An Off-Nominal Class E Amplifier—Design Oriented Analysis

Abstract

This paper analyses an off-nominal Class E ZVS amplifier, with an R.F. supply choke and the switch on-duty ratio D = 0.5, to identify conditions for its high-frequency and high-efficiency operation. Simple user-friendly analytical expressions for the essential parameters of the amplifier have been derived and subsequently validated by simulation and experimental results. It has been proven that the off-nominal amplifier can be optimized to outperform the nominal Class E amplifier with respect to crucial parameters such as, e.g., power efficiency or the operating frequency in some applications.

1. Introduction

Class E power amplifiers find applications in wireless communication systems as high-efficiency, constant-envelop-signal amplifiers or linearised amplifiers (EER, outphasing method or Doherty configuration) [1,2,3]. The high efficiency of the Class E PA has inspired its use in many other applications, such as: industrial induction heaters, dc/dc converters, plasma generators, wireless power transmission systems (WPT) supplying energy to RFID tags, medical implants and endoscopic capsules, etc. [4,5,6,7,8,9,10,11]. Such a wide range of applications often requires Class E amplifiers to be optimized to operate efficiently in the given conditions. A Class E amplifier is typically designed to operate in the nominal or optimum mode of operation with its switch commutating in ZVS and ZVDS (zero-voltage derivative switching) conditions [11,12]. The mode offers high efficiency, but it can be troublesome to adapt its parameters to requirements relating to its application. Matching circuits (LC networks or transformers) often have to be used to obtain the required output power for the specified load resistance and the dc supply voltage [11,13]. This increases the cost and complexity of the amplifier circuit. In addition, it can create new problems, such as circuit mistuning caused by the component instability in a high-Q LC matching network. Moreover, at a high frequency of operation the nominal mode in the Class E amplifier can be impossible to achieve if a transistor switch with a too large output capacitance is used. In the off-nominal (also called sub-optimum [11,14,15,16,17] or variable slope [16]) mode of operation of the Class E amplifier, the transistor switch commutates in ZVS conditions only. Then, there is a greater degree of freedom in designing and optimizing the amplifier parameters [14,18]. In this amplifier, its load resistance and output power can have their values designated independent of each other in a certain range. This helps eliminate the matching circuit [18]. Moreover, the maximum operating frequency of the off-nominal amplifier can be made much higher than the nominal one. The off-nominal Class E amplifier can be also optimized with respect to other parameters, such as power efficiency [19], power loss in the switch or maximum peak voltage/current in the switch, etc.

The off-nominal Class E amplifier with an R.F. supply choke has been analysed in several publications [12,14,15,16,18,19,20,21], with studies concentrating on selected aspects of the amplifier operation. In an earlier study [12], basic analytical relationships for sub-optimum Class E amplifiers were first given. By extending the analysis from this [12], a general expression for the maximum operating frequency for the off-nominal amplifier was obtained in a subsequent study [20]. A more general analysis presented in two different studies [14,15] demonstrated that peak switch voltage and current can be used as designated parameters in the off-nominal amplifier design. An extensive analysis of the amplifier with any off-duty ratio presented in another study [19] was based on numerical solving of the set of complex equations. It was shown that power efficiency in the off-nominal amplifier can be improved if compared to the nominal circuit, but at the expense of the significant reduction of output power. In another study [21], the analysis of the off-nominal Class E amplifier concerned output power/voltage control by frequency or reactance regulation. Power losses in the switch in the off-nominal amplifier were analysed in another work [18], but other aspects of the circuit operation were not considered.

The presented paper is a significant expansion of preliminary research and ideas published in an earlier paper [18]. In this paper, results of the analysis for the off-nominal Class E amplifier with an R.F. supply choke and on-duty ratio D = 0.5 are discussed with respect to practical application issues. Simple analytical design-oriented equations for power losses in the switch and reactive components; peak switch voltage and current; the maximum operating frequency as well as the efficiency of the amplifier have been derived. The analysis has proven that the maximum operating frequency of the off-nominal Class E amplifier can be made even a few times higher than the maximum operating frequency of the nominal/optimum Class E amplifier for both the same output power and dc supply voltage. This means that transistor switches with higher output capacitance can be used at a high operating frequency in the off-nominal amplifier. A method of reducing conduction losses in the transistor switch in the off-nominal amplifier, with designated output power by increasing its dc supply voltage, is also described and its limitations are discussed. A user-friendly design procedure for the off-nominal amplifier with D = 0.5 is proposed and illustrated with two design examples. Design Example I demonstrates the design of the off-nominal amplifier operating at a frequency that is too high for the nominal mode due to high output capacitance of the transistor switch. Design Example II shows how to reduce conduction losses in the switch by increasing the dc supply voltage of the amplifier for the given output power and load resistance. The design procedure used in the examples was validated by a simulation of the 30 W/13.56 MHz amplifier and measurements of 50 W/140 kHz experimental amplifier for Design Example I and Design Example II, respectively.

2. Circuit Analysis

A diagram of the analysed Class E amplifier is shown in Figure 1a. The amplifier is analysed for the following assumptions:

(1)

Transistor T is an ideal switch operating at the constant frequency f with zero switching times and normalized on-duty ratio D = 0.5;

(2)

All reactive components are linear and lossless, and choke L_CH inductance is large enough to neglect its ac component;

(3)

The equation i_O(ωt) = I_O·sin(ωt + φ) is a sinusoidal output current with amplitude I_O and phase φ;

(4)

At the switch turn-on instant the switch voltage v_S(ωt = 2π) = 0 and its derivative is negative dv_S(ωt = 2π)/dωt ≤ 0.

By modifying values of the components C₁, L_SR, C_SR and R_O, a nominal Class E amplifier can become an off-nominal amplifier. Hence, it significantly simplifies the analytical description of the off-nominal amplifier if its parameters are normalised to respective parameters of the nominal reference amplifier. Then, parameters of the off-nominal amplifier can be found from simple closed-form equations. This approach allows the designer to avoid using troublesome design data tables as proposed in one study [14], or retrieving some necessary parameters from a plot [15]. For both amplifiers it is also assumed that their operating frequencies f are equal and their dc supply voltages are equal E_SUPoff = E_SUPnom.

From the set of equations derived in one paper [21] for the Class E amplifier with an R.F. supply choke and D = 0.5, the following relations for the off-nominal amplifier are obtained:

$$x_{SRoff}=\frac{\pi \left({\pi }^2+4\right)\left(4+\left({\pi }^2-8\right)p_O^2\right)}{16\left(4+{\pi }^2{p}_O^2\right)}$$

(1)

$$r_O=\frac{\left({\pi }^2+4\right)p_O}{{\pi }^2{p}_O^2+4}$$

(2)

$$x_{SRoff}=\left(X_{LSR}-X_{CSR}\right)/R_{nomr}$$

(3)

$$p_O=P_{SUPoff}/P_{SUPnom}$$

(4)

$$r_O=R_{off}/R_{nomr}$$

(5)

where:

• x_SRoff—normalised resultant reactance of the L_SR–C_SR branch at the operating frequency f for the off-nominal amplifier;
• R_nomr—total load resistance of the reference nominal amplifier;
• r_O—total normalised load resistance R_off for the off-nominal amplifier;
• p_O—normalised dc supply power for the amplifier (p_O∈<0;1>, and for {p_O = 1, r_O = 1}-the amplifier operates in the nominal mode);
• P_SUPoff, P_SUPnom—dc supply power of the off-nominal and nominal Class E amplifiers, respectively.

As given by one study [11], in the nominal Class E amplifier operating with D = 0.5, its dc supply power P_SUPnom and load resistance R_nom is expressed as

$$P_{SUPnom}\cdot R_{nom}=\frac{8E_{SUPnom}^2}{{\pi }^2+4}.$$

(6)

By using Equations (2), (4) and (5) and taking into account that E_SUPnom = E_SUPoff, the Equation (6) is expressed as

$$\frac{P_{SUPoff}}{p_O}\times \frac{{\pi }^2{p}_O^2+4}{\left({\pi }^2+4\right)p_O}{R}_{off}=\frac{8E_{SUPoff}^2}{{\pi }^2+4}.$$

(7)

From (7), normalized dc supply power p_O equals

$$p_O=\frac{2}{\sqrt{8E_{SUPoff}^2/\left(P_{SUPoff}\cdot R_{off}\right)-{\pi }^2}}.$$

(8)

For the nominal reference amplifier, its total load resistance R_nomr is found from (4) and (6)

$$R_{nomr}=\frac{8E_{SUPoff}^2\cdot p_O}{\left({\pi }^2+4\right)P_{SUPoff}}.$$

(9)

Values of the reactive components for the off-nominal Class E amplifier from Figure 1 are obtained from equations for the nominal amplifier [11] and, therefore, for clarity they remain referenced to R_nomr

$$L_{SR}=Q_{SRnomr}\cdot R_{nomr}/2\pi f,$$

(10)

$$C_{SR}=1/2\pi f\left(Q_{SRnomr}-x_{SRoff}\right)\cdot R_{nomr},$$

(11)

$$C_1=\frac{8}{2{\pi }^{2}f\left({\pi }^2+4\right)R_{nomr}},$$

(12)

$$L_{CH}>7R_{nomr}/f.$$

(13)

where Q_SRnomr—loaded quality factor for the series branch L_SR–C_SR of the nominal reference amplifier.

Voltage v_S and current i_S waveforms in the transistor T are derived in the same way as for the nominal circuit [11] but expressed by means of introduced normalised parameters, and they are given as follows:

$$\frac{v_S(\omega t)}{E_{SUPoff}}=\left\{\begin{array}{cc}0,\hfill & \mathrm{for} 0\le \omega \mathrm{t}<\pi \hfill \\ \frac{\pi \cdot \sqrt{{\pi }^2+4}}{2}\sqrt{\frac{p_O}{r_O}}\times \hfill & \hfill \\ \left(\begin{array}{l}\frac{2(\omega t-\pi )}{\sqrt{{\pi }^2+4}}\sqrt{p_O\cdot r_O}\\ +\cos \left(\omega t+\phi \right)+\cos \phi \end{array}\right),& \mathrm{for} \pi \le \omega \mathrm{t}<2\pi \hfill \end{array}\right.$$

(14)

$$\phi =\pi -\arccos \left(\pi p_O/\sqrt{{\pi }^2{p}_O^2+4}\right),$$

(15)

$$\frac{i_S(\omega t)}{I_{SUPoff}}=\left\{\begin{array}{ll}1-\sqrt{\frac{{\pi }^2+4}{4p_O\cdot r_O}}\sin \left(\omega t+\phi \right),& \mathrm{for} 0\le \omega \mathrm{t}<\pi \\ 0,& \mathrm{for} \pi \le \omega \mathrm{t}<2\pi . \end{array}\right.$$

(16)

where φ—phase shift of output current i_O.

The waveform of the current i_C1 in shunt capacitor C₁ is given by:

$$\frac{i_{C1}(\omega t)}{I_{SUPoff}}=\left\{\begin{array}{cc}0,\hfill & \mathrm{for} 0\le \omega \mathrm{t}<\pi \hfill \\ 1-\sqrt{\frac{{\pi }^2+4}{4p_O\cdot r_O}}\sin \left(\omega t+\phi \right),\hfill & \mathrm{for} \pi \le \omega \mathrm{t}<2\pi .\hfill \end{array}\right.$$

(17)

Peak value V_SMAXoff of the switch voltage v_S occurs at ωt_VSMAX

$$\omega t_{VSMAX}=2\pi -\phi +\arcsin \left(2\sqrt{\frac{p_O\cdot r_O}{{\pi }^2+4}}\right).$$

(18)

Peak value I_SMAXoff of the switch current i_S equals to

$$I_{SMAXoff}=\frac{P_{SUPoff}}{E_{SUPoff}}\left(1+\frac{1}{2}\sqrt{\frac{{\pi }^2+4}{p_O\cdot r_O}}\right).$$

(19)

Power losses in transistor T are found, under assumption that they are small enough not to affect the waveforms [11,19]. Conduction power losses P_Condoff in the switch T for off-nominal operation are obtained by integrating the switch current i_S(ωt)

$$P_{Condoff}=\frac{r_{DSon}}{2\pi }{\displaystyle \underset{0}{\overset{\pi }{\int }}{i}_S^2\left(\omega t\right)d\omega t}={\left(\frac{P_{SUPoff}}{4E_{SUPoff}}\right)}^2\left(24+{\pi }^2+\frac{4}{p_O^2}\right)\cdot r_{DSon}$$

(20)

where r_DSon—on resistance of the transistor switch T.

Switching losses in transistor T at its turn-off instant are estimated, assuming that the switch current i_S falls linearly from its initial value i_S(π), charging capacitor C₁ during fall time t_f << T = 1/f [11,17]

$$P_{SWoff}=\frac{1}{2\pi }{\displaystyle \underset{\pi }{\overset{\pi +\omega t_f}{\int }}{i}_S{v}_{S}d\omega t}=\frac{p_O{\left(1+\frac{1}{p_O}\right)}^2}{4}\frac{P_{SUPoff}{\left(\omega t_f\right)}^2}{12}.$$

(21)

Power losses in the switch in the nominal amplifier are found from (20), (21) as the special case for p_O = 1 and P_SUPoff = P_SUPnom, which agrees with the estimation given in [11]

$$P_{Condnom}={\left(\frac{P_{SUPnom}}{E_{SUPnom}}\right)}^2\frac{28+{\pi }^2}{16}\cdot r_{DSon}$$

(22)

$$P_{SWnom}=\frac{P_{SUPnom}{\left(\omega t_f\right)}^2}{12}.$$

(23)

Turn-on switching losses in the transistor T are neglected as small, assuming that transistor T is turned on when v_S ≈ 0 (0 ≤ ωt < π) [11,19].

3. Analysis for Specific Conditions

Based on the presented general analytical description of the off-nominal Class E amplifier, its operation in specified conditions is analysed and relations defining the parameters of the amplifier are derived.

3.1. Equal dc Supply Voltages E_SUPoff = E_SUPnom

In the analysed conditions, the dc supply voltages of the off-nominal Class E amplifier and the nominal Class E reference amplifier are equal E_SUPoff = E_SUPnom. Off-nominal operation of the amplifier is obtained for any normalised dc power in the range p_O∈<0;1>. From (20), (22) normalised conduction losses in the switch p_CondoffE for the off-nominal amplifier are

$$p_{CondoffE}=\frac{P_{Condoff}}{P_{Condnom}}=p_O^2\frac{24+{\pi }^2+\frac{4}{p_O^2}}{28+{\pi }^2} \mathrm{for} E_{SUPoff}=E_{SUPnom}$$

(24)

The plot of conduction losses in the switch p_CondoffE vs. dc power p_O in Figure 2a shows that designing the off-nominal amplifier to operate with p_O < 1 (which also means decreased output power) reduces conduction losses in the switch. The drain efficiency η_DoffE for the off-nominal amplifier for E_SUPoff = E_SUPnom can be found from (9), (20) and (21) as

$$\begin{array}{l}{\eta }_{DoffE}=\frac{P_{SUPoff}-P_{Condoff}-P_{SWoff}}{P_{SUPoff}}=1-\frac{p_O}{2\left({\pi }^2+4\right)}\cdot \frac{r_{DSon}}{R_{nomr}}\left(24+{\pi }^2+\frac{4}{p_O^2}\right)\\ -\frac{p_O{\left(1+\frac{1}{p_O}\right)}^2}{4}\frac{{\left(\omega t_f\right)}^2}{12}. \end{array}$$

(25)

where R_nomr is found from (6).

By solving dη_DoffE/dp_O = 0, one can determine the value of normalised dc power p_O(η_DoffEMAX) for the maximum drain efficiency η_DoffEMAX for the given set of normalised switch on-resistance r_DSon/R_nom and normalised turn-off time ω·t_f as follows

$$p_O({\eta }_{DoffEMAX})=\sqrt{\frac{4\frac{r_{DSon}/R_{nomr}}{{\left(\omega \cdot t_f\right)}^2}+\frac{4+{\pi }^2}{24}}{\left(24+{\pi }^2\right)\frac{r_{DSon}/R_{nomr}}{{\left(\omega \cdot t_f\right)}^2}+\frac{4+{\pi }^2}{24}}}$$

(26)

The plots in Figure 2b demonstrate that the drain efficiency η_DoffE increases as p_O (along with output power) decreases, starting from its nominal value p_O = 1 (P_SUPnom = P_SUPoff). The maximum drain efficiency η_DoffEMAX occurs at approximately p_O(η_DoffEMAX) ≈ 0.35. If conduction losses in the switch T are significant, then the value of η_DoffEMAX can be made higher up to a few percent (6% for curve—3 in Figure 2b) beyond the drain efficiency of the nominal amplifier η_Dnom. For small values of p_O- > 0, the drain efficiency η_DoffE decreases because load resistance R_off becomes small (r_O- > 0 as p_O- > 0,see Figure 3a) and then power losses in the switch resistance r_DSon become significant if compared to the output power. When designing the amplifier, it is often necessary to take into account power losses in reactive components. Power loss in shunt capacitor C₁ can be estimated from (9), (15) and (17) as

$$P_{C1off}=\frac{r_{C1}}{2\pi }{\displaystyle \underset{\pi }{\overset{2\pi }{\int }}{i}_{C1}^2\left(\omega t\right)d\omega t}=P_{SUPoff}\frac{4+\left({\pi }^2-8\right)p_O^2}{2\left({\pi }^2+4\right)p_O}\cdot \frac{r_{C1}}{R_{nomr}}$$

(27)

where r_C1—series resistance of C₁.

Loss P_LSR in the series inductance L_SR and loss P_CSR in capacitor C_SR are caused by conducted output current i_O, and are found from Equations (2), (5) and (9) as

$$\begin{array}{l}{P}_{LSR}=\frac{I_O^2}{2}{r}_{LSR}=\frac{P_{SUPoff}\cdot {\eta }_{offE}\cdot \left({\pi }^2\cdot p_O^2+4\right)}{\left({\pi }^2+4\right)p_O}\frac{r_{LSR}}{R_{nomr}}\\ P_{CSR}=\frac{I_O^2}{2}{r}_{CSR}=\frac{P_{SUPoff}\cdot {\eta }_{offE}\cdot \left({\pi }^2\cdot p_O^2+4\right)}{\left({\pi }^2+4\right)p_O}\frac{r_{CSR}}{R_{nomr}}\end{array}$$

(28)

where η_offE—total efficiency for the off-nominal amplifier (includes switch losses and reactive component losses); r_LSR, r_CSR—series resistance of inductance L_SR and capacitor C_SR, respectively.

Power loss in R.F. supply choke L_CH results from dc supply current I_SUPoff, and using (9) it is equal to

$$P_{LCH}=I_{SUPoff}^2\cdot r_{LCH}=P_{SUPoff}\left(\frac{P_{SUPoff}}{E_{SUPoff}^2}\right)\cdot r_{LCH}=P_{SUPoff}\frac{8p_O}{{\pi }^2+4}\frac{r_{LCH}}{R_{nomr}}$$

(29)

Total efficiency η_offE for the off-nominal amplifier can be estimated as

$$\begin{array}{l}{\eta }_{offE}=\frac{P_{SUPoff}-P_{Condoff}-P_{SWoff}-P_{LCH}-P_{C1}-P_{LSR}-P_{CSR}}{P_{SUPoff}}\\ =\left(\begin{array}{l}1-\frac{p_O}{2\left({\pi }^2+4\right)}\cdot \frac{r_{DSon}}{R_{nomr}}\left(24+{\pi }^2+\frac{4}{p_O^2}\right)\\ -\frac{p_O{\left(1+\frac{1}{p_O}\right)}^2}{4}\frac{{\left(\omega \cdot t_f\right)}^2}{12}-\frac{8p_O}{{\pi }^2+4}\frac{r_{LCH}}{R_{nomr}}\\ -\frac{4+\left({\pi }^2-8\right)p_O^2}{2\left({\pi }^2+4\right)p_O}\cdot \frac{r_{C1}}{R_{nomr}}\end{array}\right)\times \frac{\left({\pi }^2+4\right)p_O}{\left({\pi }^2+4\right)p_O+\frac{\left({\pi }^2{p}_O^2+4\right)\left(r_{LSR}+r_{CSR}\right)}{R_{nomr}}}\end{array}$$

(30)

From the plot in Figure 3a, it is seen that the off-nominal amplifier can be designed to operate with p_O < 1 and load resistance R_off > R_nomr (r_O > 1) or R_off ≤ R_nomr (r_O ≤ 1). From derivative (2) dr_O/dp_O = 0, the maximum value of R_off is R_offMAX = (1/π + π/4) × R_nomr = 1.1037 × R_nomr for p_O = 2/π. To boost the efficiency of the off-nominal amplifier it can be designed to operate with load resistance R_off = R_nomr (r_O = 1) but at reduced output power (p_O < 1). By solving simple Equation (2) for r_O = 1, it is found that then normalised dc power is p_O = 4/π² ≈ 0.4053. The obtained value of p_O is consistent with the result (p_O = 0.4070 for r_O = 1) achieved in one study [19] in a much more laborious way by numerical solving of a set of complex equations. This demonstrates the efficacy of the here proposed analytical description of the amplifier. The plots of the normalised peak switch current I_SMAXoff/I_SMAXnom and normalised peak switch voltage V_SMAXoff/E_SUPoff vs. p_O are displayed in Figure 3b. From the plots it is seen that the peak switch voltage V_SMAXoff and peak switch current I_SMAXoff decrease as the output power is reduced, and their values are smaller than those for the nominal amplifier. The normalised slope s of the switch voltage v_S(ωt) at the switch turn-on instant (ωt = 2π) can be derived from (14) as s = (1/E_SUPoff) × dv_s/dωt = π × (p_O − 1). Its minimum value s = −π occurs for p_O = 0, and maximum s = 0 for p_O = 1 (optimum operation). Power-output capability defined as c_p = (I_SUPoff × E_SUPoff)/(I_SMAXoff × V_SMAXoff) can be found from (14), (15), (18) and (19). The plot of c_p versus p_O displayed in Figure 3a shows that c_p is quite high even when power p_O < 0.5 (c_p(p_O = 0.5) = 0.0865).

This is explained by peak switch current I_SMAXoff/I_SMAXnom decreasing as p_O becomes smaller (Figure 3b). The presented analysis demonstrates that by designing the off-nominal amplifier to operate with reduced dc power (p_O < 1) at E_SUPoff = E_SUPnom, its efficiency can be improved.

3.2. Equal dc Supply Voltages E_SUPoff = E_SUPnom and Equal dc Supply Power P_SUPoff = P_SUPnom

Here, it is assumed that dc supply voltages E_SUPoff = E_SUPnom = const. and dc supply power P_SUPoff = P_SUPnom = const. for both the off-nominal and the nominal amplifiers are equal, respectively. Thus, normalised conduction losses p_CondoffPE in the switch found from (20) and (22) are given by

$$p_{CondoffPE}=\frac{P_{Condoff}}{P_{Condnom}}=\frac{24+{\pi }^2+\frac{4}{p_O^2}}{28+{\pi }^2}.$$

(31)

The drain efficiency η_DoffPE for the off-nominal amplifier for E_SUPoff = E_SUPnom, P_SUPoff = P_SUPnom is estimated from (9), (20), (21) as

$${\eta }_{DoffPE}=\frac{P_{SUPoff}-P_{Condoff}-P_{SWoff}}{P_{SUPoff}}=1-\frac{1}{2\left({\pi }^2+4\right)}\cdot \frac{r_{DSon}}{R_{nom}}\left(24+{\pi }^2+\frac{4}{p_O^2}\right)-\frac{p_O{\left(1+\frac{1}{p_O}\right)}^2}{4}\frac{{\left(\omega \cdot t_f\right)}^2}{12}.$$

(32)

Since the off-nominal amplifier operates with constant dc supply power P_SUPoff = const. its parameters are presented vs. normalised load resistance R_off/R_nom, which from (5), (6) and (7) equals to

$$\frac{R_{off}}{R_{nom}}=r_O\cdot p_O.$$

(33)

In Figure 4a the plot of normalised conduction losses p_CondoffPE in the switch vs. load resistance R_off/R_nom shows that conduction losses in the switch increase as load resistance decreases. Similarly, the drain efficiency η_DoffPE is also reduced (Figure 4b) as normalised resistance R_off/R_nom decreases. This results from the assumption that P_SUPoff = const., which requires that any decrease of load resistance R_off/R_nom has to be compensated for by the increase of the amplitude I_O of the load current i_O to fulfil the assumption. Hence, the switch peak current I_SMAXoff (Figure 4a) also increases as R_off/R_nom is reduced, leading to higher losses in the switch. The plot of the normalised peak switch voltage V_SMAXoff/E_SUPoff vs. R_off/R_nom shows that the value of the maximum voltage across the switch decreases as load resistance is diminished. Hence, the switch peak current I_SMAXoff (Figure 4a) also increases as R_off/R_nom is reduced, leading to higher losses in the switch. The plot of the normalised peak switch voltage V_SMAXoff/E_SUPoff vs. R_off/R_nom shows that the value of the maximum voltage across the switch decreases as load resistance is diminished.

This is the effect of the increased value of capacitance C₁, which has to be increased to maintain P_SUPoff = const. when R_off is reduced (see (9), (12)).

A typical problem occurring while designing a nominal Class E amplifier for designated voltage E_SUPnom, dc power P_SUPnom and the operating frequency f is that the frequency f exceeds the maximum operating frequency f_MAXnom of the nominal amplifier (f > f_MAXnom). Above frequency f_MAXnom, shunt capacitance C₁ is not fully discharged prior to the switch T turn-on, which results in switching losses (non-ZVS) and reduced efficiency. For the nominal Class E amplifier (D = 0.5), the maximum operating frequency is found from (6) and (12) [12] as f_MAXnom = P_SUPnom/(2π²·C₁ × E_SUPnom²) ≈ 0.05066 × P_SUPnom/(C₁ × E_SUPnom²). The off-nominal Class E amplifier can operate in ZVS mode at frequencies that are higher than the maximum f_MAXnom. The maximum operating frequency f_MAXoff for the off-nominal amplifier (D = 0.5) is obtained from (9) and (12) as

$$f_{MAXoff}=\frac{P_{SUPoff}}{2{\pi }^2\cdot C_1\cdot E_{SUPoff}^2\cdot p_O}$$

(34)

For P_SUPoff = P_SUPnom, E_SUPoff = E_SUPnom, and an identical value of the switch shunt capacitance C₁, the ratio f_MAXnom/f_MAXoff of the maximum frequencies for off-nominal and nominal operation of the Class E amplifier can be expressed as

$$\frac{f_{MAXnom}}{f_{MAXoff}}=p_O \mathrm{for} \mathrm{D}=0.5$$

(35)

By selecting a small enough value of normalised dc power p_O < 1 the maximum operating frequency f_MAXoff can be increased even several times over the value of maximum frequency f_MAXnom of the nominal amplifier for the same values of output power, dc supply voltage and shunt capacitance C₁. However, the small value of p_O increases losses in the switch (31), decreasing the drain efficiency η_DoffPE (Figure 4b). A relation between the ratio f_MAXnom/f_MAXoff and the normalised conduction losses p_condoffPE can be found from (31) and (35) for P_SUPoff = P_SUPnom, E_SUPoff = E_SUPnom as

$$p_{CondPE}=\frac{P_{Condoff}}{P_{Condnom}}=\frac{24+{\pi }^2+4/{\left(\frac{f_{MAXnom}}{f_{MAXoff}}\right)}^2}{28+{\pi }^2}$$

(36)

Using (36) one can find out how the increase in the frequency f_MAXoff affects conduction losses in the switch T (switching losses are neglected for simplicity). From Figure 5a it is seen that, e.g., for f_MAXnom/f_MAXoff = 0.5 is p_condPE = 1.317, which means that conduction losses P_Condoff in the switch increase slower than the rise of the operating frequency f_MAXoff (two times). Thus, it is possible to design an efficient off-nominal amplifier to operate at a given P_SUPoff, E_SUPoff and a high frequency f_MAXoff>f_MAXnom if power losses in the switch are sufficiently small. This is displayed in Figure 5b by the plots of η_DoffPE vs. f_MAXoff/f_MAXnom. When designing the off-nominal amplifier to operate above f_MAXnom, it is also useful to estimate the required on-resistance r_DSon of the switch for the assumed drain efficiency η_DoffPE. A simplified estimation of the normalised on-resistance r_DSon/R_nom can be found from (32) by neglecting the usually small switching losses (P_SWoff = 0). Then using (35) one arrives at the expression

$$\frac{r_{DSon}}{R_{nom}}=\frac{2\left(1-{\eta }_{DoffPE}\right)\left({\pi }^2+4\right)}{24+{\pi }^2+4/p_O^2}$$

(37)

The equation (37) can be also used for the preliminary assessment of the attainable drain efficiency η_DoffPE for normalised frequency f_MAXnom/f_MAXoff for a switch T with known r_DSon.

3.3. Equal dc Supply Power P_SUPoff = P_SUPnom and Equal Load Resistance R_off = R_nom

In some applications Class E amplifiers are used as H.F. high-current drivers for low-resistance loads such as loop antennas (with low radiation resistance) or low-impedance power transmitting coils (e.g., RFID or an endoscopic transmitting coil). High currents in the amplifier circuit can result in high losses in its components. The transistor switch is usually most vulnerable because both switching and conduction losses occur in it. Typically, to reduce conduction losses in components a matching circuit is used that transforms low-resistance load to a higher value “seen” by the amplifier. This, however, increases the cost and complicates the amplifier circuit. In the off-nominal amplifier it is possible to reduce conduction losses in the transistor switch by increasing the dc supply voltage E_SUPoff. Then, the matching circuit may become unnecessary. To compare switch losses in the nominal and off-nominal Class E amplifiers, let us assume that both amplifiers are loaded with equal load resistance R_nom = R_off, have equal dc power P_SUPnom = P_SUPoff and dc supply voltage E_SUPoff>E_SUPnom. From (2), (7), (20) and (21)–(23), normalised conduction losses p_CondoffPR and normalised switching losses p_SWPR in the transistor T are found as

$$p_{CondoffPR}=\frac{P_{Condoff}}{P_{Condnom}}=\frac{4+{\pi }^2+24{\left(\frac{E_{SUPnom}}{E_{SUPoff}}\right)}^2}{28+{\pi }^2};for \frac{E_{SUPnom}}{E_{SUPoff}}\le 1$$

(38)

$$p_{SWPR}=\frac{P_{SWoff}}{P_{SWnom}}={\left(2+\sqrt{\frac{4+{\pi }^2}{{\left(\frac{E_{SUPnom}}{E_{SUPoff}}\right)}^2}-{\pi }^2}\right)}^2/8\sqrt{\frac{4+{\pi }^2}{{\left(\frac{E_{SUPnom}}{E_{SUPoff}}\right)}^2}-{\pi }^2}$$

(39)

The plot of normalised conduction losses p_CondoffPR vs. normalised dc supply voltage E_SUPnom/E_SUPoff is presented in Figure 6a. From the plot it can be seen that increasing the dc supply voltage E_SUPoff decreases the ratio E_SUPnom/E_SUPoff and it consequently reduces conduction losses P_Condoff in the transistor T if compared to transistor losses P_Condnom in the nominal amplifier. There is, however, a limit in reducing conduction losses in the switch T by this method. As the ratio (E_SUPnom/E_SUPoff) → 0, then the normalised conduction losses p_CondoffPR → (π² + 4)/(π² + 28) ≈ 0.3663. This is explained by the fact that then E_SUPoff → ∞, the dc supply current I_SUPoff → 0 and the conduction losses in transistor T result only from conducting output current i_O (then P_Condoff = I_O² × r_DSon/4). The case E_SUPoff → ∞ is of no practical use, but if one assumes, e.g.,p_CondoffPR = 0.5 (50% decrease of conduction losses in the switch T), then from (38) is E_SUPnom/E_SUPoff = 0.4594 and from (14) is V_SMAXoff/E_SUPoff = 3.178. Increasing E_SUPoff also causes switching losses p_SWPR to rise (39), but usually for t_f << T switching losses are much smaller than conduction losses.

From (2) and (25), the drain efficiency η_DoffPR of the off-nominal amplifier for P_SUPoff = P_SUPnom and R_off = R_nom is expressed as

$${\eta }_{DoffPR}=1-\frac{\left(4+{\pi }^2+24{\left(\raisebox{1ex}{$E_{SUPnom}$}\!\left/ \!\raisebox{-1ex}{$E_{SUPoff}$}\right.\right)}^2\right)}{2\left(4+{\pi }^2\right)}\frac{r_{DSon}}{R_{nom}}-\frac{{\left(2+\sqrt{\left(4+{\pi }^2\right)/{\left(\raisebox{1ex}{$E_{SUPnom}$}\!\left/ \!\raisebox{-1ex}{$E_{SUPoff}$}\right.\right)}^2-{\pi }^2}\right)}^2\times {\left(\omega t_f\right)}^2}{96\sqrt{\left(4+{\pi }^2\right)/{\left(\raisebox{1ex}{$E_{SUPnom}$}\!\left/ \!\raisebox{-1ex}{$E_{SUPoff}$}\right.\right)}^2-{\pi }^2}}$$

(40)

From (40) by neglecting switching losses one obtains an estimation of the normalized switch on-resistance r_DSon/R_nom for the assumed value of drain efficiency η_DoffPR.

$$\frac{r_{DSon}}{R_{nom}}=\frac{2\left(4+{\pi }^2\right)\left(1-{\eta }_{DoffPR}\right)}{4+24{\left(E_{SUPnom}/E_{SUPoff}\right)}^2+{\pi }^2}$$

(41)

The plots in Figure 6a,b demonstrate that increasing dc supply voltage E_SUPoff reduces conduction losses p_CondoffPR in the switch T, and it can improve drain efficiency η_DoffPR by 5–6%. However, the increase of E_SUPoff more than 3–4 times over E_SUPnom can lead to drop in the efficiency η_DoffPR because of switching losses in T (curve 3 in Figure 6b). The plots of peak switch current I_SMAXoff/I_SMAXnom and peak switch voltage V_SMAXoff/E_SUPoff displayed in Figure 6a show that they tend towards finite values I_SMAXoff/I_SMAXnom → (4 + π²)^0.5/(2 + (4 + π²)^0.5) ≈ 0.6505 and V_SMAXoff/E_SUPoff → π.

4. Design Examples, Simulations and Experimental Verification

4.1. Design Example I

To verify the efficacy of the presented analysis in the H.F. range, a practical example of an off-nominal Class E amplifier (as in Figure 1a) is designed and verified by simulation with LTspice. The simulation is carried out for two types of switches: (a) an ideal bi-directional switch T with zero switching times, on-resistance r_DSon = 16 mΩ and (b) a model of eGaN FET transistor EPC2016-Infineon (V_DSMAX = 100 V, r_DSon = 16 mΩ, estimated equivalent output capacitance of the transistor C_T = 340 pF). Calculated and simulated parameters of the amplifier are presented in

Table 1. Design example I—specification and comparison of results.

Design Example I Specification: ESUPoff = 24 V, PSUPoff = 30 W, foff = 13.56 MHz, QSRnomr = 10, rDSon = 16 mΩ, ωtf = 0.05
Parameter	Calculated CT = 0	Simulation with Ideal Switch CT = 0	Simulation with EPC2016 CT = 340 pF
C1ext = (C1−CT) [pF]	556	556	216
Switch Losses PToff = PCondoff + PSWoff	0.112	0.100 (PSWoff = 0)	0.120
Output Power PO [W]	30	31.56	33.56
Switch Peak Voltage VSMAXoff [V]	76.86	81.2	89.8
Drain Efficiency ηDoffPE	0.996	0.997	0.996
LCH [μH]	>2	5	10

.

The off-nominal amplifier is designed for the following specification: dc supply voltage E_SUPoff = 24 V, dc supply power P_SUPoff = 30 W (to simplify the design, power losses in reactive components are assumed small and neglected), operating frequency—f_off = 13.56 MHz, Q_SRnomr = 10—loaded quality factor, normalized fall-time of the drain current ωt_f = 0.05, drain efficiency η_DoffPE > 0.99. For the given data, the maximum operating frequency f_MAXnom of the nominal Class E amplifier is too low (from (34) f_MAXnom = f_MAXoff (p_O = 1) = 7.76 MHz). Sincef_MAXnom < f_off, the nominal Class E amplifier with D = 0.5 cannot be used in this design. It is, therefore, necessary to use the off-nominal amplifier. To design the off-nominal amplifier, the normalised dc power p_O has to be chosen. Its maximum value is p_OMAX = f_MAXnom/f_off = 7.76/13.56 = 0.5723, but the preferred value of p_O, ensuring high efficiency (Figure 2b), is p_O = 0.35. An advantage of using lower p_O is also that the shunt capacitance C₁ is increased, which reduces its nonlinearity resulting from transistor output capacitance C_T (C₁ = C_ext(linear) + C_T). Nonlinear C_T is responsible for the increased value of the peak switch voltage V_SMAXoff [22,23]. Values of the calculated basic parameters for the example I amplifier are given in

Table 2. Design example I—calculated values of the basic parameters.

Parameter	Value	Equations Used to Find the Parameter
Rnomr [Ω]	3.876	(9)
C1 [pF]	556	(12)
LSR [μH]	0.4549	(10)
xSroff	2.211	(1)
rO	0.9319	(2)
Roff [Ω]	3.612	(5)
CSR [pF]	388.7	(11)
LCH [μH]	>2	(13)
rDsonmax [mΩ]	46.2	(6), (37)
PCondoff [W]	0.104	(20)
PSwoff [mW]	8.1	(21)
ηDoffPE	0.996	(32)
VSMAXoff [V]	76.86	(14), (15), (18)
ISMAXoff [A]	5.326	(19)

. The calculated drain efficiency is η_DoffPE = 0.996, which is better than its preliminary assumed value of 0.99. This results from the lower r_DSon resistance of EPC2016. The maximum allowable value of the switch on-resistance r_Dsonmax for the assumed η_DoffPE > 0.99 is given in Table 2.

Simulated waveforms for the amplifier with the model of eGaN FET EPC2016 are depicted in Figure 7a. The gate driver of the transistor was a voltage source generating a rectangular waveform (D = 0.5), changing from 0 V to 5 V with the rise/fall times equal to 0.5 ns and its internal resistance R_DRV = 1.5 Ω. Higher than calculated values of simulated peak voltage V_SMAXoff of the switch and higher output power P_O result from the use of finite values of inductance L_CH and loaded quality factor Q_SRnomr. Moreover, in the simulation with the model of eGaN FET EPC2016, the nonlinearity of the transistor output capacitance C_T additionally increased the value of the peak drain voltage V_SMAXoff. The example I amplifier can operate at even a much higher frequency, but with increased power losses. Assuming drain efficiency η_DoffPE = 0.99, ωt_f = 0.05, r_DSon = 16 mΩ from (32) is p_O = 0.236, which results in f_off = f_MAXnom/p_O = 7.76 MHz/0.236 = 32.88 MHz and power losses P_Condoff = 0.165 W, P_SWoff = 0.132 W. Thus, the maximum operating frequency of the off-nominal Class E amplifier can be flexibly adjusted to the designer’s needs.

4.2. Design Example II

To verify the method of reducing conduction losses in the switch, an off-nominal Class E amplifier was designed, built and tested. The circuit specifications were as follows: f = 140 kHz, D = 0.5, output resistor R_O = 2 Ω, output power P_O = 50 W, assumed total efficiency of the amplifier (driving losses excluded) η_offassum = 0.95, drain efficiency η_DoffPR > 0.99, loaded quality factor Q_SRnomr = 8, dc supply power P_SUPoff = P_SUPnom = P_O/η_offassum = 50/0.95 = 52.63 W, total load resistance of the off-nominal amplifier R_off = R_nom = R_O/η_offassum= 2/0.95 = 2.105 Ω, fall time of the switch current t_f = 50 ns. If, for the given specifications, a nominal Class E amplifier (p_O = 1, r_O = 1)is used, then it has to operate with a dc supply voltage E_SUPnom = 13.86 V (6), losses: conduction P_Condnom = 0.410 W (22) and switching P_SWnom = 8.484 mW (23), as well as peak switch voltage V_SMAXnom = 49.37 V (14).

To reduce conduction losses in transistor T, a dc supply voltage for the off-nominal amplifier was arbitrarily chosen to be a standard value E_SUPoff = 24 V (it can be any value higher than E_SUPnom).

By comparing estimated values of conduction losses P_Condnom in the nominal amplifier and losses P_Condoff in the example II amplifier (

Table 3. Design example II—calculated values of the basic parameters.

Parameter	Value	Equations Used to Find the Parameter
pO	0.3551	(8)
Rnomr [Ω]	2.242	(9)
C1 [nF]	93.11	(12)
LSR [μH]	20.39	(10)
xSRoff	2.199	(1)
rO	0.9389	(2)
CSR [nF]	87.41	(11)
LCH [μH]	>112	(13)
rDSonmax [mΩ]	26.7	(41)
PCondoff [W]	0.237	(38)
PSWoff [mW]	10.96	(39)
ηDoffPR	0.996	(40)
VSMAXoff [V]	76.9	(14), (15), (18)
ISMAXoff [A]	9.265	(19)

), one can notice an over 42% reduction of conduction losses in the off-nominal amplifier.

The actual values of components used in the experimental amplifier were as follows: L_CH= 364.8 μH, L_SR = 20.63 μH, C_SR = 87.31 nF, C₁ = 93.03 nF, R_O = 2 Ω (RNP-50s/Nikkom), transistor T-IFRB4228 150 V/12 mΩ. To possibly reduce losses in reactive components, both inductive components were wound with Litz wire 270 × 0.071 (L_CH—38 turns, L_SR—13 turns)on custom gapped ETD44/22/15 3F3 ferrite cores and, as capacitors,C_SR and C₁ low-loss MLCC C0G TDK capacitors were used. The measured efficiency of the built circuit (excluding driver loss) was η_meas = 0.95 = η_offassum, which agrees well with the calculated value of efficiency η_offcalc = 0.954 obtained from (30) for r_C1 = r_CSR = 2 mΩ, r_LSR = 0.07 Ω and r_LCH = 0.1 Ω.

In Figure 7b, voltage waveforms in the experimental circuit are shown. Higher than predicted peak switch voltage V_SMAXoff resulted from: the finite value of the used choke L_CH inductance, as well as from distortion of output current i_O caused by a finite Q_SRnomr value. The schematic of the experimental circuit and the photograph of the actual amplifier are shown in Figure 8a,b, respectively.

5. Conclusions

In the paper, an off-nominal Class E amplifier with an R.F. supply choke and duty ratio D = 0.5 was analysed to assess its potential to operate with high efficiency at a high operating frequency. Simple closed-form analytical expressions have been derived to estimate essential parameters of the amplifier such as: power losses in the transistor switch, its maximum operating frequency, power efficiency, peak voltage across the switch, etc. The derived equations provide an effective, easy and flexible way to optimize the amplifier in respect to its various parameters, largely eliminating the need to use data tables, plots or numerical solvers to extract amplifier parameters.

By means of design examples, simulation and measurements it has been demonstrated that the off-nominal Class E amplifier can operate at a frequency much higher than the nominal Class E amplifier, and that conduction losses in the transistor switch can be effectively reduced by increasing the dc supply voltage of the amplifier. The obtained results can be useful in designing Class E amplifiers for applications in transmitters for wireless communication systems, in wireless power transmission systems supplying energy to RFID tags, sensors or medical instruments such as wireless endoscopic capsules, etc., as well as in resonant dc/dc converters or induction heaters.