Announcements

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Announcements

• Homework 2 due today for on-campus students.
  Off-campus students submit according to your own
  schedule.
• Homework 3 is posted and is due NEXT Friday for
  on-campus students (Friday Feb. 8)

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19.4 Load-dependent propertiesof 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
  doesnt!)

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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.
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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? lt Zi0 !
  The no-load transistor current is therefore
  greater than the short-circuit transistor current.

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Example Zi of LCC

• for f lt f m, Zi increases with increasing R
  .
• for f gt f m, Zi decreases with increasing R
  .
• for f fm, Zi constant for all R .
• at a given frequency f,  Zi is a monotonic
  function of R.
• Its not necessary to draw the entire plot just
  construct Zi0 and  Zi? .

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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.
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LCC example

• f gt f? ZVS occurs for all R
• f lt f0 ZCS occurs for all R
• f0 lt f lt f?, ZVS occurs for Rlt Rcrit, and ZCS
  occurs for Rgt 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.

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LCC example, continued
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.
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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

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Solve for the ellipse which meets requirements
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Calculations
The required short-circuit current can be found
by solving the elliptical output characteristic
for Isc
hence
Use the requirements to evaluate the above
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Solve for the open circuit transfer function

• The requirements imply that the inverter tank
  circuit have an open-circuit transfer function of

Note that Voc need not have been given as a
requirement, we can solve the elliptical
relationship, and therefore find Voc given any
two required operating points of ellipse. E.g.
Isc could have been a requirement instead of Voc
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Solve for matched load (magnitude of output
impedance )

• Matched load therefore occurs at the operating
  point

Hence the tank should be designed such that its
output impedance is
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Solving for the tank elementsto give required
Zo0 and Hinf

• Lets design an LCC tank network for this example

The impedances of the series and shunt branches
can be represented by the reactances
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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
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Analysis in terms of Xs and Xp
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Hinf
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Zo0
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Zo0
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Analysis in terms of Xs and Xp
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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
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Evaluate tank element values
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DiscussionChoice of series branch elements

• The series branch is comprised of two elements L
  and Cs, but there is only one design parameter
  Xs 733 ?. 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 ?
  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
mH
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Requirements met at one frequency
25
What if Cs infinity?
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DiscussionChoice of series branch elements

• The series branch is comprised of two elements L
  and Cs, but there is only one design parameter
  Xs 733 ?. 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 ?
  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
mH
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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 ?. Hence ZVS for
R lt 1466 ?, and the nominal operating point with
R 900 ? has ZVS.
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Rcrit
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Ellipse again with Rcrit, Rmatched, and
RnomShowing ZVS and ZCS
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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
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Extending ZVS range
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Extending ZVS range
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Extending ZVS range
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Discussion wrt ZVS and transistor current
scalingSeries and parallel tanks

• fs above resonance
• No-load transistor current 0
• ZVS
• fs below resonance
• No-load transistor current 0
• ZCS

• fs above resonance
• No-load transistor current greater than short
  circuit current
• ZVS
• fs below resonance but gt fm
• No-load transistor current greater than short
  circuit current
• ZCS for no-load ZVS for short-circuit
• fs lt fm
• No-load transistor current less than short
  circuit current
• ZCS for no-load ZVS for short-circuit

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Discussion wrt ZVS and transistor current
scalingLCC tank

• fs gt finf
• No-load transistor current greater than short
  circuit current
• ZVS
• fm lt fs lt finf
• No-load transistor current greater than short
  circuit current
• ZCS for no-load ZVS for short-circuit
• f0 lt fs lt fm
• No-load transistor current less than short
  circuit current
• ZCS for no-load ZVS for short-circuit
• fs lt f0
• No-load transistor current less than short
  circuit current
• ZCS