Non-Linear Average Value Modeling
of a Three-Phase Bridge With Losses
Naval Combat Survivability
October 21, 2001
S.D. Sudhoff
Purdue University
Abstract – This brief document sets forth the average value
model of a 3-phase bridge converter modulated with a voltage
source based control strategy. The model represents an
improvement over the standard model in that conduction
losses are included in a rigorous way. The chief assumption
of the model is that the current waveform is sinusoidal.
I. SEMICONDUCTOR MODEL
In order to include semiconductor and diode losses in the
NLAM model, it is assumed that the voltage drop of a turned
on and conducting active semiconductor switch and the
forward biased diodes be expressed as
v = v sw + rsw i
(1)
v = vd + rd i
(2)
and
d ag =
where
1
2
(1 + d as )
(4)
d as = d cos(θ ci − φ ) − d 3 cos(3(θ ci − φ ))
(5)
and where d , d 3 , and φ are constants in the steady-state.
Now an expression for the dc link current may be derived.
From circuit and fast-average definition considerations, the dc
current into the a-phase leg of the inverter, ia , dc may be
expressed
iˆa,dc = i ai d ag
(6)
The average a-phase current may thus be expressed
ia , dc =
respectively, where v is the forward voltage across the device
and i is the current through the device.
II. DERIVATION OF LOSS MODEL
The objective of this section is to establish an expression
for the average-value of the q- and d-axis voltage and dc
current given information about the q- and d-axis current and
the modulation.
To derive an NLAM model it is convenient to work in a
reference frame in which all of the current is in the positive qaxis. The will be referred to as the converter current reference
frame. The position of this reference frame will be denoted
θ ci . By the inverse transformation [1], the a-phase current
c
iai = iqi
cos θ ci
(3)
The next step in the analysis is to specify the duty-cycle
of the a-phase leg of the inverter. In particular, the a-phase
duty cycle (that is the relative time the upper switch of the aphase leg is on relative to the lower switch on a fast-average
value basis), which is denoted d ag , is given by
(7)
The limits of integration in (6) are arbitrary (except for the
selection of one period) but have been selected here to be
consistent with later parts of this work. Manipulating (3-7)
yields
i a,dc =
1 ci
d i qi cos(φ )
4
(8)
Multiplying (8) by 3 to obtain the total dc current yields
i dc =
our of the inverter, iai , may be expressed in the from in terms
c
of the q-axis current in the converter reference frame, iqi
, as
1 3π / ˆ2
∫ ia , dc (θ ci )dθ ci
2π −π / 2
3 ci
d i qi cos(φ )
4
(9)
In order to calculate an expression for the q- and d-axis
voltages, the most convenient way is to begin by finding the
fundamental component of the a-phase line-to-ground voltage
vag . The fast average of vag may be expressed as
vˆag +
vˆag = 
vˆag −
iai > 0
iai < 0
(10)
where
vˆ ag + = (v dc − v sw − rsw i ai )d ag + (−v d − rd i ai )(1 − d ag )
(11)
vˆ ag − = (v dc + v d − rd i ai )d ag + (v sw − rsw i ai )(1 − d ag )
(12)
and where vdc is the dc inverter voltage.
The fundamental component of v̂ag may be expressed as
vˆag
fund
= a1 cos θ ci + b1 sin θ ci
(13)
where
π
−π / 2
π /2
(14)
1 π /2
1 3π / 2
ˆ
b1 =
v
sin(
θ
)
d
θ
+
∫ ag +
∫ vˆag − sin(θ ci )dθ ci
ci
ci
π
π
−π / 2
π /2
(15)
In (13-14) the limits are chosen so to separate the
integrands into the regions where the a-phase current is
positive and negative, respectively.
Manipulation of (4),(5),(11),(12),(14) and (16) yields
1
2
a1 = (v dc − v sw + v d )d cos φ − (v d + v sw ) − L
π
2
4
1
 ci
d cos(φ ) − 15 d 3 cos(3φ ) (rsw − rd ) i qc
 (rsw + rd ) +
3π
2

(16)
and
1
b1 = (v dc + v d − v sw )d sin(φ ) − L
2
(17)
2(rsw − rd ) 
3
 ci
 d sin(φ ) − d 3 sin(3φ )  i qc +
3π
5


(
)
Since the fundamental component of the line-to-ground
voltage is equal to the fundamental component of the line-toneutral voltage, from (13) is follows that
ci
vqi
= a1
(18)
= b1
(19)
ci
vdi
v q*
vdc
v*
md* = d
vdc
which is the desired result.
III. DERIVATION OF MODULATION SIGNAL
Before applying (9), (18) and (19) it is necessary to know
the values of d , d 3 , and φ . To this end it is assumed that a
model input is the commanded modulation indices that are
define as:
(20)
(21)
Given the commanded modulation indices in the arbitrary
reference frame, it is convenient to calculate these quantities
in the converter current reference frame. This is readily
accomplished using the frame-to-frame transformation
mqci *  cos(θ ci − θ ) − sin(θ ci − θ )  mq* 
 ci *  = 
 * 
md   sin(θ ci − θ ) cos(θ ci − θ )  md 
1 π /2
1 3π / 2
a1 =
∫ vˆag + cos(θ ci )dθ ci +
∫ vˆag − cos(θ ci )dθ ci
π
mq* =
(22)
where θ ci is the position of the converter reference frame and
θ is the position of the arbitrary reference frame in which the
modulation signals are specified. Given the current in the
arbitrary reference frame, the angle θ ci − θ may be readily
expressed
θ ci − θ = angle(iqi − jidi )
(23)
where iqi and idi are the q- and d-axis currents in the same
frame of reference as in with the original modulation
commands mq* and md* are specified. As an aside note that
with this choice of reference frames
ci
2
2
iqi
= iqi
+ idi
(24)
Once the modulation indices in the converter current
reference frame are calculated using (20) and (21), the next
step is to calculate d , d 3 , and φ . To this end, it is convenient
to define the converter voltage reference frame θ cv such that
all the voltage (at least in the presence of ideal
semiconductors) is in the q-axis. The position of this
reference frame relative to the arbitrary reference frame is
readily expressed as
θ cv − θ = angle(mq* − jmd* )
(25)
and in this reference frame
mqcv* = mq*2 + md*2
(26)
mdcv* = 0
(27)
In terms of the converter voltage reference frame the aphase duty cycle may be expressed
d as = d * cos(θ cv ) − d 3* cos(3θ cv )
(28)
Comparing (23) to (25),
is chosen where
φ = θ ci − θ cv
x<a
a

bound( a, b, x) ≡  x a ≤ x ≤ b
b
x>b

(29)
Substitution of (23) and (25) into (29) yields
φ = angle(iqi − jidi ) − angle(mq* − jmd* )
(30)
Minimization of the objective function yields
The calculation of d and d 3 remains. The commanded
duty cycle magnitude is related to the modulation index by
d * = 2mqcv*
d 3* =
1 *
d
6
(32)
where d * is given by (31).
The final calculation of d and d 3 is a function of the
magnitude of the commanded duty cycle d * . Provided that
d* <
2
3
d = d* +
(33)
4
π
sin θ cv1 +
d = d*
(34)
d 3 = d 3*
(35)
and
In the event (33) becomes exceeded, then the a-phase
modulation signal is clipped. As an approximate means to
analyze this condition, an effective value of d and d 3 are
introduced so that the approximate artificial a-phase duty
cycle
d a , aprx = d cos(θ cv ) − d 3 cos(3θ cv )
(36)
d 3 = d 3* −
is a best fit to (28) after it has been bounded to have a
magnitude of less than one. To this end, the objective function
π /2
(bound(−1,1, d a (θ cv )) − d a,aprx (θ cv ))2 dθ cv
sin(2θ cv1 )(d 3* − d * ) +
1
3π
(39)
2 *
d 3θ cv1 + K
π
 (9d * − 2d 3* ) cos θ cv1 + (3d * − 2d 3* ) cos(3θ cv1 ) − K

 sin θ
cv1
 2d * cos(5θ ) − 4 − 8 cos(2θ )

cv1
cv1
 3

(40)
In (39) and (40), the angle θ cv1 is the value of θ cv where
the a-phase duty cycle becomes small enough that it is no
longer clipped. This angle may be found by setting (28) equal
to one. In particular, solving
d cos(θ cv1 ) − d 3 cos(3θ cv1 ) = 1
(41)
one solution to (41) in [0, π / 2] then the largest solution in that
range should be chosen.
It should be noted that there is one approximation
involved in (39-41). In particular, it is assumed here that the
a-phase duty cycle waveform is clipped at 1 until θ cv = θ cv1 at
which time the a-phase duty cycle is no longer clipped.
However there is a very narrow range of duty cycles d a is less
than unity initially, reaches the limit at 1, and then moves
away from the limit. Assuming d 3 is set to d / 6 , then the
range over which this occurs is from
2
3
∫0
π
yields a value for θ cv1 . In the event that there are more than
then the modulator operation is not saturated so
f (d , d 3 ) =
1
1 *
2
d 3 sin( 4θ cv1 ) − d *θ cv1
π
2π
(31)
Assuming third harmonic modulation is used, and the
amplitude of the third harmonic term is proportional to the
fundamental, and that the constant of proportionality is such
that the maximum possible voltage may be obtained without
overmodulating, we have that
(38)
(37)
<d <
6
5
(42)
Numerically, the range of d is from 1.16 to 1.2. Since this
range is fairly narrow, and the effect of the approximation is
surmised to be very small within that range, the effect has
been ignored herein.