19.4 Load-dependent properties of resonant converters. Resonant inverter design objectives: 1. Operate with a specified load characteristic and range of operating points With a nonlinear load, must properly match inverter output characteristic to load characteristic
and let Zo0 be the output impedance (with vishort-circuit). Then,
This result can be rearranged to obtain
The output voltage magnitude is:
Hence, at a given frequency, the output characteristic (i.e., the relation between ||vo|| and ||io||) of any resonant inverter of this class is elliptical.
This result is valid provided that (i)the resonant network is purely reactive, and (ii) the load is purely resistive.
Expressing the tank input impedance as a function of the load resistance R:
ZD is equal to the tank output impedance under the condition that the tank input source vs1 is open-circuited. ZD = Zo
ZN is equal to the tank output impedance under the condition that the tank input source vs1 is short-circuited. ZN = Zo0
If the tank network is purely reactive, then each of its impedances and transfer functions have zero real parts, and the tank input and output impedances are imaginary quantities. Hence, we can express the input impedance magnitude as follows:
So the resonant network input impedance is a monotonic function of R, over the range 0 < R < .
In the special case || Zi0 ||=|| Zi||,|| Zi || is independent of R.
then || Zi0|| < || Zi|| ; hence transistor current decreases as load current decreases
then || Zi0|| > || Zi|| ; hence transistor current increases as load current decreases, and transistor current is greater than or equal to short-circuit current for all R
It is assumed that zero-current switching (ZCS) occurs when the tank input impedance is capacitive in nature, while zero-voltage switching (ZVS) occurs when the tank is inductive in nature. This assumption gives a necessary but not sufficient condition for ZVS when significant semiconductor output capacitance is present.
Note that Zi, Zo0, and Zo have zero real parts. Hence,
If ZCS occurs when Ziis capacitive, while ZVS occurs when Ziis inductive, then the boundary is determined by Zi= 0. Hence, the critical load Rcrit is the resistance which causes the imaginary part of Zito be zero:
Solution for Rcrit yields
Typical dependence of Rcrit and matched-load impedance || Zo0 || on frequency f, LCC example.
Typical dependence of tank input impedance phase vs. load R and frequency, LCC example.
The required short-circuit current can be found by solving the elliptical output characteristic for Isc:
Use the requirements to evaluate the above:
Hence the tank should be designed such that its output impedance is
The impedances of the series and shunt branches can be represented by the reactances
The output impedance is given by the parallel combination:
Solve for Xs and Xp:
The reactance of the series branch should be
For example, Cs = 3Cp = 3.2 nF leads to L = 1.96 µH
Since Zo0 and H are determined uniquely by the operating point requirements, then Rcrit is also. Other, more complex tank circuits may have more degrees of freedom that allow Rcrit to be independently chosen.
Evaluation of the above equation leads to Rcrit = 1466 Ω. Hence ZVS for R < 1466 Ω, and the nominal operating point with R = 900 Ω has ZVS.
The open-circuit tank input impedance is
So when the load is open-circuited, the transistor current is
Similar calculations for a short-circuited load lead to