Fig. 1 (a) The main circuit topology of DC/AC inverter with
LCLLC resonant boost tank. (b) The simplified equivalent circuit of the
LCLLC resonant circuit.
The simplified ideal equivalent circuit of the resonant inverter is
shown in Fig. 1b [33-36]. Assumed that the isolated transformer with
ratio of 1:1 is an ideal component, which is considered as an ideal
inductance. Therefore, LCLC circuit with adding an excitation inductance
(LCLLC) was analyzed. A rectangular bipolar pulse was generated through
the full-bridge inverter, which can be expressed via Fourier
decomposition in the following equation:
The equation defines n as a natural number and ω as the
switching angular frequency of the full-bridge inverter, with a square
wave amplitude of A. According to this equation, Vs contains only odd harmonics, such as the 3rd, 5th, and 7th, etc. By
filtering out the higher harmonics and amplifying the fundamental
component, the desired sinusoidal AC voltage output is achieved.
Based on the circuit in Fig. 1b, according to the Kirchhoff’s law of
voltage and current, the output voltage can be listed as Eq. (2):
The voltage gain is derived in the following equation:
The Analysis of the LCLLC Circuit
Assumed that the Lt is approaching infinity, the
circuit can be considered as an LCLC topology and the voltage gain can
be simplified by the following equation:
As previously stated in the introduction, the bio-load by TTFields is a
capacitive impedance. The First Harmonic Approximation (FHA) analysis
offers a valuable reference for the selection of parameters for LCLC
circuit structures without significantly high-order harmonics [37].
A preliminary parameter selection with Lr = 6.8
μH, Lm = 22 μH, Cp =100
nF, Cw =10 nF is obtained based on parameter
selection method of reference [37]. The load impedance chooses 90 Ω.
Fig. 2a shows the voltage gain G with the variations of frequency
at the load (Zload ) of 90 Ω. As shown in Fig. 2a,
the attenuation multiplier at the third harmonic of the target frequency
(600 kHz) has already reached to -32.1 dB. The voltage gain shows a
bimodal distribution with frequency change:fp1 (167 kHz) and fp2 (287
kHz) are the two resonant gain frequencies. Fig. 2a shows that the LCLC
circuit designed in this study could output 200 kHz AC field with high
order harmonic.
Fig. 2a indicates that the voltage gain does not reach the maximum value
at the target frequency (200 kHz). In order to obtain higher voltage
gain, one way is that the target frequency could be shifted near the
first resonant frequency by parameter sweeps ofLr on the condition without high-order harmonics.
Fig. 2b shows the frequency response of the voltage gains with different
values of Lr . The Lr varies in the range form 0.1 μH to 100 μH. The first resonant frequency
is gradually moved left with increased value ofLr . For Lr of 1 μH, the
first resonant frequency is 241 kHz. However, the voltage gain of third
harmonics reaches -6.27 dB, which means that the high order harmonics
may occur. The first resonant frequency is shifted from 241 kHz to 46
kHz, when the values of Lr range from 1 μH to 100
μH.
To choose the proper Lr to make target frequency
shift to first resonant frequency, a least-square fitting is simply used
to process the simulated data between first resonant frequency and
values of Lr (R 2 is
0.995) . The mathematical function is as follow:
Where the variable F represents the first resonant frequency
(kHz). Based on the above mathematical function, the target frequency
(200 kHz) could be shifted to first resonant frequency whenLr is 4.1 μH.
Since the excitation inductance of transformer also involves the
resonant step-up tank (Eq. (3)), this paper also tries to adjust the
excitation inductance to shift the first resonant frequency, which is
shown in Fig. 2c The Lt also varies in the range
form 0.1 μH to 100 μH. The first resonant frequency is also gradually
moved left with increased value of Lt . The first
resonant frequency is shifted from 248 kHz to 171 kHz, when the values
of Lt range from 1 μH to 100 μH. The adjustment
of excitation inductance of transformer could also shift the targeted
frequency to the first resonant frequency. Moreover, compared with the
results on the adjustment of Lt (Fig. 2b) , the
first resonant frequency changes in a relatively narrow change.
A least-square fitting is also used to process the simulated data
between first resonant frequency and values ofLt . The mathematical function is as follow(R 2 is 0.986) :