classe2-calculator

classe2-calculator calculates Class-E Power Amplifier circuit component values in order to assist with the practical design of Class-E Power Amplifier based circuits for RF communication, plasma control, or any other application. It is pronounced "classi er", as an English-Mandarin portmanteau.

Example

Suppose we wish to generate a Class E power amplifier for a radio transceiver with the following parameters:

Nominal 12V supply
5 watt output
Operation on 20m, 30m and 40m bands
Using the MOSFET SI2304DDS
Fixing the design to 10uH L1 inductor

$ ./classe2 -v 12 -p 5 -b 20m -m SI2304DDS --l1 10uh
Command line: ./classe2 -v 12 -p 5 -b 20m -m SI2304DDS --l1 10uh
UTC datetime: Sat Oct 26 23:27:11 UTC 2024

Input parameters:
 Output Power = 5.00W
 Supply Voltage = 12.00V
 MOSFET = SI2304DDS (Vdss=30V, Id=2.7A, Rds_on=0.049Ω, Ciss=235pF, Coss=45pF, Crss=17pF, Vgs_th=2.3375V)
 Q Factor = None

[20m] @ 14MHz-14MHz (c=14MHz, nbw=350kHz, bw=420kHz, q=33.75)
 Power Amplifier components:
  Topology = infinite
  L1 = 10µH (user-specified)
  L2 = 6.2µH
  C1 = 136.39pF
  C2 = 21.02pF
  R = 16Ω (before matching)

  Top 3 recommended Power Amplifier component configurations:

   Configuration 1: (0% error, estimated SRF margin 0 (Concern), 6 parts)
   (SRF Concern: Consider using a different inductor or parallel combination for L2)
     L2 = 6.2µH (1.5µH + 4.7µH) [ESR=2.00x] [I=0.50x]
     C1 = 136.36pF (150.00pF || 1.50nF) [ESR=0.50x] [I=2.00x]
     C2 = 21.02pF (22.00pF || 470.00pF) [ESR=0.50x] [I=2.00x]

   Configuration 2: (0.01% error, estimated SRF margin 0 (Concern), 6 parts)
   (SRF Concern: Consider using a different inductor or parallel combination for L2)
     L2 = 6.2µH (1.5µH + 4.7µH) [ESR=2.00x] [I=0.50x]
     C1 = 136.00pF (68.00pF + 68.00pF) [ESR=2.00x] [I=0.50x]
     C2 = 21.02pF (22.00pF || 470.00pF) [ESR=0.50x] [I=2.00x]

   Configuration 3: (0.01% error, estimated SRF margin 0 (Concern), 6 parts)
   (SRF Concern: Consider using a different inductor or parallel combination for L2)
     L2 = 6.2µH (6.8µH || 68µH) [ESR=0.50x] [I=2.00x]
     C1 = 136.36pF (150.00pF || 1.50nF) [ESR=0.50x] [I=2.00x]
     C2 = 21.02pF (22.00pF || 470.00pF) [ESR=0.50x] [I=2.00x]

Matching Network Configurations (pi topology):
    Precise configuration (0.03% error, 9 parts):
      C1_match: 980.00pF (150.00pF + 150.00pF + 680.00pF)
      L_match: 0.8µH (0.003µH + 0.12µH + 0.68µH)
      C2_match: 321.50pF (1.50pF + 100.00pF + 220.00pF)
    Approximate configuration 1 (2.05% error, 4 parts):
      C1_match: 1.00nF (1.00nF)
      L_match: 0.82µH (0.82µH)
      C2_match: 320.00pF (100.00pF + 220.00pF)
    Approximate configuration 2 (2.67% error, 3 parts):
      C1_match: 1.00nF (1.00nF)
      L_match: 0.82µH (0.82µH)
      C2_match: 330.00pF (330.00pF)

Low Pass Filter Configurations:
 7th = (0.20% error, 21 parts): C1_lpf=66.70pF (4.70pF + 15.00pF + 47.00pF) L1_lpf=0.47µH (0.82µH || 1.5µH || 3.9µH) C2_lpf=270.30pF (3.30pF + 47.00pF + 220.00pF) L2_lpf=0.75µH (0.008µH + 0.18µH + 0.56µH) C3_lpf=270.30pF (3.30pF + 47.00pF + 220.00pF) L3_lpf=0.47µH (0.82µH || 1.5µH || 3.9µH) C4_lpf=66.70pF (4.70pF + 15.00pF + 47.00pF)
 7th ≈ (18.44% error, 7 parts): C1_lpf=68.00pF L1_lpf=0.47µH C2_lpf=220.00pF L2_lpf=0.68µH C3_lpf=220.00pF L3_lpf=0.47µH C4_lpf=68.00pF
 6th = (0.14% error, 18 parts): C1_lpf=77.40pF (4.70pF + 4.70pF + 68.00pF) L1_lpf=0.53µH (0.003µH + 0.056µH + 0.47µH) C2_lpf=289.00pF (1.00pF + 68.00pF + 220.00pF) L2_lpf=0.72µH (0.033µH + 0.22µH + 0.47µH) C3_lpf=212.00pF (15.00pF + 47.00pF + 150.00pF) L3_lpf=0.19µH (0.005µH + 0.039µH + 0.15µH)
 6th ≈ (14.11% error, 6 parts): C1_lpf=68.00pF L1_lpf=0.56µH C2_lpf=330.00pF L2_lpf=0.68µH C3_lpf=220.00pF L3_lpf=0.18µH
 5th = (0.34% error, 14 parts): C1_lpf=92.20pF (2.20pF + 22.00pF + 68.00pF) L1_lpf=0.61µH (0.007µH + 0.039µH + 0.56µH) C2_lpf=300.00pF (150.00pF + 150.00pF) L2_lpf=0.61µH (0.007µH + 0.039µH + 0.56µH) C3_lpf=92.20pF (2.20pF + 22.00pF + 68.00pF)
 5th ≈ (10.22% error, 5 parts): C1_lpf=100.00pF L1_lpf=0.56µH C2_lpf=330.00pF L2_lpf=0.56µH C3_lpf=100.00pF
 4th = (0.14% error, 12 parts): C1_lpf=114.70pF (4.70pF + 10.00pF + 100.00pF) L1_lpf=0.69µH (0.002µH + 0.01µH + 0.68µH) C2_lpf=277.00pF (10.00pF + 47.00pF + 220.00pF) L2_lpf=0.29µH (0.002µH + 0.015µH + 0.27µH)
 4th ≈ (19.30% error, 4 parts): C1_lpf=100.00pF L1_lpf=0.68µH C2_lpf=330.00pF L2_lpf=0.27µH
 3rd = (0.10% error, 7 parts): C1_lpf=149.86pF (220.00pF || 470.00pF) L1_lpf=0.75µH (0.008µH + 0.18µH + 0.56µH) C2_lpf=149.86pF (220.00pF || 470.00pF)
 3rd ≈ (9.15% error, 3 parts): C1_lpf=150.00pF L1_lpf=0.68µH C2_lpf=150.00pF
 2nd = (0.14% error, 6 parts): C1_lpf=212.00pF (15.00pF + 47.00pF + 150.00pF) L1_lpf=0.53µH (0.003µH + 0.056µH + 0.47µH)
 2nd ≈ (5.80% error, 2 parts): C1_lpf=220.00pF L1_lpf=0.56µH
 1st = (0.20% error, 2 parts): C1_lpf=300.00pF (150.00pF + 150.00pF)
 1st ≈ (10.22% error, 1 parts): C1_lpf=330.00pF

                                                  ANTENNA
                                                  SYSTEM
                                                   |  |
======= Class E Power Amplifier ===================|==|===
                                                   |  |
   VCC--[L1]--[C1]--[C2]--[L2]--.       DRV--[MOSFET] |
               |                |               |     |
              GND               |              GND    |
                                |                     |
                                |                     |
 .------------------------------'                     |
 | Z<50Ω                                              |
=|===== Pi Matching Network ==========================|===
 |                                                    |
 `--[C1_match]--[L_match]--[C2_match]--.              |
       |                      |        |              |
      GND                    GND       |              |
                                       |              |
 .-------------------------------------'              |
 | Z=50Ω                                              |
=|===== Low Pass Filter ==============================|===
 |                                                    |
 `--[C1_lpf]--[L2_lpf]--[C3_lpf]--[L4_lpf]--[C5_lpf]--'
       |                   |                   |
      GND                 GND                 GND

==========================================================

Why?

The motivation for writing this software was increasing frustration attempting to understand and explore the solution space for the design of surface mount component based systems using existing tools, coupled with the intent to perform similar calculations in the future. What distinguishes it from other options?

Open source
- Reliable tool for exploration of the solution space
- Transparent algorithms
- Well documented design
Portable (runs on all platforms)
SI / metric units (no cubit foot-yard waveform eighth-of-a-bodypart handspannery here)
SMD design as a primary use case, thus the following are in scope:
- SRF
- Current
- Zener protection against voltage peaks
Produces explainable (well documented) designs which can be readily audited.

Potential output / feature-set / goals

Primary schematic
- Classic "Infinite" inductor subtype
  - "Current driven": Based on zero voltage switching (ZVS) and zero voltage derivative switching (ZVDS) conditions
  - Identification: L1 > (~10-15)×L2
  - Invoke with -t infinite
  - Benefits:
    - Simple to design
    - Well understood in amateur circles
    - Excellent harmonic suppression.
  - Drawbacks:
    - Larger physical size (due to the large RF choke inductor)
    - Poorly suited to surface mount components, therefore not very manufacturable
    - More expensive components and assembly
    - Higher ESR losses
- "Finite" L1 inductor subtype
  - "Current driven": Based on zero voltage switching (ZVS) and zero voltage derivative switching (ZVDS) conditions
  - Identification: L2 < L1 < 5×L2
  - Key differences:
    - Smaller and cheaper inductor, more suited to surface mount manufacturing
  - Invoke with -t finite
  - Benefits:
    - Compromise between bandwidth and efficiency
    - Moderate selectivity and good harmonic suppression
    - Better suited to surface mount components
    - Physically smaller than classic "infinite" class E
    - Cheaper components
  - Drawbacks:
    - More complex to design
    - Less well understood in amateur circles
- Shunt capacitance and shunt filter subtype
  - "Current driven": Based on zero voltage switching (ZVS) and zero voltage derivative switching (ZVDS) conditions
  - Identification: Addition of L0, sometimes no L1 at all
  - Key differences:
    - Replaces the series filter with a shunt filter (L0), typically an LC circuit (ie. inductor plus capacitor), which can serve a dual purpose as DC-feed inductance.
    - Incorporates both shunt capacitance C (to absorb output capacitance) and series inductance L (to absorb output inductance)
    - In some cases, removes the RF Choke (L1) entirely
  - Invoke with -t shunt
    - -o norfc — remove the RF Choke (L1) entirely, by having L0 take on a dual purpose role as DC-feed inductance, result in a more compact design
    - -o srx — include a series reactance X compensation, to increase suitability for broadband operation
    - -o nocx — remove the capacitance CX that is usually placed in parallel with the shunt filter and load resistance
  - Benefits:
    - Higher maximum operating frequency (fmax) compared to the classical Class-E PA and other variants.
    - Peak switch voltage does not grow as fast with increases in the quality factor under load (QL) as with traditional topologies, allowing higher frequency or power operations.
    - Better harmonic suppression
    - Option to remove the RF Choke (L1) entirely
    - Smaller inductance values result in a physically smaller and cheaper design with lower ESR
  - Drawbacks:
    - Complexity
    - Bandwidth limitations
    - Harmonic control
    - Impedance matching complexity
    - Manufacturability
    - Parasitic effects
    - Power handling limitations
    - Thermal management
- "Inverse" Class-E designs
  - "Voltage driven": based on zero current switching (ZCS) and zero current derivative switching (ZCDS) conditions
  - Identification: L1 < L2
  - Invoke with -t inverse
  - Benefits:
    - 20% lower peak switching voltage
    - Lower inductance values
    - Physically smaller (a significant benefit in MMIC)
    - Higher load resistance
    - Inherent absorption of the transistor’s output inductances including bond wires
    - Better scope for optimization for high efficiency at higher output power level
    - Higher peak output power
    - Cheaper components
    - Suitable for high frequency applications such as microwave bands
  - Drawbacks:
    - Most complex to design
    - Poor selectivity and degraded harmonic suppression
- Maybe later
  - Non-50% duty cycles
  - Push-Pull Class-E: A dual-ended design that uses two transistors operating 180 degrees out of phase, offering higher power and better even-harmonic suppression.
  - Differential Class-E: Similar to push-pull.
  - Outphasing Class-E: Dual Class-E amplifiers to implement a derivative of the Chireix outphasing technique allowing for efficient power control and linearization.
  - Class-E with parallel circuit: Parallel resonant circuit in the load network offer advantages in certain applications.
  - Class-E with reactive impedance: Additional reactive components shape the load impedance, potentially improving performance or flexibility.
  - Class-E with resistive impedance: A resistive component in the load network, useful for certain design constraints or applications.
  - Class-E with second harmonic peaking: Deliberately introduces second harmonic components to reshape the voltage waveform and potentially improve efficiency.
  - Class-E/F: Hybrid topology combining features of Class-E and Class-F, improving efficiency by shaping both voltage and current waveforms.
  - Class-E/F Push-Pull: Dual-ended variant of the previous topology.
  - Class-E/F2,3: Specific variant of the Class-E/F family that uses second and third harmonic tuning to shape the waveforms.
  - Class-EF: Hybrid topology that incorporates aspects of both Class-E and Class-F designs to achieve high efficiency over a wider bandwidth.
  - Class-EM: Transmission line replaces some lumped elements in the load network, potentially allowing higher frequency operation.
  - Class-E3: Third harmonic peaking variant that aims to further optimize the switching conditions and efficiency.
  - Class-F design
Antenna matching networks
- L type
- Pi type
Antenna tuning unit (ATU) recommendations (maybe later)
SPICE simulation source (maybe later)
Coil core / winding instructions (maybe later)

Class-E Schematic

Classic Sokal / "Infinite" Class-E Topology

Taken from Sokal (1975).

L1 = RF Choke inductor with high reactance at the fundamental frequency of operation.
C1 = Shunt capacitor. Practically affected by switching MOSFET device output capacitance C(OSS).
- According to classeradio: The function of the shunt capacitance is to reduce the peak voltage appearing across the MOSFET when the device is in the off state, and to spread the width of the "off" pulse. The shunt capacitor is also part of the output matching network. This capacitor must be a high quality, high current component. Typically, multi-layer ceramic capacitors (MLCC) work best, however silver-mica capacitors can also be used in this function. The actual value of the shunt capacitor is important for a couple of reasons. First, if the capacitance is too small, you will see a very high RF peak voltage across your MOSFETs. If the value is too large, the efficiency can suffer. This is not an extremely critical value. [...] The shunt capacitor value is correct if the peak RF voltage across the MOSFETs during the "off" cycle is around 3.5 times the DC voltage applied to the stage. If the voltage is higher than this value, you may need to increase your shunt capacitor.
C2 / L2 = Resonator
- C2 = "Tuning capacitor" (or "Series capacitor")
  - According to classeradio: The series tuning capacitor is subject to very high RF voltage [...] Using a lower inductance and higher capacitance in the resonant circuit will reduce the voltage across the tuning capacitor somewhat. A useful rule of thumb for figuring the tuning capacitor voltage rating is 3 to 5 times the peak to peak RF voltage fed to the resonant network plus a safety factor.
Q = Active Device Switch (MOSFET)
- According to classeradio: The peak voltage across the MOSFETs is going to be a little less than 4x the DC applied voltage for a proper class E transmitter. This will vary somewhat with tuning and your exact circuit. If you have very low, or NO shunt capacitor, the ratio of peak RF voltage to applied DC can be 6x or 8x the DC or MORE. For class E transmitters with proper shunt capacitors, figure 4x the DC, plus a safety factor.
- In the ideal situation, the efficiency of a class-E amplifier is 100%. However, in practice:
  - The switch has a finite on-resistance (RDS(ON)); and
  - The transition times (both from the off-state to the on-state (t(on)) and vice-versa (t(off)), and their related switching latencies (td(on) and td(off)) and rise/fall times (t(r) and t(f))) are not negligible.
  - Both of these factors result in power dissipation in the switch and reduce the efficiency.
- Key selection criteria include:
  - On resistance (RDS(ON); lower is better)
  - Capacitances (C(RSS), C(ISS), C(OSS); lower is better)
  - Transition times (td(on), td(off), t(r), t(f); though apparently usually not modeled in practice)
  - Drain current (I(D); higher is better; high enough is OK)
  - Voltage (VGS; higher is better, >4x VCC and/or add low capacitance zener diodes which are typically those rated for higher voltages; high enough is OK)
Non component-specific design variables
- R = Optimal load resistance
- QL = Quality factor under load
  - Higher QL values mean higher peak voltages.
  - Certain bands require higher QL factors.
- VCC = DC supply and nominal operating voltage
- P = target output power in watts

Classic Sokal was an infinite or ideal L1 (RF choke) based design, series tuned, with a shunt capacitor.

A cleaner schematic from Mury (2005).

"Finite" Class-E Topology

From Mury (2005).

"Inverse" Class-E Topology

From Mury (2005).

A still-cleaner schematic from Thian (2009).

Comprised of a transistor, an RF choke inductor (RFC), a DC-blocking capacitor, a series inductor (L), a parallel-tuned resonator (C(P) and L(P)), a shunt capacitor (C), and a load resistance (R).
- The shunt capacitor (C) is required in order to compensate for the fundamental-frequency phase shift.
- The DC current passing through the RF choke (RFC) is transformed by transistor (Q) on–off switching action into a series of related fundamental-frequency and harmonic currents.
- The harmonic currents are then filtered out by the high-parallel-tuned resonator (C(P) and L(P)), leaving only the fundamental-frequency current.

Class-E Design equations

Forenotes

The two most common types of Class-E power amplifiers are classified based on the scale of L1 DC feed inductance. When L1 is large (to the point of approximating "infinite"), it is termed "infinite". Otherwise, it is termed "finite".

	"Infinite"	"Finite"
`L1` vs `L2`	10-15× (or more)	<10×
Bandwidth	Broadband	Narrowband
Selectivity	Poorly selective	Strongly selective
Harmonic suppression	Poor	Good
Losses due to ESR	Higher	Lower
Load resistance	Lower (harder matching network)	Higher (easier matching network)
Inter component dependence	Independent	Strongly interdependent
Ease of design	Relatively easy	Relatively difficult
Physical size	Large	Small
Manufacturing cost	Expensive	Inexpensive
Hand-wound coils	Suitable	Relatively unsuitable
Surface-mount devices	Relatively unsuitable	Suitable
Popularity	Most amateur radio projects	Relatively less popular
Load transformation	No	Yes

Generalities:

The sharper the resonance (higher Q) the greater the sensitivity, but the more critical it is to have accurate C1 and C2.
Some bands require higher Q than others
Most amateur radio projects seem to assume hand-wound coils which are relatively unsuitable for finite induction variants

Design equations

Forenotes for non physics people because terminology can be confusing:

f is the frequency in Hz (cycles per second).
- 2π represents one full rotation (ie. 360°) in radians.
- Multiplying by 2 ⋅ π converts the frequency to radians per second.
- Therefore 2πf represents the angular frequency (ω) in radians per second.
ω is the angular frequency in radians per second.
- To convert to angular frequency in radians per second from Hz, ω = 2πf
- To convert angular frequency in radians per second to Hz, f = ω ÷ 2π
φ means phase.

The equations:

R — Load resistance
- This is the optimal load resistance required to produce the maximum output power P.
- VK1SV variant (2001 or later)
  - Mathjax: $$R = 0.576801 \cdot \frac{(V_{CC}-V_o)^2}{P} \cdot [1.0000086 - \frac{0.414396}{Q} - \frac{0.577501}{Q^2} + \frac{0.205967}{Q^3}]$$
  - Unicode: R = 0.576801 ⋅ ((VCC-Vo)² ÷ P) ⋅ (1.0000086 - (0.414396/Q) - (0.577501/Q²) + (0.205967/Q³))
  - Javascript: var R=0.576801*((Vcc-Vo)*(Vcc-Vo)/P)*(1.0000086-(0.414396/Q)-(0.577501/(Q*Q))+(0.205967/(Q*Q*Q)));
  - Judging from source code of the VK1SV calculator, this seems to be a variant implementation of Sokal's improved formula (2001), with the polynomial formula f(Q) substituted in to provide the complete set of terms. The only known difference is that 0.414396 is used in place of the original's 0.414395 — a functionally insignificant difference, yet worth noting.
- Sokal's improved formula (2001)
  - Mathjax: $$R = K \cdot \frac{(VCC-Vo)^2}{P} \cdot (1 + f(Q))$$
  - Unicode: R = K ⋅ ((VCC-Vo)² / P) ⋅ (1 + f(Q))
  - Where:
    - K is a constant and f(Q) is a third-order polynomial function of Q to provide a closer fit to empirical data.
    - f(Q) is a polynomial approximation to account for the effect of the loaded quality factor (Q) on the load resistance calculation.
      - Unicode: f(Q) = 1.0000086 - 0.414395/Q - 0.577501/Q² + 0.205967/Q³
      - Mathjax: $$f(Q) = 1.0000086 - \frac{0.414395}{Q} - \frac{0.577501}{Q^2} + \frac{0.205967}{Q^3}$$
  - Advancement over Raab (2001) by providing improved accuracy across a wider range of Q values (especially lower Q values).
  - It accounts for non-ideal switch behavior by including the Vo term (voltage drop across the switch when on).
  - Originally published as Class-E High Efficiency RF/Microwave Power Amplifiers: Principles of Operation, Design Procedures, and Experimental Verification by Nathan O. Sokal in QEX (2001), since republished with corrections (2006).
- Raab formula (2001)
  - Mathjax: $$R = \frac{(V_{CC} - V_o)^2}{P}[k_1][1 + \frac{k_2}{Q_L} + \frac{k_3}{Q_L^2}]$$
  - Unicode: R = ( ((VCC - Vo)^2) / P) ⋅ k1 ⋅ (1 + (k2 / QL) + (k3 / QL^2))
  - Where k1 is Raab's first curve-fitting constant of 0.576801, k2 is Raab's second curve-fitting constant of -0.451759, k3 is Raab's third curve-fitting constant of -0.402444, VCC is the supply voltage in volts, P is the desired output power in watts, QL is the loaded quality factor, and Vo is the minimum voltage across the switch ("saturation voltage").
  - Improves accuracy by including the effects of QL on the relationship between supply voltage, output power, and load resistance.
  - The inclusion of QL allows the equation to be used for both high-Q and low-Q designs.
    - For very high QL values, the QL terms become negligible and the equation approaches the ideal Class-E case.
    - For lower QL values, desirable for bandwidth or other reasons, the additional terms provide the necessary corrections to maintain accuracy.
- Basic (infinite) equation
  - Mathjax: $$R = \frac{0.5768 \cdot V_{DD}^2}{P_{out}}$$
  - Unicode: R = 0.5768 ⋅ VDD² ÷ P
  - Assumes a 50% duty cycle.
  - Practical tolerance: ±10-15%
- Finite variant
  - Mathjax: $$R = \frac{0.5768 V_{DD}^2}{P_{out}} \cdot f(Q_L)$$
  - Unicode: R = 0.5768 ⋅ VDD² ÷ P ⋅ f(QL)
  - Where f(QL) is a correction factor derived from numerical simulations and based on loaded Q.
  - Allows for more accurate designs with finite choke inductance.
  - Practical tolerance: ±5-10%
C1 — Shunt capacitance
- Optimal shunt capacitance is a function of the operating frequency and the optimal load resistance, and therefore in turn the supply voltage and the desired output power.
- Raab formula (2001)
  - Mathjax: $$C_1 = \frac{1}{34.2219fR}[0.99866 + \frac{0.91424}{Q_L} - \frac{1.03175}{Q_L^2}] + \frac{0.6}{(2\pi f)^2L_1}$$
  - Unicode: C1 = (1 / (34.2219 ⋅ f ⋅ R)) ⋅ (0.99866 + (0.91424 / QL) - (1.03175 / QL²)) + (0.6 / (2πf)²) ⋅ L1
  - Where f is the operating frequency in Hz, R is the output load resistance in ohms, QL is the loaded quality factor, and L1 is the RF choke or DC feed inductor value in henries.
  - This equation improves accuracy by including second-order effects of QL and accounting for the finite inductance of L1. The constants were derived from curve-fitting to exact numerical solutions.
- Basic (infinite) equation
  - Mathjax: $$C1 = \frac{0.1836}{2\pi f R}$$
  - Unicode: C1 = 0.1836 ÷ 2πfR
  - Calculates the minimum shunt capacitance value at 50% duty cycle and infinite choke inductance.
  - Practical tolerance: ±15-20%
- Duty cycle variant
  - Mathjax: $$C1 = \frac{k_C}{2\pi f R}$$
  - Unicode: C1 = kC ÷ 2πfR
  - Where kC is a coefficient dependent upon duty cycle.
  - Allows for non-50% duty cycle designs.
  - Practical tolerance: ±10-15%
- Finite DC feed inductance variant
  - Mathjax: $$C1 = \frac{0.1836}{2\pi f R} \cdot g(Q_L)$$
  - Unicode: C1 = kC ÷ 2πfR ⋅ g(QL)
  - Where g(QL) is a correction factor based on loaded Q.
  - Provides more accurate results for real-world choke inductances.
  - Practical tolerance: ±8-12%
L2 — Variously termed series/output/resonator/tank inductance
- Optimal series inductance is a function of frequency and of the optimal load resistance, and therefore in turn the supply voltage and the desired output power.
- Raab formula (2001)
  - Mathjax: $$L_2 = \frac{Q_L \cdot R}{2\pi f}$$
  - Unicode: L2 = (QL ⋅ R) ÷ 2πf
  - Where QL is the loaded quality factor, R is the load resistance, and f is the operating frequency in Hz.
  - This equation represents an improvement by explicitly including QL, allowing more accurate calculation of L2 for different loaded Q values. Previously, L2 was often calculated assuming an infinite Q.
- Basic (infinite) equation
  - Mathjax: $$L_2 = \frac{1.15R}{2\pi f}$$
  - Unicode: L2 = 1.15R ÷ 2πf
  - Exact value for 50% duty cycle, infinite choke inductance
  - Practical tolerance: ±10-15%
- Duty cycle variant
  - Mathjax: $$L_2 = \frac{k_L R}{2\pi f}$$
  - Unicode: L2 = (kL ⋅ R) ÷ 2πf
  - Where kL is a coefficient dependent on duty cycle.
  - Enables designs with non-50% duty cycles.
  - Practical tolerance: ±8-12%
  - Inductance has an inverse relationship to duty cycle, ie. it increases as the duty cycle lowers, and decreases as the duty cycle increases.
- Finite DC feed inductance variant
  - Mathjax: $$L_2 = \frac{1.15R}{2\pi f} \cdot h(Q_L)$$
  - Unicode: L2 = 1.15R ÷ 2πf ⋅ h(QL)
  - Where h(QL) is a correction factor based on loaded Q. It is usually less than one, reducing the required inductance.
  - Provides more accurate results for real-world choke inductances.
  - Practical tolerance: ±5-10%
L1 — RF Choke or DC feed inductance
- Choi formula (2001)
  - Mathjax: $$L_1 = \frac{k_C \cdot V^2}{f \cdot P} \cdot \left[1 + \frac{k_1}{Q_L} + \frac{k_2}{Q_L^2}\right]$$
  - Unicode: L1 = kC ⋅ V²/(f ⋅ P) ⋅ (1 + k₁/QL + k₂/QL²)
  - Where kC is Choi's primary constant 0.2085, k₁ is Choi's first QL adjustment constant 1.789, k₂ is Choi's second QL adjustment constant -1.481, V is the supply voltage in volts, P is the desired power output in watts, f is the frequency in Hz, and QL is the loaded quality factor.
  - This formula by Jaehyeong Choi better accounts for variations in QL.
  - Tolerance: Approximately ±2-5%
  - Increasingly used in modern designs where accuracy is required.
- Raab improved infinite formula (1990s)
  - Mathjax: $$L_1 \geq \frac{5 \cdot R}{2\pi f}$$
  - Unicode: L1 ≥ 5R ÷ 2πf
  - Calculates the minimum value of the "infinite" choke inductance approximation.
  - Here RF choke infinite inductance minimum value is a function of frequency and of the optimal load resistance, and therefore in turn the supply voltage and the desired output power.
  - Practical tolerance: +20-30% (higher values generally acceptable)
- Kazimierczuk-Puczko simplified formula (1987) — NRND
  - Mathjax: $$L_1 \geq 10 \cdot L_2$$
  - Unicode: L1 ≥ 10 ⋅ L2
  - Simplified formula for approximating the mimimum L1 inductance.
  - First ratio-based approach to simplify the design process.
  - Widely referenced in the amateur radio community.
- Kazimierczuk-Puczko formula (1987) — NRND
  - Mathjax: $$L_1 = \frac{K_{KP} \cdot V^2}{f \cdot P}$$
  - Unicode: L1 = kKP ⋅ V²/(f ⋅ P)
  - Where kKP is the Kazimierczuk-Puczko constant (0.2116) which is an optimized coefficient based on ideal Class-E operation, V is the supply voltage in volts, P is the desired power output in watts and f is the frequency in Hz.
  - Antique formula from Kazimierczuk and Puczko's 1987 paper which is no longer recommended.
  - At the time of its publication, the formula provided an improvement over Sokal's formula by considering the loaded quality factor (QL).
- Sokal formula (1975) — NRND
  - Mathjax: $$L_1 = \frac{K_S \cdot V^2}{2\pi \cdot f \cdot P}$$ or $$L_1 = \frac{K_S \cdot V^2}{\omega \cdot P}$$
  - Unicode: L1 = kS ⋅ V² ÷ (2πf ⋅ P) or L1 = kS ⋅ V² ÷ (ω ⋅ P)
  - Where kS is "Sokal's constant" (generally a value between 1.8 and 2, but observed to sometimes be higher in later use), V is the supply voltage in volts, P is the desired power output in watts, f is the frequency in Hz, and ω is the angular frequency in radians per second.
  - Antique formula from Sokal's 1975 paper which is no longer recommended.
  - Doesn't account for many factors considered in modern class E design, such as loaded Q factor, duty cycle, and device characteristics.
- Finite DC feed inductance rule of thumb variant
  - Mathjax: $$L_1 = k_{LRF} \cdot \frac{R}{2\pi f}$$
  - Unicode: L1 = kLRF ⋅ R ÷ 2πf
  - Where kLRF is a coefficient based on desired loaded Q value, typically smaller than 5, allowing for more compact designs at the cost of some efficiency.
  - Allows for intentional finite choke inductance designs.
  - Practical tolerance: ±10-15%
- Optimized efficiency rule of thumb variant
  - Mathjax: $$L_1 = \frac{0.732R}{2\pi f} \cdot j(Q_L)$$
  - Unicode: L1 = (0.732 ⋅ R) ÷ 2πf ⋅ j(QL)
  - Where j(QL) is a correction factor based on loaded Q, typically being greater than 1.
  - Optimizes efficiency for finite choke inductance designs.
  - Provides a balance between size and performance.
  - Practical tolerance: ±5-10%
C2 — Series capacitance
- Optimal series capacitance is a function of frequency and of the optimal load resistance, and therefore in turn the supply voltage and the desired output power.
- Raab formula (2001)
  - Mathjax:$$C_2 = \frac{1}{2\pi fR}\frac{1}{Q_L - 0.104823}[1.00121 + \frac{1.01468}{Q_L - 1.7879}] - \frac{0.2}{(2\pi f)^2L_1}$$
  - Unicode: C2 = (1 ÷ (2πfR)) ⋅ (1 / (QL - 0.104823)) ⋅ (1.00121 + (1.01468 / (QL - 1.7879)) ) - (0.2 / (2πf)² ⋅ L1)
  - Where QL is the loaded quality factor, R is the load resistance, f is the operating frequency in Hz, and L1 is the RF choke or DC feed inductor value in henries.
- Basic (infinite) equation
  - Mathjax: $$C_2 = \frac{1}{2\pi f(5.4466R)}$$
  - Unicode: C2 = 1 ÷ 2πf(5.4466R)
  - Exact value for 50% duty cycle, infinite choke inductance design.
  - Practical tolerance: ±15-20%
- Duty cycle variant
  - Mathjax: $$C_2 = \frac{1}{2\pi f(k_{C0}R)}$$
  - Unicode: C2 = 1 ÷ 2πf(kC0 ⋅ R)
  - Where kC0 is a coefficient dependent on duty cycle.
  - Allows for non-50% duty cycle designs.
  - Practical tolerance: ±10-15%
- Finite DC feed inductance variant (subsequent to Raab (2001))
  - Mathjax: $$C_2 = \frac{1}{2\pi f(5.4466R)} \cdot i(Q_L)$$
  - Unicode: C2 = 1 ÷ 2πf(5.4466R) ⋅ i(QL)
  - Where i(QL) is a correction factor based on the loaded Q factor QL. It is usually less than 1, reducing the required capacitance.
  - Provides more accurate results for real-world choke inductances.
  - Practical tolerance: ±8-12%
Q — Quality factor
- Typical values in Class-E amplifiers range from 1.8 to 5 for practical implementations, with 3 to 5 being common for narrow-band applications.
- Lower values (around 1.8 to 3) are often used for broadband applications or when component tolerances are a concern.
- Higher values (3-5) are often used for narrowband applications or when precision components are available.
- The choice affects other circuit parameters, such as the shunt capacitance and the overall power output capability of the amplifier.
- Mathjax: $$Q_L = \frac{\omega L}{R}$$
- Unicode: QL = (ωL) ÷ R
- Where ω is the angular frequency in radians per second, L is the resonator inductance, and R is the load resistance (usually 50 ohms).

Additional formulae

Q(BAND) — Band quality factor
- Used to calculate the nominal Q factor of a radio band.
- A function of band center frequency and aggregate band bandwidth.
- Mathjax: $$Q_{BAND} = f_0 / \Delta f$$
- Unicode: Q(BAND) = f₀ / Δf
- English: "The Band Quality Factor is equal to the center frequency (in Hz) divided by the aggregate bandwidth (in Hz)"
- Where:
  - Q(BAND) is the band quality factor.
  - f₀ is the center frequency of the band (in Hz).
  - Δf is the aggregate bandwidth of the band (in Hz), typically measured at -3dB points.
C(DRAINMAX) — Maximum MOSFET Drain Capacitance
- Maximum MOSFET drain capacitance is a function of frequency and load resistance.
- Mathjax: $$C_{DRAINMAX} = \frac{k}{2\pi f \cdot R_{LOAD}}$$
- Unicode: C(DRAINMAX) = k / (2πf ⋅ R(LOAD))
- English: "The Maximum MOSFET drain capacitance (in farads) is equal to the scaling factor divided by the frequency (in radians per second) multiplied by the load resistance (in ohms)."
- Where:
  - k is a scaling factor (typically between 1 (possibly unworkable at high frequencies) and 4, with 3 as a suggested default at high frequencies)
  - f is the operating frequency in Hz
  - R(LOAD) is the load resistance in ohms, typically 50Ω
- Note this formula is merely suggested as a design guideline and has no theoretical basis in the literature.
Transistor Selection Formulae
- Figure of Merit (FOM) ("Figure of Malady")
  - Mathjax: $$FOM = R_{DS(ON)} \cdot Q_g$$
  - Unicode: FOM = RDS(ON) ⋅ Q(g)
  - English: "The Figure of Merit is equal to the product of the on resistance (in ohms) and the gate charge (in nanocoulombs)"
  - Where:
    - RDS(ON) is the on resistance of the device, in ohms
    - Q(g) is the gate charge, in nanocoulombs (nC)
    - FOM is expressed in the unit ohm-coulombs (Ω⋅C) or volt-seconds (V⋅s) which are equivalent. As lower values are better, perhaps the formula would be better termed "Figure of Malady".
  - Traditional metric for MOSFET selection - falling out of favour, as it does not function well at high frequencies and does not take in to account other parameters which may be more significant in many applications.
- Baliga's Figure of Merit (BFOM)
  - Mathjax: $$BFOM = \frac{1}{R_{DS(ON)} \cdot A}$$
  - Unicode: BFOM = 1 ÷ (RDS(ON) ⋅ A)
  - English: "Baliga's Figure of Merit is equal to the inverse of the product of the device on resistance (in ohms) and the active area of the device (in square centimeters)"
  - Where:
    - The result BFOM is the figure of merit. Higher values are better.
    - RDS(ON) is the on resistance of the device, in ohms
    - A is the active area of the device, in cm²
  - Note the above is a practical adaptation, the original formula is BFOM = ε ⋅ μ ⋅ Ec^3 where:
    - ε is the dielectric constant of the semiconductor material
    - μ is the electron mobility
    - Ec is the critical electric field strength
- Baliga's High Frequency Figure of Merit (BHFFOM)
  - Mathjax: $$BHFFOM = \frac{1}{R_{DS(ON)} \cdot C_{iss}}$$
  - Unicode: BHFFOM = 1 ÷ (RDS(ON) ⋅ C(ISS))
  - English: "Baliga's High Frequency Figure of Merit (in hertz-per-ohm) is equal to the inverse of the product of the device on resistance (in ohms) and the device input capacitance (in farads)"
  - Where:
    - The result BHFFOM is expressed in units of 1/(Ω⋅F) or Hz/Ω. Higher values are better.
    - RDS(ON) is the on resistance of the device, in ohms
    - C(ISS) is the input capacitance of the device, in farads (F)
      - However, as the figure is usually provided in the device datasheet as a value expressed in picofarads (pF), first divide the value in pF by 1,000,000,000,000 (1 trillion).
- Conduction Figure of Merit (CFOM) ("Conduction Figure of Malady")
  - Mathjax: $$CFOM = \frac{1}{R_{DS(ON)} \cdot Q_g \cdot C_{OSS}}$$
  - Unicode: CFOM = (1 / RDS(ON)) ⋅ Qg ⋅ C(OSS)
  - English: "The Conduction Figure of Merit (in siemens-coulomb-farads) is equal to the inverse of the on resistance (in ohms) times the gate charge (in coulombs) times the output capacitance (in farads)"
  - Where:
    - The result CFOM is expressed in units of SCF (siemens-coulomb-farad). As lower values are better, perhaps the metric would be better termed Conduction Figure of Malady.
    - RDS(ON) is the on resistance of the device, in ohms
    - Qg is the gate charge, usually expressed in coulombs (C)
    - C(OSS) is the output capacitance of the device, in farads (F)
      - However, as the figure is usually provided in the device datasheet as a value expressed in picofarads (pF), first divide the value in pF by 1,000,000,000,000 (1 trillion).
- Effective Capacitance
  - Mathjax: $$C_{EFF} = \frac{Q_{OSS}}{V_{DS}}$$
  - Unicode: C(EFF) = Q(OSS) ÷ V(DS)
  - English: "The effective output capacitance (in farads) is equal to the output charge (in coulombs) divided by the drain-to-source voltage (in volts)"
  - Where:
    - C(EFF) is the effective output capacitance, in farads (F)
    - Q(OSS) is the output charge, in coulombs (C)
    - V(DS) is the drain-to-source voltage, in volts (V)
  - Provides a more accurate representation of the MOSFET's behavior at high frequencies than the static C(OSS) value.
- Switching Loss Formula
  - Mathjax: $$P_{sw} \approx 0.5 \cdot V_{DS}^2 \cdot f \cdot C_{oss_{eff}}$$
  - Unicode: P(SW) ≈ 0.5 ⋅ V(DS)² ⋅ f ⋅ C(OSS_EFF)
  - English: "The switching loss (in watts) is approximately equal to half of the drain-to-source voltage (in volts) times the switching frequency (in hertz) times the effective output capacitance (in farads)"
  - Where:
    - P(SW) is the switching loss, in watts (W)
    - V(DS) is the drain-to-source voltage, in volts (V)
    - f is the switching frequency, in hertz (Hz)
    - C(OSS_EFF) is the effective output capacitance, in farads (F)
  - Therefore:
    - Losses increase with any of the three input variables, but this relationship is linear for frequency and effective output capacitance, wherease it is square (quadratic) with with voltage.
  - Estimates power dissipation for a given device and operating frequency.

Zener diode selection

Zener diodes act as shunt capacitors.

If zener diodes are selected to provide a safety for voltage peaks, note that their capacitance should be a concern, and is frequently not specified in the datasheet.

General rules of thumb:

Zener diodes are less effective at voltage regulation at higher frequencies.
- It may be necessary to use multiple diodes or to physically distribute them around the circuit depending upon your application.
Zener diodes degrade rapidly in function with thermal stress.
- For systems which may see use with high transmit duty cycles, especially high power applications with a single zener diode, consider thermals during layout and/or consider using multiple zener diodes.
Select the physically smallest zener diodes possible.
- Physically larger zener diodes will have greater capacitance.
Select the highest voltage rated zener diodes possible.
- Capacitance is inversely related to voltage rating, very broadly as follows:
  - Low voltage zeners (< 5V): 200-400 pF (generally to be avoided)
  - Medium voltage zeners (5-20V): 50-200 pF (potentially acceptable)
  - High voltage zeners (> 20V): 5-50 pF (significantly preferable)

For example:

VCC is 12V
Using a "~5x" rule of thumb we can expect peaks of >60V
However, certain components selected are only be available at 30V ratings
Consequently, we seek to add zener diode based protection.

In this case, zener based protection would be best selected with a voltage rating as close to 30V as possible. If we find there are multiple options available, we should select the physically smallest zener diode with that voltage rating.

In a different case where a negligible capacitance value zener diode is unavailable and you must use a higher capacitance component (eg. perhaps if the desired voltage rating is only available at larger physical size, or multiple capacitors will be in use creating a higher aggregate effective shunt capacitance), you might try to reduce the shunt capacitor C1 in value somewhat to compensate.

Architecture

python
Unix style command-line execution (for maximum compatibility, integration, and ease of automation)

It pulls in various information from external JSON files.

components.json: Contains common sizes for capacitors and inductors, and common zener diode voltage cutoffs used in generating proposed circuits.
transistors.json: Contains detailed properties of MOSFETs used as the Class-E Power Amplifier switching device as follows:

Name	Unit	Property
`lcsc`	N/A	Component ID at LCSC
`vdss_v`	V	Maximum drain-source voltage
`id_a`	A	Maximum drain current
`rds_on_o`	Ω	RDS(On) resistance
`ciss_pf`	pF	Input capacitance
`coss_pf`	pF	Output capacitance
`crss_pf`	pF	Reverse transfer capacitance
`vgs_th_v`	V	Gate threshold voltage

inductors.json: Contains inductor rating, tolerance, resistance, current and SRF information.

Name	Unit	Property
`inductance`	µH	Nominal rated inductance
`tolerance`	±%	Tolerance in percent above/below nominal rated value
`dcr_ohms`	Ω	DC resistance
`idc_ma`	mA	DC current limit
`srf_mhz`	MHz	Self-resonant frequency

frequency_bands.json: Contains frequency band information.

Name	Unit	Property
`start`	MHz	Lowest frequency at the beginning of the band
`end`	MHz	Highest frequency at the end of the band
`q`	-	Band quality factor (ratio of bandwidth to frequency)

Note on interface approach

The traditional unix style command-line interface is typically transactional, which is to say that a particular execution goal is explicitly specified (setting the program "mode" and often all related inputs) and then the program runs and returns its output. This is an approach which works well for specific tasks and is well suited to automation but begins to prove tedious when very high numbers of iterative executions may be required to explore a potential solution space.

Recognising that the design problem for Class-E power amplifiers is essentially a system of interdependant equations which may be approached with differing design goals, it would be feasible to turn to mathematics for some high level toolkits for exploring the solution space and determining potentially interesting areas which may be worth reporting. For example, using iterative methods.

However, because we're really just interested in our specific problem at this stage, we'll keep things transactional for simplicity.

There should be at least two levels of verbosity, a standard output-focused mode (default) and a learning-focused verbose mode which shows working.

Contributing

All contributions gladly accepted.
- Bugfixes
- Documentation
- New features
- New transistors
- Translations
- Unit tests
- Use cases

Python is not my usual language, so I'm very happy to take recommendations on library selection, unit testing, etc.

Name		Name	Last commit message	Last commit date
Latest commit History 135 Commits
images		images
theory		theory
LICENSE		LICENSE
README.md		README.md
classe2		classe2
components.json		components.json
frequency_bands.json		frequency_bands.json
inductors.json		inductors.json
mosfets.json		mosfets.json

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