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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
2. Obtain zero-voltage switching or zero-current switching
• Preferably, obtain these properties at all loads
• Could allow ZVS property to be lost at light load, if necessary
3. Minimize transistor currents and conduction losses
• To obtain good efficiency at light load, the transistor current should
scale proportionally to load current (in resonant converters, it often
doesn’t!)
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Topics of Discussion
Section 19.4
Inverter output i-v characteristics
Two theorems
• Dependence of transistor current on load current
• Dependence of zero-voltage/zero-current switching on load
resistance
• Simple, intuitive frequency-domain approach to design of resonant
converter
Examples and interpretation
• Series
• Parallel
• LCC
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Inverter output characteristics
Let H be the open-circuit (R)
transfer function:
and let Zo0 be the output impedance
(with vi short-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.
with
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Inverter output characteristics
General resonant inverter
output characteristics are
elliptical, of the form
This result is valid provided that (i) the resonant network is purely reactive,
and (ii) the load is purely resistive.
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Matching ellipse
to application requirements
Electronic ballast
Fundamentals of Power Electronics
Electrosurgical generator
5
Chapter 19: Resonant Conversion
Input impedance of the resonant tank network
Appendix C: Section C.4.4
Expressing the tank input impedance as a function of the load resistance R:
where
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
ZN and ZD
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
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Magnitude of the tank input impedance
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:
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
A Theorem relating transistor current variations
to load resistance R
Theorem 1: If the tank network is purely reactive, then its input impedance
|| Zi || is a monotonic function of the load resistance R.




So as the load resistance R varies from 0 to , the resonant network
input impedance || Zi || varies monotonically from the short-circuit value
|| Zi0 || to the open-circuit value || Zi ||.
The impedances || Zi || and || Zi0 || are easy to construct.
If you want to minimize the circulating tank currents at light load,
maximize || Zi ||.
Note: for many inverters, || Zi || < || Zi0 || ! The no-load transistor current
is therefore greater than the short-circuit transistor current.
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Proof of Theorem 1
Previously shown:
 Differentiate:
 Derivative has roots at:
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.
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Zi0 and Zi for 3 common inverters
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Example: || Zi || of LCC
• for f < f m, || Zi ||
increases with
increasing R .
• for f > f m, || Zi ||
decreases with
increasing R .
• at a given frequency f,
|| Zi || is a monotonic
function of R.
• It’s not necessary to
draw the entire plot: just
construct || Zi0 || and
|| Zi ||.
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Discussion: LCC
LCC example
|| Zi0 || and || Zi || both represent
series resonant impedances,
whose Bode diagrams are easily
constructed.
|| Zi0 || and || Zi || intersect at
frequency fm.
For f < fm
then || Zi0 || < || Zi || ; hence
transistor current decreases as
load current decreases
For f > fm
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
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Discussion —series and parallel
• No-load transistor current = 0, both above
and below resonance.
• ZCS below resonance, ZVS above
resonance
• Above resonance: no-load transistor current
is greater than short-circuit transistor
current. ZVS.
• Below resonance: no-load transistor current
is less than short-circuit current (for f <fm),
but determined by || Zi ||. ZCS.
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
A Theorem relating the ZVS/ZCS boundary to
load resistance R
Theorem 2: If the tank network is purely reactive, then the boundary between
zero-current switching and zero-voltage switching occurs when the load
resistance R is equal to the critical value Rcrit, given by
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.
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Proof of Theorem 2
Previously shown:
Note that Zi, Zo0, and Zo have zero
real parts. Hence,
If ZCS occurs when Zi is capacitive,
while ZVS occurs when Zi is
inductive, then the boundary is
determined by Zi = 0. Hence, the
critical load Rcrit is the resistance
which causes the imaginary part of Zi
to be zero:
Solution for Rcrit yields
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Discussion —Theorem 2





Again, Zi, Zi0, and Zo0 are pure imaginary quantities.
If Zi and Zi0 have the same phase (both inductive or both capacitive),
then there is no real solution for Rcrit.
Hence, if at a given frequency Zi and Zi0 are both capacitive, then ZCS
occurs for all loads. If Zi and Zi0 are both inductive, then ZVS occurs for
all loads.
If Zi and Zi0 have opposite phase (one is capacitive and the other is
inductive), then there is a real solution for Rcrit. The boundary between
ZVS and ZCS operation is then given by R = Rcrit.
Note that R = || Zo0 || corresponds to operation at matched load with
maximum output power. The boundary is expressed in terms of this
matched load impedance, and the ratio Zi / Zi0.
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
LCC example




f > f: ZVS occurs for all R
f < f0: ZCS occurs for all R
f0 < f < f, ZVS occurs for
R< Rcrit, and ZCS occurs for
R> Rcrit.
Note that R = || Zo0 ||
corresponds to operation at
matched load with maximum
output power. The boundary
is expressed in terms of this
matched load impedance,
and the ratio Zi / Zi0.
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
LCC example, continued
Typical dependence of Rcrit and matched-load
impedance || Zo0 || on frequency f, LCC
example.
Fundamentals of Power Electronics
Typical dependence of tank input impedance
phase vs. load R and frequency, LCC example.
19
Chapter 19: Resonant Conversion
19.4.4 Design Example
Select resonant tank elements to design a resonant inverter that meets the
following requirements:
• Switching frequency fs = 100 kHz
• Input voltage Vg = 160 V
• Inverter is capable of producing a peak open circuit output voltage
of 400 V
• Inverter can produce a nominal output of 150 Vrms at 25 W
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Preliminary calculations
The requirements imply that the inverter tank circuit have an open-circuit
transfer function of:
The required short-circuit current can be found by solving the elliptical
output characteristic for Isc:
hence
Use the requirements to evaluate the above:
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Matched load
Matched load therefore occurs at the operating point
Hence the tank should be designed such that its output impedance is
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Solving for the tank elements
Let’s design an LCC tank network for this example
The impedances of the series and shunt branches can be represented by
the reactances
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Analysis in terms of Xs and Xp
The transfer function is given by the voltage divider equation:
The output impedance is given by the parallel combination:
Solve for Xs and Xp:
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Evaluate tank element values
The capacitance Cp should therefore be chosen as follows:
The reactance of the series branch should be
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Discussion
Choice of series branch elements
The series branch is comprised of two elements L and Cs, but there is only
one design parameter: Xs = 733 Ω. Hence, there is an additional
degree of freedom, and one of the elements can be arbitrarily chosen.
This occurs because the requirements are specified at only one operating
frequency. Any choice of L and Cs, that satisfies Xs = 733 Ω will meet
the requirements, but the behavior at switching frequencies other than
100 kHz will differ.
Given a choice for Cs, L must be chosen according to:
For example, Cs = 3Cp = 3.2 nF leads to L = 1.96 µH
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Rcrit
For the LCC tank network chosen, Rcrit is determined by the parameters of
the output ellipse, i.e., by the specification of Vg, Voc, and Isc. Note that
Zo is equal to jXp. One can find the following expression for Rcrit:
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.
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Converter performance
For this design, the salient tank frequencies are
(note that fs is nearly equal to fm, so the
transistor current should be nearly
independent of load)
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
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion