Passive Electronic Components
Lecture 7
Page 1 of 11
01-May-2015
Pulse loading of resistors
1. Resistive element temperature estimation.
2. Physical phenomena related to pulse loading.
3. Pulse-proof chip resistors.
Pulse loading of a resistor refers to short-time (commonly less than 1 second) voltage application to
the resistor terminals. Safe limit of instantaneous power dissipated by the resistor for a short time
may significantly exceed the rated steady state power dissipation. For example, steady-state power
rating of a standard 1.6mm  0.8mm (RR 1608M) thick-film chip resistor at a temperature below
70C is 0.1W. The same resistor may safely survive single pulses with power of 1W, 3W, 10W and
respectively 10ms, 1ms, 10s pulse duration.
Excessive power applied to a resistor may cause one or more of the following phenomena in
resistive element material:
 thermal decomposition,
 oxidation,
 electrical break-down,
 mechanical cracking.
1. Resistive element temperature estimation.
Let us split the span of possible pulse durations as the following:



Long-time pulses (from several milliseconds to several seconds). Significant part of
generated heat escapes from the resistor to ambiance during pulse application.
Maximum temperature in the resistor will depend mainly on heat transfer process from
resistor to ambiance.
Short-time pulses (from several nanoseconds to several microseconds). During pulse
application almost all generated heat remains in the resistive element because pulse
duration time is non-sufficient for significant amount of heat to escape out of resistive
element. By other words, resistive element may be approximately regarded as thermally
isolated (adiabatic process).
Medium-time pulses (several milliseconds). Heat transfer inside the resistor (for
example from film resistive element to ceramic substrate) cannot be ignored while
insignificant heat flow out of the resistor (to PCB) may be ignored.
1.1. Long-time pulses (from several milliseconds to several seconds).
Lumped resistor model will be used for its temperature estimation. Suppose that resistor is a
uniform solid body. Resistor itself and its relationship with the ambiance are characterized by
the following parameters:
c – specific heat capacity of its material [J/(kgK)],
m – mass of the resistor [kg],
Rt – thermal resistance between the resistor and the ambiance [K/W],
P – electrical power dissipated inside the resistor starting from the moment of time t = 0 [W],
T  T t  –temperature rise in the resistor [K]. It is the difference between resistor temperature and
ambient temperature and depends on time t.
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Lecture 7
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Thermal energy dissipated inside the resistor during short time dt is P  dt . According to the
definition of thermal resistance Rt  T P1 , where P1 is the power of heat escape outside the
resistor. Therefore, P1  T Rt . Thermal energy that escaped from the resistor during time dt
will be P1  dt  T Rt   dt . The balance of thermal energy ensures temperature change dT in
the resistor. Differential equation that represents the energy balance in the resistor will be:
cm  dT  P  dt 
1
T  dt
Rt
or
cm 
dT 1
 TP
dt Rt
with initial condition
T 0  0 .
The solution of the above differential equation is


t
T t   P  Rt 1  exp  
 cmRt




It is valid for all types of resistors: both having 2-dimensional resistive element (thick-film,
thin-film, foil) and resistors with 3-dimensional resistive element made of bulk material
(wirewound, metal strip, carbon composition).
When t   the above solution transforms into T t   P  Rt , i.e. into expression for
temperature rise in the case of steady-state power dissipation in the resistor (see plot below).
T t 
P  Rt


t
T t   P  Rt 1  exp  
cmR
t





1.2. Short-time pulses (from several nanoseconds to several microseconds).
Let us consider the case of very short pulse duration time t. In this case almost all generated heat
remains inside the resistive element (adiabatic process). For small t values the above solution of
differential equation may be linearized:
T t   T 0   t 
P
t
mc
Passive Electronic Components
Lecture 7
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The temperature rise in resistive element is linear function of pulse duration t. The slope of liner
function is expressed through pulse power P, resistive element mass m, specific heat capacity c
of resistive element material. The above equation is valid for all types of resistors: both having
2-dimensional resistive element (thick-film, thin-film, foil) and resistors with 3-dimensional
resistive element made of bulk material (wirewound, metal strip, carbon composition).
1.3. Medium-time pulses (several milliseconds).
In many cases heat capacity of 3-dimensional resistive element (wirewound, metal strip, carbon
composition) is the major part of heat capacity of the entire resistor. At that, heat transfer inside the
resistor may be neglected. Or by other words temperature rise in resistive element may be estimated
using the equation for short-time pulses.
Thermal capacity of 2-dimensional resistive element is negligible when compared to heat capacity
of the entire resistor. When pulse duration is in milliseconds range or longer there is significant
internal heat transfer between resistive element that generates the heat and substrate that absorbs the
heat. Therefore, the equation for short-time pulses that is based on adiabatic heating of resistive
element is not applicable.
Let us consider temperature rise distribution in flat resistor with 2-dimensional resistive element
loaded by medium duration pulse. Suppose that a flat resistor has a uniform (non-trimmed)
resistive film layer on the surface of its substrate. The substrate is commonly much thicker than the
resistive film. Assume that the resistor dissipates a square-wave (constant voltage) pulse. Assume
also that the pulse parameters (duration, power/energy) and thermo-mechanical properties of the
resistor substrate (thermal conductivity, heat capacity and density) are known. The temperature rise
distribution in the resistor is to be found as a function of coordinates and time.
Proposed mathematical model is a substrate with essentially infinite outline dimensions and
thickness (half-space substrate). It is topped by resistive film of thickness d that has uniform heat
generation capability over its volume (Fig.1). The uniformity of the heat source results in constant
temperature in each plane parallel to the surface of the infinite substrate. It means that the substrate
temperature rise may be represented as a function of two independent variables: time t and spatial
coordinate z: T  T( z, t ) . Axis Oz has its origin on external surface of the resistive layer, and is
normal to it. Assume that T( z,0)  0 (the initial temperature rise in each point of the substrate is
zero) and that heat generation starts at the time t  0 with constant rate. Temperature rise
distribution T( z , t ) has to be determined.
Heat source
layer
d
0
Substrate
z
Fig.1
It may be shown that in the case of transient heat conduction process [1, p.10]
Passive Electronic Components
Lecture 7
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
1 T
A
 2T  
 ,
a t
K
(1)
where:
t - time [s],
A - heat source power per unit volume [W/m3] (in general case it is a function of coordinates and
time),
a
K
- thermal diffusivity [m2/s],
c
K - thermal conductivity [W/(mK)],
 - density [kg/m3],
c - specific heat capacity [J/(kgK)].
The solution T( z, t ) of equation (1) that corresponds to the described condition is given in [1, p.80].
In the substrate volume (z > d) :
T( z, t ) 
2aAt   z  d
 f 
K   2 at
  zd
  f 
  2 at

.

where:
T( z , t ) – temperature rise, K;
K – thermal conductivity of the substrate material, W/(mK);
c – specific heat capacity of the substrate material, J/(kgK);
 – density of the substrate material, kg/m3;
d – thickness of the heat source layer, m;
A(t) – volumetric density of heat power production in the heat source layer, W/m3;
A(t )  A, t   ;
A(t )  0, t   .
a – thermal diffusivity of the substrate material, m2/s:
The function f( x) is the following:
f( x) 
1 
2

  1  2 x 2  erfc( x) 
 x  exp(  x 2 ).
4 




where:
x –dimensionless real variable.
erfc( x) – complementary error function:
erfc( x) 

 exp   d

2
2
x
(2)
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Lecture 7
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Assume that the thickness of the heat source layer is infinitesimal, so that d0. Then the solution
(2) can be simplified:
T( z , t ) 
2 at  w   z 2 
 z
 z
 
 exp  
 erfc 
K    4at  2 at
 2 at

.

(3)
Here
w – surface density of heat power generation in the two-dimensional heat source layer, W/m2.
Equation (3) may be presented in the following form:
T( z , t ) 
2
w at
 Θ(θ),
K



(4)

Θ(θ)  exp  θ 2    θ erfc θ ,
θ( z , t ) 
z
2 at
(5)
.
Suppose that z  0 . Then   0 , erfc 0  1 , 0  1 and the equation (4) will transform into
expression for the surface (resistive film) temperature as a function of time:
T( 0, t ) 
2
w at
.
K


(6)
It is evident from comparing (4) and (6), that function Θθ  Tz, t  T0, t  represents the relative
temperatures in the substrate with respect to the surface (resistive film) temperature T0, t  . As
shown in Fig.2, Θθ  is a descending function. Thus, at every instant of time maximum temperature
in the substrate is reached in its surface
z 0.
() 1
0.5
0
0
1
Fig. 2
2
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Lecture 7
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Fig. 2 demonstrates that almost all of the heat energy absorbed by the substrate is related to
0  θ  1 range. Replacement of the variable  with the variables z and t using (5), results in:
0  z  2 at .
(7)
The significance of the last result is that at any instant of time vast majority of the heat energy
absorbed by the substrate is contained in a substrate layer defined by (7). The remaining portion of
the infinitely thick substrate, 2 at  z   has negligible influence on the temperature distribution
in the layer (7). In other words, the relatively simple infinite model of the resistor is applicable to
the temperature analysis of film resistor loaded by pulse when substrate thickness h and pulse
duration  satisfy the following condition:
h  2 a ,
(8)
Assume, now, that a square-wave pulse of power P or energy E and duration time  is applied to a
real resistor with finite area S of its resistive element. Assume also that the resistor thickness
satisfies (8).
The power density w of two-dimensional heat source in the model (4) can be calculated as the
following:
w
P
E

.
S S
(9)
Suppose that heat transfer on direction parallel to resistor’s surface plane is negligible. Then the
resistor may be regarded as a portion of infinite “resistor” that dissipates the same pulse power per
unit of surface area. Substitution of (9) into (4) yields an expression for the distribution of the
temperature rise in the substrate in terms of the given parameters:
T( z,  ) 
P a
2
E
 Θ(θ) 

 Θ(θ).
 KS
 cS a
2

(10)
Substitution of (9) into (6) yields an expression for the distribution of the temperature rise in the
resistive film (z = 0) in terms of the given parameters:
T( 0, ) 
2P

2E


.
S
Kc S Kc  
(11)
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Lecture 7
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All the discussed models are systemized in the following table.
Type of resistor load
2-dimensional
resistive element
3-dimensional
resistive
element
Pt
T t  
mc
Short-time pulse
Medium-time pulse
Long-time pulse
Steady state
T( 0,  ) 
2P


S
Kc
T t  


t
T t   P  Rt 1  exp  
 cmRt

T t   P  Rt
Pt
mc



Comments
Heat remains inside resistive
element (no heat transfer to
ambiance)
Heat remains inside resistor
(no heat transfer to ambiance)
Heat transfer to ambiance is
significant
For comparison
2. Physical phenomena related to pulse loading.
2.1. Burnout of resistive film.
Suppose that chip resistor with bare and uniform (non-trimmed) thick-film resistive element (Fig. 3)
is loaded by square-wave pulse.
h
Fig.3
Suppose that resistor substrate is made of 96% alumina (typical substrate for thick-film resistor),
and, therefore, K = 24 W/(mK), c = 8.8102 J/(kgK), = 3.72103 kg/m3, a = 7.310-6 m2/s.
Suppose too that duration  of applied pulse satisfies (8) where h is resistor substrate thickness.
Some example values of  and respective minimum values of h calculated per (8) are presented in
the table below:
Minimum alumina substrate thickness
when solution (11) for infinite model is applicable
h, mm
, ms
1
3
10
It follows from (11) that
0.17
0.30
0.54
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P
T( 0,  )  S
Kc

.
2

(12)
Functions P  or E  are common characteristics of resistor’s withstandability to pulse load.
They may be found in resistor datasheets and standards. Suppose that T(0,) = 250C, S = 1.7 mm2
(1206 size of chip resistor). The results of calculations per (12) are presented in Fig.4. Experimental
verification of relationship (12) was performed using 1206 non-trimmed chip resistors. The resistors
were loaded by 10,000 square-wave pulses characterized by selected pair of P,  values at +70C
ambient temperature. Period of pulses was selected to keep average power much lower than rated
power of the resistor. The plotted in Fig.4 experimental data represents maximum pulse power at
the given pulse duration that results in resistance shift less than 1%. Resistance shift in this case
results from thermal destruction (burnout) of resistive film. Basing on disclosed theory maximum
temperature of resistive film in the described experiment may be estimated as 250C + 70C =
320C (Temperature rise plus ambient temperature).
Pulse power, W
Single Pulse Load
of 1206 Non-Trimmed Chip Resistor
Calculated (T=250K)
Experimental
1.E-07
1.E-05
1.E-03
1.E-01
Pulse duration, s
Fig.4
3.2. Substrate cracking.
Failure analysis of the resistors failed after application of single pulse reveals burnt resistor film in
all the cases (Fig.5a). But sometimes substrate cracking is observed too (Fig. 5b).
The theory of substrate cracking phenomenon was developed in [2].
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Lecture 7
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Fringe area
a)
b)
Fig.5
In qualitative level it may be explained as the following. A short time of pulse load is insufficient
for appreciable heating up of substrate fringes – areas not covered by resistive film (see Fig.5).
Nevertheless, the fringes are forced to follow the thermally induced expansion of the central part of
the resistor. This situation produces significant tensile stresses in the fringes that may result in
substrate cracking. It was shown both theoretically and experimentally in [2] that cracking takes
place when pulse duration is about 1ms and longer.
3.3. Fatigue cracking of soldering joints.
There are many applications where chip resistors are subjected to a series of pulses. There is
specific effect related to multiple pulses impact - fatigue cracking of soldering joints.
Fig.6
It commonly happens when pulse duration is close to time of heat propagation all way through the
substrate. In this case as it follows from (8)
h  2 a ,

h2
.
4a
(13)
In this case mechanical “actuation” of the substrate is more efficient compared to longer or shorter
pulses: temperature gradient is high all way through substrate thickness as shown in Fig.2. Thermal
expansion of substrate material results in about spherical shape of its top and bottom surfaces (see
Fig.6). At that, PCB remains flat therefore high level of mechanical stress is induced in solder joints
Passive Electronic Components
Lecture 7
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of the resistor. A solder is known to be plastic material susceptible to fatigue fracture when
mechanical stress is applied repeatedly hundreds and thousands times.
Fatigue cracking of solder joins that results from ambient temperature changes and TCE mismatch
between chip component and PCB may develop for months and years because common duration of
ambient temperature cycle is several hours. In the case of pulse loading of resistor temperature
cycle duration may be seconds or less. That is why crack development in this case may happen after
several hours.
3. Pulse-proof chip resistors.
Pulse-proof chip resistor may withstand significantly higher electrical pulse load when compared to
standard chip resistor of the same size. It is achieved by the following.
3.1.Non-trimmed resistors. Resistive element without laser trimming has no current
concentrators (hot spots).
Chip with laser trim
Chip without laser trim
(Red lines – direction of electrical current)
Fig.7
3.2.Cylindrical chips (MELF. It follows from (12) that pulse power rating of film resistor is
proportional to surface area of its fusing element. Cylindrical chip (Fig.8a) has  times more
resistive element element area than flat chip resistor (Fig.8b) that has the same projection.
a) MELF chip resistor
b) Flat chip resistor
Fig.8
3.3.Symmetrical chip with 2 resistive elements. Chip resistor shown in Fig.9 has twice more
fusing element area than flat chip resistor having the same projection and respectively has
twice higher pulse power rating. Symmetrical construction prevents thermally induced
bending of chip (see Fig.6) and significantly increases withstandability of solder joints to
fatigue cracking.
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Lecture 7
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Resistive element 1
Substrate
Resistive element 2
Fig.9
Literature
1. Carslaw H.S., Jaeger J. C., Conduction of heat in solids. Oxford at the Clarendon Press,
1959.
2. Belman M., Fillion P., Safe pulse loading of thick-film chip resistors. CARTS 2002 (22nd
Capacitor and Resistor Technology Symposium) Proc., pp. 43-51.
3. Belman M., Kadim Y., and Akhtman L., Reliable Operation of Thick-Film Chip Resistors
under Pulsed Conditions. CARTS 2003 (23rd Capacitor and Resistor Technology
Symposium) Proc., pp. 117-123.
4. Belman M., Kadim Y., and Akhtman L., Surpassing Design of Surge Current Chip Resistor.
CARTS- EUROPE 2004 (18th European Passive Components Conference) Proc., pp. 182187.