products can be extrapolated in the same manner as in Figure 1.4 to meet the extrapolation of the fundamental curve at the third-order and fifth-order intercept points. These may also be used to specify the linearity of the amplifier.

The functions A and $\Phi$ required for the evaluation of the integrals in (1.53) to (1.55) may be specified numerically using interpolation on data points which have been determined experimentally or by large-signal modelling [Reference Stette45]. Alternatively a suitable function may be fitted to the data points [Reference Berman and Mahle42, Reference Saleh43, Reference Kunz46]. Polynomial expansions do not give a good fit to the data unless a large number of terms is used. In particular these models fail for drive levels approaching saturation. They do, however, reveal that the fundamental, and intermodulation, amplitudes depend only on the odd terms of the series. These can be specified independently of the terms describing the dependence of the even harmonics on the input RF power [Reference Kunz46]. Simple functions which are a good fit to experimental data for TWTs have been proposed by Saleh [Reference Saleh43]:

$A\left(r\right)=\frac{{\alpha }_{a}r}{1+{\beta }_a{r}^2}$(1.56)

and

$\Phi \left(r\right)=\frac{{\alpha }_{\varphi }{r}^2}{1+{\beta }_{\varphi }{r}^2},$(1.57)

where ${\alpha }_a$, ${\beta }_a$, ${\alpha }_{\varphi }$ and ${\beta }_{\varphi }$ are empirical constants. If the amplifier has unity gain at saturation, so that $A\left(1\right)=1$, it can be shown that ${\alpha }_a=2$ and ${\beta }_a=1$. These values are close to the empirical figures so that it is possible to use them as a useful approximation. In the same way it may be tentatively suggested that the function required to generate the even harmonics might be written

$B\left(r\right)=\frac{{\alpha }_h{r}^2}{{\left(1+{\beta }_h{r}^2\right)}^2}.$(1.58)

The curves shown in Figures 1.4 and 1.15 were generated using this non-linear model (see Worksheet 1.3).

The ratio $C/I3$ is commonly used for specifying the intermodulation distortion in an amplifier, not least because it is straightforward to measure it experimentally. However, it is at best, a proxy for the intermodulation effects because, when there are more than two carriers, the overall performance depends upon the signal levels of all the carriers. Since the phases of the phase-modulated carriers vary in a quasi-random manner it may be assumed that all phase combinations are equally probable. Where only two carriers are used it is sufficient to consider unmodulated sine waves because all possible phases arise in one period of the difference frequency. Then the peak power is twice the mean power. However, with three or more carriers this is not the case, and it is necessary to use Monte Carlo methods to examine all possible phases [Reference Carter47]. As the number of carriers increases, the probability that all the signals will be in phase simultaneously decreases. In the limit, a large number of uncorrelated phase-modulated carriers can be represented by band-limited white noise [Reference Stette45, Reference Guida48]. This leads to an alternative measure of intermodulation distortion in which the input to the amplifier is band-limited white noise from which the portion in a small frequency range has been removed by a notch filter. The non-linearity of the amplifier causes some signals to be generated in this frequency range. The ratio of the amplitude of these to that of the white noise at the output of the amplifier is the Noise Power Ratio (in dB). This is a measure of the level of interference which would be experienced by a signal at the centre of the notch [Reference Reveyrand49, Reference Katz and Gray50]. An approximation to this can be achieved experimentally, or in simulations, by using a comb of carriers at equal frequency intervals such that one carrier has been omitted from the centre of the set as shown in Figure 1.17(a) [Reference Carter47, Reference Loo51]. Figure 17(b) shows a typical output spectrum and the definition of the noise power ratio. Intermodulation products are also generated on the edges of the input spectrum so increasing its bandwidth. If the power in these is great enough then it may cause interference with other systems operating in adjacent frequency bands. For this reason the permitted bandwidth may be expressed as a spectrum envelope [Reference Amoroso52].

Figure 1.17: Amplification of multiple carriers: (a) input spectrum, and (b) output spectrum.

1.7.1 Dimensionless Parameters and Scaling

Nearly all tubes are developed in organisations in which the design process can be based upon accumulated knowledge and expertise. Many tubes are designed by scaling from existing designs, or by other modifications to them, rather than starting from a clean sheet of paper. It is, however, important that this process should be underpinned by fundamental understanding.

The basis for scaling is dimensional analysis [Reference Buckingham53, Reference Wilson54]. Let us suppose that some aspect of the performance of a tube is governed by an equation which can be written in the general form

$F\left(Q_1,Q_2\dots Q_n,r^{\prime },r^{″}\dots \right)=0,$(1.59)

where $Q_1,Q_2\dots \text{etc}.$ are dimensional variables and $r^{\prime },r^{″}\text{etc}.$ are dimensionless ratios. Very commonly these are the ratios of the leading dimensions of the tube to a single dimension or to a dimensional variable such as the free-space wavelength. If two tubes are described by the same set of dimensionless ratios, so that all dimensions are changed by the same multiplying factor, they are said to be geometrically similar to one another. If geometrical similarity is assumed then the ratios are constants and (1.59) can be written

$F\left(Q_1,Q_2\dots Q_n\right)=0.$(1.60)

This equation is a complete description of the problem, provided that none of the variables required has been overlooked. The choice of variable for inclusion is based upon an understanding of the physics of the problem. When it involves electromagnetics then the primary electric and magnetic constants ${\epsilon }_0$ and ${\mu }_0$ may have to be included in the list. The ratio of the charge to the rest mass of an electron $\left(e/m_0\right)$ is also required when the dynamics of electrons are part of the problem. Equation (1.60) remains valid even if the algebraic form of the function is unknown. The dimensions of the variables $Q$ can be expressed in terms of the fundamental dimensions mass [M], length [L], and time [T], together with voltage [V] or charge for electrical problems. Thus there are, at most, four fundamental dimensions. Table 1.7 shows the dimensions of some quantities relevant to vacuum tubes.

Table 1.7: Typical independent and dependent parameters for vacuum tubes

	Parameter	Dimensions
Constants	Velocity of light in free space (c) 0.299792 × 109 m s−1	$\left[{\text{LT}}^{-\text{1}}\right]$
	Primary electric constant $\left({\epsilon }_0\right)$ 8.854187 × 10−12 F m−1	$\left[{\text{MLT}}^{-\text{2}}{\text{V}}^{-\text{2}}\right]$
	Primary magnetic constant $\left({\mu }_0\right)$ 1.256637 × 10−6 H m−1	$\left[{\text{M}}^{-\text{1}}{\text{L}}^{-\text{3}}{\text{T}}^{\text{4}}{\text{V}}^{\text{2}}\right]$
	Elementary charge (e) 1.602176 × 10−19 C	$\left[{\text{ML}}^2{\text{T}}^{-\text{2}}{\text{V}}^{-\text{1}}\right]$
	Rest mass of the electron (m0) 9.109382 × 10−31 kg 0.510998 MeV	$\left[\text{M}\right]$
	Electron charge to mass ratio $\left(e/m_0\right)$ 1.758820 × 1011 C kg−1	$\left[{\text{L}}^{\text{2}}{\text{T}}^{-\text{2}}{\text{V}}^{-\text{1}}\right]$
Parameters	Length	$\left[\text{L}\right]$
	Voltage	$\left[\text{V}\right]$
	Magnetic flux density	$\left[{\text{L}}^{-\text{2}}\text{TV}\right]$
	Frequency	$\left[{\text{T}}^{-\text{1}}\right]$
	R.F. power	$\left[{\text{ML}}^{\text{2}}{\text{T}}^{-\text{3}}\right]$
	Current	$\left[{\text{ML}}^{\text{2}}{\text{T}}^{-\text{3}}{\text{V}}^{-\text{1}}\right]$
	Electron velocity	$\left[{\text{LT}}^{-\text{1}}\right]$
	Charge density	$\left[{\text{ML}}^{-\text{1}}{\text{T}}^{-\text{2}}{\text{V}}^{-\text{1}}\right]$

The variables Q may be combined together to form dimensionless groups denoted by $\Pi$ so that (1.60) can be written

$\psi \left({\Pi }_1,{\Pi }_2,\dots {\Pi }_i\right)=0.$(1.61)

The number of groups required is related to the number of variables by

$i=n-k,$(1.62)

where k is the number of dimensions required. This is known as Buckingham’s theorem. An immediate effect of using dimensionless variables (or dimensionless groups) is that the number of variables required to define the problem has been reduced. Now let us choose one of the groups to be a dependent variable whose value is determined by the values of the others. Then (1.61) becomes

${\Pi }_1=\Psi \left({\Pi }_2,{\Pi }_3,\dots {\Pi }_i\right).$(1.63)

If the values of the independent variables $\left({\Pi }_2,{\Pi }_3,\dots {\Pi }_i\right)$ are fixed then so is the value of the dependent variable ${\Pi }_1$. Two tubes which are described by the same values of the dimensionless variables are said to be dynamically similar to one another. Thus the basis of scaling is the maintenance of both geometrical and dynamic similarity.

The process of forming dimensionless groups is aided by choosing normalisations which are based on the underlying physics of the problem. Some examples are shown in Table 1.8.

Table 1.8: Examples of the formation of dimensionless groups

Variable	Normalising quantity	Dimensionless group
Dimension (d)	Wavelength $\left(\lambda \right)$ Propagation constant $\left(\beta \right)$	$\frac{d}{\lambda }$ or $\beta d$
Velocity (u)	Velocity of light (c) Phase velocity $\left(v_p\right)$	$\frac{u}{c}$ or $\frac{u}{v_p}$
Voltage (V)	Rest energy of the electron in eV $\left(V_0=m_0{c}^2/e\right)$	$\frac{eV}{m_0{c}^2}$
Electron plasma frequency $\left({\omega }_p\right)$ Electron cyclotron frequency $\left({\omega }_c\right)$	Frequency $\left(\omega \right)$	$\frac{{\omega }_p}{\omega }$ and $\frac{{\omega }_c}{\omega }$

A simple illustration is provided by the calculation of the velocity of an electron $\left(u_0\right)$ from an applied voltage $\left(V_a\right)$ and the dimensional constants. The dependent dimensionless group is

${\Pi }_1=\frac{u_0}{c}$(1.64)

and the independent dimensionless group is

${\Pi }_2=\frac{e{V}_a}{m_0{c}^2}.$(1.65)

In this case there are five parameters and three independent dimensions: $\left[\text{M}\right]$, $\left[\text{V}\right]$ and $\left[{\text{LT}}^{-\text{1}}\right]$ so that the problem is completely described by the two dimensionless groups which have been formed. Note that in this case the dimensions $\left[\text{L}\right]$ and $\left[\text{T}\right]$ do not occur separately so that the number of independent dimensions is reduced by one. It follows that we can write

${\Pi }_1=\Psi \left({\Pi }_2\right),$(1.66)

where $\Psi$ is an unknown function which can be determined empirically, even if its form cannot be found by mathematical analysis. The value of the reduction of the description of a problem to the form in (1.63) is greater than just that of scaling. If the form of the function $\Psi$ is known, even over a limited range of variation of the variables then it can be used to explore the effects of design choices on ${\Pi }_1$. In cases where geometrical similarity is not maintained then one, or more, of the ratios r are added as dimensionless variables. Examples of the application of this method to particular problems in tube design are given later in this book.

1.7.2 Modelling

The trend in tube design is towards achieving near-ideal RF performance coupled with high reliability and low cost of ownership. For this reason, the design of a state-of-the-art vacuum tube is a complex business which involves considerable use of computer simulations to achieve right-first-time design [Reference Antonsen55, Reference Carter56]. The advanced computer codes used for detailed design can take many hours to run, even on the most powerful computers available. They are invaluable tools for accurate modelling, but are not well-suited to the conceptual design stage which is the focus of this book. For that, the engineers must have a sound understanding of the theory of vacuum tubes, together with simple computer models which can be used for rapid design iterations.

The availability of advanced computer modelling tools such as Particle in Cell (PIC) codes can mislead the unwary into supposing that the best way of modelling a tube is to create a complete model in such a code. This approach is wrong on two counts:

1. i) A PIC code finds a self-consistent numerical solution of Maxwell’s equations and the equations of motion in a series of time steps. However, the solution is linear in the part of the problem space which is not occupied by electrons. It is therefore more efficient to compute that part of the solution once and store it for future use. The problem is then reduced to the solution of the non-linear model in the space containing electrons, subject to matching the stored solution on the boundary. This is the method used in the large-signal models of tubes which are described later in this book.

2. ii) Even the best computer codes cannot be wholly accurate models of reality because the complexity of the problem is too great, so that some parts of it may not be known precisely. Assumptions and approximations are always required. For this reason the mathematical modelling of tubes is as much an art as it is a science [Reference Johns57]. The best results are obtained when the physics of the problem are well understood so that the assumptions and approximations are valid. Any model is only useful when it has been validated by comparison with experimental results. Even then it can only be used with caution outside the range of parameters for which it has been validated.

A large number of simple models were created in Mathcad14® during the writing of the book and used to generate many of the figures. These models can be used to explore the behaviour of different aspects of tubes by changing the parameters. They are available in electronic form (see the list of worksheets in the Appendix, available online at www.cambridge.org/9780521198622). The comments included in them should be sufficient for users to create their own models using other software if they wish. The models are intended as educational tools and have not been validated for use in tube design.