5.1 Classes of Amplification, Conduction Angles, and Operating Efficiency

5.1 Classes of Amplification, Conduction Angles, and Operating Efficiency

Radio frequency (RF) power amplification represents one of the most critical stages in transmitter design. An RF power amplifier must accept a low-power modulated or unmodulated carrier signal from an exciter or frequency synthesiser and boost its power level to drive an antenna system, often converting tens of milliwatts into hundreds or thousands of watts. In doing so, the amplifier designer faces a fundamental engineering trade-off: linearity versus energy conversion efficiency.

Linearity dictates how accurately the output RF waveform reproduces the amplitude and phase characteristics of the input signal. Efficiency ($\eta$) defines the proportion of direct-current (DC) power drawn from the station power supply or high-voltage mains transformer that is converted into useful RF power delivered to the load, with the remainder converted into waste thermal heat that must be dissipated by heatsinks or cooling fans.

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1. Operating Point, Quiescent Bias, and Transfer Characteristics

Every active amplifying device—whether a bipolar junction transistor (BJT), a field-effect transistor (such as an LDMOS or VDMOS device), or a thermionic valve (vacuum tube)—possesses a dynamic transfer characteristic that governs the relationship between the controlling input voltage and the resulting output current:

• Bipolar Junction Transistors (BJTs): Governed by the exponential relationship between collector current ($I_C$) and base-emitter voltage ($V_{BE}$):

$$I_C = I_S \left( e^{\frac{V_{BE}}{V_T}} - 1 \right)$$

where $I_S$ is the transport saturation current and $V_T = kT/q \approx 26\text{ mV}$ is the thermal voltage at room temperature. Conduction begins when $V_{BE}$ reaches approximately $0.65\text{ V}$ to $0.70\text{ V}$ in silicon.

• Field-Effect Transistors (FETs / MOSFETs / LDMOS): Governed by a square-law transfer characteristic above the threshold voltage ($V_{\text{th}}$):

$$I_D = \frac{1}{2} \mu C_{\text{ox}} \frac{W}{L} (V_{GS} - V_{\text{th}})^2$$

where $I_D$ is the drain current and $V_{GS}$ is the gate-to-source voltage. Modern laterally diffused metal-oxide semiconductor (LDMOS) devices exhibit high transconductance ($g_m$) once $V_{GS}$ exceeds the gate threshold (typically $2.0\text{ V}$ to $4.5\text{ V}$).

• Thermionic Triodes: An approximate space-charge model relates anode current ($I_a$) to control-grid voltage ($V_g$) and plate voltage ($V_a$):

$$I_a = k \left( V_a + \mu V_g \right)^{\frac{3}{2}}$$

Here $\mu$ is the triode amplification factor and $k$ is a geometry-dependent constant. In tetrodes and pentodes, the control-grid and screen-grid voltages chiefly determine current; this triode expression must not be used as their complete model.

The Quiescent Operating Point ($Q$-Point)

The Quiescent operating point ($Q$-point) is the steady-state DC voltage and current condition established in the active device with zero AC input signal applied. The $Q$-point and RF drive amplitude together determine the conduction angle ($\theta$).

Mathematical Definition of Conduction Angle

The conduction angle ($\theta$) is defined as the angular portion of a full $360^\circ$ ($2\pi$ radians) sinusoidal input cycle during which the active device conducts output current ($I_{\text{out}} > 0$). Assuming an input driving voltage $v_{\text{in}}(t) = V_{\text{bias}} + V_{\text{drive}} \cos(\omega t)$, conduction occurs when $v_{\text{in}}(t) \ge V_{\text{cutoff}}$. For $V_{\text{drive}}>0$, define $x=(V_{\text{cutoff}}-V_{\text{bias}})/V_{\text{drive}}$. This ideal threshold model gives:

$$\theta=\begin{cases} 360^\circ,&x\le-1\\ 2\arccos(x),&-1<x<1\\ 0^\circ,&x\ge1 \end{cases}$$

Use degrees for the arccos result. The middle expression applies only when the waveform crosses the cutoff threshold.

Depending on the position of $V_{\text{bias}}$ relative to $V_{\text{cutoff}}$, amplifiers are categorized into distinct operational classes.

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2. Class A Amplification: Full-Cycle Conduction and Pure Linearity

In a Class A amplifier, the $Q$-point is biased in the center of the linear region of the active device's dynamic transfer characteristic. The active device conducts current continuously throughout the entire input cycle.

• Conduction Angle: $\theta = 360^\circ$ ($2\pi\text{ radians}$).
• Quiescent Current ($I_Q$): Non-zero standing current flows continuously at all times. To achieve maximum symmetrical dynamic voltage swing without clipping into cutoff or saturation, $I_Q$ is set to half the peak saturation current: $I_Q = I_{\text{max}} / 2$.

Mathematical Derivation of Theoretical Efficiency

Efficiency ($\eta$) is defined as the ratio of RF output power delivered to the load ($P_{\text{RF}}$) to the DC input power drawn from the supply ($P_{\text{DC}}$):

$$\eta = \frac{P_{\text{RF}}}{P_{\text{DC}}} \times 100\%$$

1. Resistive Load (Series-Fed): If an active device feeds a pure resistive collector/plate load $R_L$ directly from a supply $V_{CC}$ without inductive storage, the maximum unclipped peak AC voltage is $V_{\text{peak}} = V_{CC} / 2$, and the maximum theoretical efficiency is:

$$\eta_{\text{max(resistive)}} = 25\%$$

1. Inductor- or Transformer-Coupled Load: In practical RF stages, an RF choke (RFC) or tuned transformer connects the supply to the device. The magnetic field of the inductor allows the instantaneous plate/collector voltage to swing from near $0\text{ V}$ up to twice the supply voltage ($2V_{CC}$), so $V_{\text{peak}} = V_{CC}$ and $I_{\text{peak}} = I_Q$. The power equations become:

$$P_{\text{DC}} = V_{CC} \cdot I_Q = V_{CC} \cdot \frac{I_{\text{max}}}{2}$$ $$P_{\text{RF}} = \frac{1}{2} V_{\text{peak}} \cdot I_{\text{peak}} = \frac{1}{2} V_{CC} \cdot \frac{I_{\text{max}}}{2} = \frac{1}{4} V_{CC} \cdot I_{\text{max}}$$ $$\eta_{\text{max(Class A)}} = \frac{P_{\text{RF}}}{P_{\text{DC}}} = \frac{\frac{1}{4} V_{CC} I_{\text{max}}}{\frac{1}{2} V_{CC} I_{\text{max}}} = \frac{1}{2} = 50\%$$

Thermal Characteristics of Class A

A fundamental and counter-intuitive characteristic of Class A operation is that internal device power dissipation is at its absolute maximum when NO input signal is present:

$$P_{\text{diss(no-signal)}} = P_{\text{DC}} = V_{CC} \cdot I_Q$$

When an RF drive signal is applied, a portion of the continuous DC power is converted into RF power and delivered to the external load, causing the internal device dissipation to decrease:

$$P_{\text{diss(operating)}} = P_{\text{DC}} - P_{\text{RF}}$$

Practical Amateur Applications

Because the device never enters cutoff, Class A generates virtually zero crossover distortion and exhibits the highest possible linearity and lowest intermodulation distortion. However, because at least $50\%$ (and practically $70\%$ to $80\%$) of the supplied DC power is turned directly into heat, Class A requires substantial cooling at high output power. Common applications include:

• Receiver low-noise preamplifiers (LNAs) and active antenna front-ends.
• Small-signal RF and IF voltage amplifier stages.
• Low-level exciter driver stages where pristine linearity is required before subsequent power amplification.

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3. Class B Amplification: Half-Cycle Conduction and Crossover Distortion

In a Class B amplifier, the active device is biased precisely at cutoff ($V_{\text{bias}} = V_{\text{cutoff}}$). With zero RF drive applied, the quiescent current is zero ($I_Q = 0$).

• Conduction Angle: $\theta = 180^\circ$ ($\pi\text{ radians}$).
• Quiescent Current ($I_Q$): $0\text{ A}$. Current flows only during the positive half-cycle of the input AC voltage.

Mathematical Derivation of Theoretical Efficiency

During the conducting half-cycle, current flows as a half-wave rectified sinusoid with peak amplitude $I_{\text{peak}}$. The average DC current drawn from the power supply is:

$$I_{\text{DC}} = \frac{1}{2\pi} \int_0^\pi I_{\text{peak}} \sin(\omega t) d(\omega t) = \frac{I_{\text{peak}}}{\pi}$$

The total DC input power is:

$$P_{\text{DC}} = V_{CC} \cdot I_{\text{DC}} = \frac{V_{CC} \cdot I_{\text{peak}}}{\pi}$$

Assuming the fundamental sinusoidal output voltage swing reaches $V_{\text{peak}} = V_{CC}$ (reconstructed via a push-pull complementary device or a high-$Q$ resonant tank circuit), the fundamental RF power delivered to the load is:

$$P_{\text{RF}} = \frac{V_{\text{peak}} \cdot I_{\text{peak}}}{4} = \frac{V_{CC} \cdot I_{\text{peak}}}{4}$$

Calculating maximum theoretical efficiency:

$$\eta_{\text{max(Class B)}} = \frac{P_{\text{RF}}}{P_{\text{DC}}} = \frac{\frac{V_{CC} I_{\text{peak}}}{4}}{\frac{V_{CC} I_{\text{peak}}}{\pi}} = \frac{\pi}{4} \approx 0.7854 = 78.54\%$$

Crossover Distortion

Half-cycle device current is normal in Class B. A tuned RF load extracts its fundamental, and an ideal Class B RF stage can preserve a varying signal envelope. The chopped device-current waveform alone is not proof of crossover distortion.

In practical push-pull stages, the transfer characteristics near cutoff can create a notch as conduction transfers between devices. Inadequate bias produces a region where neither contributes enough current. This crossover distortion creates harmonics and, with varying-envelope signals, intermodulation products. Class AB uses overlapping conduction to reduce this nonlinearity.

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4. Class AB Amplification: The Amateur Linear Power Amplifier Standard

To reduce crossover distortion while retaining useful efficiency, RF linear power amplifiers commonly use Class AB bias.

• Conduction Angle: $180^\circ < \theta < 360^\circ$ (typically between $200^\circ$ and $220^\circ$).
• Quiescent Current ($I_{CQ}$): Biased slightly above cutoff. A small standing quiescent current (typically $50\text{ mA}$ to $500\text{ mA}$ depending on the power rating of the valve or transistor) flows under zero-signal conditions.
• Practical Efficiency: Typically $60\%$ to $70\%$.

Overcoming Crossover Distortion

By maintaining a slight forward bias at idle, the active device conducts continuously during the low-amplitude zero-crossing region of the input waveform. In a push-pull Class AB amplifier, device conduction overlaps smoothly: one device begins conducting before the opposite device turns completely off. This reduces the crossover notch; overall linearity still depends on bias, loading and drive level.

Preservation of Amplitude Envelopes

SSB, AM, filtered PSK31 and 16-QAM have varying RF envelopes and require linear amplification after modulation. Compressing or clipping their envelopes creates distortion and unwanted spectrum. FT8 differs: its continuous-phase 8-GFSK waveform has constant amplitude during the message; information is conveyed by frequency/phase, while shaped start/end ramps control keying transients. Those ramps must also be preserved.

Class AB provides the linear transconductance required to amplify amplitude-varying signals cleanly. Consequently, Class AB is widely used in amateur HF and VHF linear power amplifiers.

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5. Class C Amplification: Resonant Tank Flywheel Effect and Restrictions

In a Class C amplifier, the active device is biased significantly below its cutoff threshold. The device conducts current only during the very crest of the positive input cycle.

• Conduction Angle: $\theta < 180^\circ$ (typically between $120^\circ$ and $140^\circ$).
• Quiescent Current ($I_Q$): $0\text{ A}$ in the ideal model. With no RF drive, bias holds the device below cutoff; the required voltage and polarity depend on the device type.
• Practical Efficiency: Typically $75\%$ to $85\%$, with theoretical efficiency approaching $100\%$ as the conduction angle approaches $0^\circ$ (though output power simultaneously approaches zero).

The Resonant Tank Circuit Flywheel Effect

Because current flows through the active device only as brief, highly non-sinusoidal pulses, a single-ended Class C amplifier cannot reproduce an intact AC waveform directly into a resistive load. Instead, the plate or collector load is a parallel $LC$ resonant tank circuit tuned precisely to the fundamental operating frequency.

When a short pulse of collector/plate current strikes the tank circuit, it shock-excites the tuned network into oscillation, delivering energy to the tank's electric and magnetic fields. During the remainder of the RF cycle (when the active device is completely cut off), the circulating reactive current between the inductor and capacitor maintains continuous sinusoidal voltage oscillations across the load—a phenomenon known as the flywheel effect, analogous to a heavy mechanical flywheel that continues spinning smoothly between brief engine piston strokes. The high loaded quality factor ($Q_L$) of the tank circuit attenuates the harmonic content of the current pulses, ensuring a clean sinusoidal voltage waveform at the antenna.

Non-Linearity and Modulation Compatibility

Because the active device remains completely cut off until the input RF drive exceeds the deep cutoff bias threshold, the relationship between input voltage and output current is intensely non-linear. Small-amplitude signals produce zero output, while larger signals are severely clipped.

Class C is commonly used to amplify constant-envelope CW, FM and FSK signals. It can also generate AM by high-level modulation, varying the collector/anode supply (and screen supply where required) in step with the audio. This differs from amplifying an existing AM envelope. Spectral compliance depends on the resulting transmission, not a regulatory ban on an amplifier class.

Common constant-envelope applications include:

• Continuous Wave (CW): Morse code telegraphy.
• Frequency Modulation (FM): VHF/UHF voice and packet radio.
• Frequency-Shift Keying (FSK): RTTY and constant-envelope digital modes.

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6. Switched-Mode RF Amplifiers: Class D, Class E, and Class F

Traditional linear and tuned amplifiers (Classes A, B, AB, and C) operate the active device as a controlled transconductance current source. Because the device simultaneously supports finite voltage across its terminals ($v(t)$) while carrying finite current ($i(t)$), instantaneous internal power dissipation is always present:

$$p_{\text{diss}}(t) = v(t) \cdot i(t)$$

Modern high-efficiency solid-state transmitters utilize switched-mode amplifier topologies, where active devices (high-speed power MOSFETs, LDMOS, or Gallium Nitride GaN FETs) act strictly as binary switches:

• When the MOSFET switch is ON (low-resistance state): Current flows, but the voltage across the device is near zero ($v(t) \approx 0 \implies p_{\text{diss}} \approx 0$).
• When the switch is OFF (cutoff): Voltage across the device is high, but current is zero ($i(t) = 0 \implies p_{\text{diss}} = 0$).

Because voltage and current waveforms are engineered never to overlap in time, theoretical device dissipation is zero, yielding theoretical efficiencies of $100\%$ and practical laboratory/commercial efficiencies between $88\%$ and $95\%$.

Switching amplifiers are inherently non-linear. However, they can be utilized in high-efficiency linear transmitters using Envelope Elimination and Restoration (EER) (also known as the Kahn technique or polar modulation), where the input signal is split into phase-modulated and amplitude-modulated components: the phase component drives the switching PA, while the amplitude component modulates the DC supply voltage to the switching stage.

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7. Comprehensive Comparison of Amplifier Classes

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8. Worked Engineering Examples

Worked Example 5.1.1: Calculating Conduction Angle and DC Input Power

An RF power transistor stage operates from a $V_{CC} = 28.0\text{ V}$ DC rail. The device has a cutoff threshold of $V_{\text{cutoff}} = 0.70\text{ V}$. A DC base bias of $V_{\text{bias}} = 0.50\text{ V}$ is applied, and the sinusoidal input RF drive has a peak amplitude of $V_{\text{drive}} = 1.20\text{ V}$.

1. Calculate the conduction angle ($\theta$) of the stage:

$$\cos\left( \frac{\theta}{2} \right) = -\frac{V_{\text{bias}} - V_{\text{cutoff}}}{V_{\text{drive}}} = -\frac{0.50\text{ V} - 0.70\text{ V}}{1.20\text{ V}} = -\frac{-0.20}{1.20} = +0.1667$$ $$\frac{\theta}{2} = \arccos(0.1667) \approx 80.4^\circ \implies \theta = 2 \times 80.4^\circ = 160.8^\circ$$

Because $\theta = 160.8^\circ < 180^\circ$, this amplifier operates in Class C.

1. If the amplifier draws an average DC current of $I_{\text{DC}} = 4.50\text{ A}$ from the $28.0\text{ V}$ supply and delivers $95.0\text{ W}$ of fundamental RF power into a matched $50\ \Omega$ load via its tuned resonant tank, calculate the DC input power, circuit efficiency, and thermal heat dissipation:

$$P_{\text{DC}} = V_{CC} \cdot I_{\text{DC}} = 28.0\text{ V} \times 4.50\text{ A} = 126.0\text{ W}$$ $$\eta = \frac{P_{\text{RF}}}{P_{\text{DC}}} \times 100\% = \frac{95.0\text{ W}}{126.0\text{ W}} \times 100\% \approx 75.4\%$$ $$P_{\text{diss}} = P_{\text{DC}} - P_{\text{RF}} = 126.0\text{ W} - 95.0\text{ W} = 31.0\text{ W}$$

The heatsink must safely dissipate $31.0\text{ W}$ of heat to prevent the transistor junction from exceeding its maximum rated thermal limit.

Worked Example 5.1.2: Thermal Dissipation Comparison for a 500 W Transmitter

An amateur constructs an HF transmitter intended to deliver $P_{\text{RF}} = 500\text{ W}$ output power. Compare the required DC input power and heat dissipation across three different design implementations:

• Implementation A (Class A): Operating efficiency $\eta = 30\%$.
• Implementation B (Class AB Linear): Operating efficiency $\eta = 65\%$.
• Implementation C (Class C for CW/FM): Operating efficiency $\eta = 80\%$.

$$P_{\text{DC}} = \frac{P_{\text{RF}}}{\eta}, \quad P_{\text{diss}} = P_{\text{DC}} - P_{\text{RF}}$$

• Implementation A (Class A):

$$P_{\text{DC}} = \frac{500\text{ W}}{0.30} = 1667\text{ W}, \quad P_{\text{diss}} = 1667\text{ W} - 500\text{ W} = 1167\text{ W}$$

• Implementation B (Class AB):

$$P_{\text{DC}} = \frac{500\text{ W}}{0.65} = 769.2\text{ W}, \quad P_{\text{diss}} = 769.2\text{ W} - 500\text{ W} = 269.2\text{ W}$$

• Implementation C (Class C):

$$P_{\text{DC}} = \frac{500\text{ W}}{0.80} = 625.0\text{ W}, \quad P_{\text{diss}} = 625.0\text{ W} - 500\text{ W} = 125.0\text{ W}$$

This comparison demonstrates the cooling burden of high-power Class A operation: the amplifier must safely dissipate $1167\text{ W}$ of heat. The design must balance linearity, efficiency and available cooling. Class AB reduces thermal dissipation to a manageable $269.2\text{ W}$ while maintaining the linearity necessary for clean SSB operation. For constant-carrier CW or FM, Class C reduces thermal dissipation further to just $125.0\text{ W}$.