**6. RF transmitter current harmonics and life expectancy**

The simplified schematic of the two module switch-mode outphasing transmitters is shown in **Figure 15** [35]. Two full-bridge converters, named leading and lagging, have their output voltages combined using their output transformers TX1 and TX2. The load is a resonant antenna consisting of the antenna inductor La, parallel-tuned capacitor Ca and resistor Ra representing antenna losses and defying antenna Q. The load impedance has its maximum at the operating frequency of the parallel resonance equal to Ra. For the higher harmonics of the output voltage, antenna impedance is capacitive and drops at higher frequency. Higher harmonics of the transmitter output current are limited by the output filters Lf and Cf. This filter is tuned to the series resonance at the operating frequency to introduce minimum output filter voltage drop which is proportional to the output current.

**Figure 15.** Switch-mode outphasing transmitter.

The transferring function transmitter output voltage to the antenna voltage is shown in **Figure 16**. The "saddle characteristic" has two poles reflecting two resonances at lower and higher frequencies than the operating frequency. The amplitude of those peaks depends upon the antenna and filter losses as shown for Q = 10 and Q = 100. The box at the operational frequency 0.5 MHz sets the acceptable system bandwidth which is necessary for correct restoring of the envelope of RF pulses used in NMR tools for the proton excitation. For RF pulses with Hann envelope with duration of 25us minimum bandwidth to correctly recover amplified antenna pulses, the bandwidth of the system output filter plus antenna should be 40 kHz minimum; with reliable margin for wide temperature range, it should be 80 kHz with flatness 5% or 0.5db (**Figure 17**).

Increase of the system bandwidth benefits the metrological parameters of the tool and accuracy in the replication of the shortest RF pulses, used for the hydrogen nuclei excitation, and makes possible to use the same filter for more than one frequency without the filter capacitor switched. But there is another drawback: the higher-frequency resonance (the right pole) amplifies higher harmonics of the output current due to the drop of the filter/antenna input impedance (**Figure 18**). The biggest harmonic in the 50% duty cycle module output voltage is the third harmonic, and the third harmonic of the output current may significantly exceed the amplitude of the fundamental harmonic. The smaller is output filter inductance the higher are values of the unwanted higher harmonics in the output current and the higher are conductive losses resulting in the component temperature rise. The output filter design is a compromise between the bandwidth and accuracy vs. losses and reliability. Higher harmonic elimination removes the cause of the excessive output current and significantly reduces power losses for the same number of switches [28].

Standard NMR excitation pulse sequence consists of the multiple identical pulses with the equal period and the first pulse with half duration and the same amplitude Vp =1000 V as

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The harmonics of the transmitter output current depend upon the voltage harmonics in the transmitter output signal. **Figure 20** shows the values of the first seven most important harmonics of the output current for transmitters based on the simplest two module Case A configurations [35], four module-enhanced transmitter Case B with eliminated the 3rd harmonic [30] and eight module transmitter Case C with eliminated the 3rd and 5th harmonics. Proper waveforms of the leading or lagging voltages marked as A, B and C are provided in **Figure** 

*out* per switch and

<sup>2</sup> ∗ *Rdson* (12)

**12**. Elimination of higher harmonics decreases the total output RMS current *I*

:

**Figure 18.** Transmitter output current of the third harmonic compared to the first one vs. filter inductance.

shown on **Figure 19**.

**Figure 17.** Bandwidth vs. filter inductance.

resulting conductive power dissipation *Ps*

*Ps* = *Psw* + *Iout*

To assess the advantage of the discussed harmonic elimination method, three 500 kHz transmitters operating identical loads (2uH, 52 nF output filter and 1uH, 100 nF, 90ohm, Q 30 antenna) and using the same number of power MOSFET switches (32 total) but different topologies and control signals corresponding to Case A, Case B and Case C configurations (**Figure 10**) were simulated for the standard for NMR pulse train with 25us and 50us Hann envelope, 1000Vmax and 20% duty cycle (**Figure 19**) [29].

**Figure 16.** Transferring function filter plus antenna.

**Figure 17.** Bandwidth vs. filter inductance.

The transferring function transmitter output voltage to the antenna voltage is shown in **Figure 16**. The "saddle characteristic" has two poles reflecting two resonances at lower and higher frequencies than the operating frequency. The amplitude of those peaks depends upon the antenna and filter losses as shown for Q = 10 and Q = 100. The box at the operational frequency 0.5 MHz sets the acceptable system bandwidth which is necessary for correct restoring of the envelope of RF pulses used in NMR tools for the proton excitation. For RF pulses with Hann envelope with duration of 25us minimum bandwidth to correctly recover amplified antenna pulses, the bandwidth of the system output filter plus antenna should be 40 kHz minimum; with reliable margin for

Increase of the system bandwidth benefits the metrological parameters of the tool and accuracy in the replication of the shortest RF pulses, used for the hydrogen nuclei excitation, and makes possible to use the same filter for more than one frequency without the filter capacitor switched. But there is another drawback: the higher-frequency resonance (the right pole) amplifies higher harmonics of the output current due to the drop of the filter/antenna input impedance (**Figure 18**). The biggest harmonic in the 50% duty cycle module output voltage is the third harmonic, and the third harmonic of the output current may significantly exceed the amplitude of the fundamental harmonic. The smaller is output filter inductance the higher are values of the unwanted higher harmonics in the output current and the higher are conductive losses resulting in the component temperature rise. The output filter design is a compromise between the bandwidth and accuracy vs. losses and reliability. Higher harmonic elimination removes the cause of the excessive output

wide temperature range, it should be 80 kHz with flatness 5% or 0.5db (**Figure 17**).

26 Power System Harmonics - Analysis, Effects and Mitigation Solutions for Power Quality Improvement

current and significantly reduces power losses for the same number of switches [28].

envelope, 1000Vmax and 20% duty cycle (**Figure 19**) [29].

**Figure 16.** Transferring function filter plus antenna.

To assess the advantage of the discussed harmonic elimination method, three 500 kHz transmitters operating identical loads (2uH, 52 nF output filter and 1uH, 100 nF, 90ohm, Q 30 antenna) and using the same number of power MOSFET switches (32 total) but different topologies and control signals corresponding to Case A, Case B and Case C configurations (**Figure 10**) were simulated for the standard for NMR pulse train with 25us and 50us Hann Standard NMR excitation pulse sequence consists of the multiple identical pulses with the equal period and the first pulse with half duration and the same amplitude Vp =1000 V as shown on **Figure 19**.

The harmonics of the transmitter output current depend upon the voltage harmonics in the transmitter output signal. **Figure 20** shows the values of the first seven most important harmonics of the output current for transmitters based on the simplest two module Case A configurations [35], four module-enhanced transmitter Case B with eliminated the 3rd harmonic [30] and eight module transmitter Case C with eliminated the 3rd and 5th harmonics. Proper waveforms of the leading or lagging voltages marked as A, B and C are provided in **Figure 12**. Elimination of higher harmonics decreases the total output RMS current *I out* per switch and resulting conductive power dissipation *Ps* :

$$\mathbf{P}\_s = \mathbf{P}\_{sw} + \mathbf{I}\_{out}^2 \ast \mathbf{R}\_{down} \tag{12}$$

**Figure 18.** Transmitter output current of the third harmonic compared to the first one vs. filter inductance.

**Figure 19.** Antenna voltage (Vp(t)) and RF pulses with Hann envelope.

where *Ps* is the power switch total power dissipation, *Psw* is the power of switching losses, *Iout* is the RMS value of switch current and *Rdson* is the switch resistance drain to source, which for silicon MOSFET is the function of the temperature and current [36].

Additional temperature rise *∆T<sup>j</sup>* of the switch die over the ambient temperature *TA* depends on the thermal resistance *R*th from the switch die to the ambient temperature *TA* and current *I out*:

$$T\_j = T\_A + \Delta T = T\_A + \mathbf{R}\_{th} \* \left(\mathbf{P}\_{sw} + \mathbf{I}\_{out}^2\right) \tag{13}$$

The die temperature *T<sup>j</sup>* affects the switch life expectancy *tv* according to the Arrhenius law:

$$\mathbf{t}\_p = \mathbf{C} \ast \mathbf{e} \overset{\mathbf{x}\_i}{\underset{i}{\mathbf{x}}} \tag{14}$$

where *tv*

Additional temperature rise *∆T<sup>j</sup>*

compared to the life expectancy at *TA*:

*AF* =

**Figure 21.** Life expectancy tv vs. transmitter topology.

to the spread in time switch operation [29].

**7. Conclusion**

is the switch life expectancy in time units, years or hours; *T* is the absolute tempera-

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by the accelerator factor (AF)

*TA*) (15)

ture; *EA* is the apparent activation energy in general depending on *T*, recommended value

decreases life expectancy at *T*<sup>j</sup>

Implementation of the sequential harmonic elimination decreases output current and related temperature rise, which results into the improvement of the transmitter reliability in terms of the life expectancy. The life expectancy (shown in **Figure 21**) changes from 1000 h at 175°C, taken as a reference point, down to 296 h for Case A, to 770 h for Case B and to 788 h for Case C. All transmitters use 32 power MOSFET switches; the difference is in the topology, control method and number of the additional RF output transformers, which easily fit the pressure

Switching from classic transmitter topology Case A to Case B increases life expectancy 2.6 times, from Case A to Case C—2.66 times. It is clear that Case B transmitter is the best solution in terms of reliability improvement vs. extra cost and complexity. The benefits include not only increases of the life expectancy but also increases of tool accuracy, improving EMI conditions in the confined space of the power train placed in the high-pressure housing due

The sequential harmonic elimination provides a simple method of removing higher harmonics from the output voltage of the multimodule multilevel converters operating with the fundamental switching frequency and using identical modules, generating 50% duty cycle output

*tv*(*<sup>T</sup>* \_\_\_*\_j*) *tv(TA)* = *e E\_\_\_A k* ( *\_\_1 TJ −\_\_\_1*

0.8 eV [37]; *C* is a constant and *k* is the Boltzmann constant [38].

housing and do not generate significant amount of heat.

**Figure 20.** Higher harmonic (blue) elimination decreases total output current (red).

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**Figure 21.** Life expectancy tv vs. transmitter topology.

where *tv* is the switch life expectancy in time units, years or hours; *T* is the absolute temperature; *EA* is the apparent activation energy in general depending on *T*, recommended value 0.8 eV [37]; *C* is a constant and *k* is the Boltzmann constant [38].

Additional temperature rise *∆T<sup>j</sup>* decreases life expectancy at *T*<sup>j</sup> by the accelerator factor (AF) compared to the life expectancy at *TA*:

$$AF = \frac{t\_{v(\mathbf{I}\_i)}}{t\_{v(\mathbf{I}\_j)}} = \mathcal{e}^{\frac{E}{k} \left(\frac{1}{T\_i} \frac{1}{T\_s}\right)}\tag{15}$$

Implementation of the sequential harmonic elimination decreases output current and related temperature rise, which results into the improvement of the transmitter reliability in terms of the life expectancy. The life expectancy (shown in **Figure 21**) changes from 1000 h at 175°C, taken as a reference point, down to 296 h for Case A, to 770 h for Case B and to 788 h for Case C. All transmitters use 32 power MOSFET switches; the difference is in the topology, control method and number of the additional RF output transformers, which easily fit the pressure housing and do not generate significant amount of heat.

Switching from classic transmitter topology Case A to Case B increases life expectancy 2.6 times, from Case A to Case C—2.66 times. It is clear that Case B transmitter is the best solution in terms of reliability improvement vs. extra cost and complexity. The benefits include not only increases of the life expectancy but also increases of tool accuracy, improving EMI conditions in the confined space of the power train placed in the high-pressure housing due to the spread in time switch operation [29].
