Page 11 of 20 www.irf.com 09/09/08
THERMAL COMPENSATION FOR INDUCTOR DCR CURRENT
SENSING
The positive temperature coefficient of the DCR can
be compensated if R
T
varies inversely proportional to
the DCR. DCR of a copper coil, as a function of
temperature, is approximated by
))(1()()(
CuRR
TCRTTTDCRTDCR ⋅
+⋅= . (2)
T
R
is some reference temperature, usually 25 °C, and
TCR
Cu
is the resistive temperature coefficient of
copper, usually assumed to be 0.0039 near room
temperature. Note that equation 2 is linearly
increasing with temperature and has an offset of
DCR(T
R
) at the reference temperature.
If R
T
incorporates a negative temperature coefficient
thermistor then temperature effects of DCR can be
minimized. Consider a circuit of two resistors and a
thermistor as shown below.
Rs
RthRp
Figure 3 R
T
Network
If Rth is an NTC thermistor then the value of the
network will decrease as temperature increases.
Unfortunately, most thermistors exhibit far more
variation with temperature than copper wire. One
equation used to model thermistors is
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−⋅
⋅=
0
11
0
)()(
TT
thth
eTRTR
β
(3)
where R
th
(T) is the thermistor resistance at some
temperature T, R
th
(T
0
) is the thermistor resistance at
the reference temperature T
0
, and is the material
constant provided by the thermistor manufacturer.
Degrees Kelvin are used in equation 3. If R
S
is large
and R
P
is small, the curvature of the effective network
resistance can be reduced from the curvature of the
thermistor alone. Although the exponential equation 3
can never compensate linear equation 2 at all
temperatures, a spreadsheet can be constructed to
minimize error over the temperature interval of
interest. The resistance R
T
of the network shown as a
function of temperature is
)(
+
+=)(
TR
1
R
1
1
RTR
thp
sT
(4)
using R
th
(T) from equation 3.
Equation 1 of the last section may be rewritten as a
new function of temperature using equations 2 and 4
as follows:
()
)(
+
⋅
)(
=)(
TDCR
RR
TR
V
TI
2CS1CS
T
IG
FS
. (5)
With Rs and Rp as additional free variables, use a
spreadsheet to solve equation 5 for the desired full
scale current while minimizing the I
FS
(T) variation
over temperature.
