### Analysis of Hot Swap Circuits with Foldback Current Limit

```Analysis of Hot Swap Circuits with Foldback Current Limit
A reliable analog circuit guarantees proper operation within the parametric tolerances of the
active controlling IC and passive components. For a Hot Swap circuit to perform properly,
the minimum and maximum values of a number of parameters must be gathered from the
data sheets of all components. From these, the Hot Swap circuit’s behavior in the face
loads are calculated for Hot Swap circuits with foldback current limit characteristics.
OVERVIEW
For a Hot Swap circuit, as shown in
Figure 1, the critical parameters are
operating voltage (VOPER) maximum current limit (ILIMIT) timer period (T) and the
maximum output voltage slew rate (SO),
which happens when the Hot Swap circuit
starts to operate with no-load. These
parameters are selected initially based on
and MOSFET’s drain-source on-resistance
(RDS(ON)) and its safe operating area (SOA).
ILIMIT • T
VOPER
For an R-C load it is easy to define a
surplus current, which is allowed for
the capacitive component of the load,
current limit is constant (ILIMIT = constant).
The voltage VFIX is lower than VOPER .
The current limit value as a function of
the output voltage, shown in Figure 1b,
is expressed by three separate equations
for different output voltage levels:
There are two common problems
that require solutions when charging a load with a Hot Swap circuit:
for a successful power-up transient.
ILOAD(VOUT ) = IINIT + • VOUT (t)
While slewing, the MOSFET acts as a source
follower so the maximum output voltage
slew rate, SO , is the same as the GATE pin
slew rate SG , and it is defined by the circuit
components (gate current divided by gate
to ground capacitance), and has a strong
influence on the power up transient. The
timer period, T, is the time allowed for
the Hot Swap circuit to operate in current limit mode before a fault is generated. A successful power-up transient
is one that does not generate a fault.
can be added in parallel to a resistive
load, RL , for a successful power-up
transient.
I
I
where = LIMIT INIT
VFIX VINIT
The problem of proper operation over the
full variation of the circuit parameters is
relatively simple for a circuit with constant
current limit, ILIMIT. The relationship of
parameters of a purely capacitive load at
the constant current limit for time T is:
This current limit value persists until the
output voltage reaches VLIMIT (point A),
after which the current limit increases linearly, which occurs after the output voltage reaches VFIX (point B). After point B the
A linear approximation of the foldback
characteristic is shown in Figure 1b. It is
used in all the following considerations.
The main points of this characteristic are:
•The operating voltage is VOPER .
•The initial current limit value, when the
output voltage 0 ≤ VOUT ≤ VINIT, is IINIT.
when
(1)
0 VOUT VINIT
when VINIT VOUT VFIX
(2)
when
(3)
VFIX VOUT VOPER
The value of the parameters marked on
Figure 1b, the timer period T value and the
slew rate SO are all known, with tolerance,
and used in the following solutions. For
some circuits, the slew rate SO is fast
enough that it has a negligible effect on
the inrush transient, and for others, it
is significant. The two loads above are
solved for these two cases of slew rate.
design ideas
Figure 1. (a) Major functional components of a
Hot Swap circuit and (b) linear approximation of
the foldback characteristic
(a)
(b)
RSENSE
Q1
CL
Hot Swap
CONTROLLER
CNO_FLT and CFLT. CNO_FLT is the maximum capacitive load with which the
circuit passes the power-up transient
without a fault for any possible combination of circuit parameters. CFLT is the
the power-up transient is always unsuccessful, and a fault is generated. From
these, the capacitive load range can be
divided into three groups. The powerup transient is successful for capacitive
loads from zero to CNO_FLT. Power-up is
unsuccessful for loads larger than CFLT.
The power-up transient is unpredictable for loads from the CNO_FLT to CFLT.
The following Hot Swap circuit parameters can be initially defined with tolerance: VINIT, VFIX , IINIT, ILIMIT, T.
The function ILOAD (VOUT) shown in
Figure 1b and equations (1–3) has three
distinct operating regions for current limit.
The slew rate, SO , can either cause the circuit to leave a current limit mode in any of
these operating regions (before the timer
period T expires), or it can have no effect
(i.e., SO is very fast). Each of these scenarios, transients, should by analyzed for any
Hot Swap circuit. Each is described below.
Some transients allow finding an analytical
expression for the worst-case parameters.
RL
A
IINIT
CGATE
CALCULATING THE MAXIMUM PURE
One important parameter to know for a
Hot Swap circuit is the maximum pure
capacitive load that a circuit can successfully power up into without a fault.
VOPER
B
ILIMIT
CTIMER
VINIT
However, the general or universal solution can be obtained in a numerical form.
Case 1: S O Never Limits Current
Suppose that natural slew rate SO is
fast enough to keep the operating point
in current limit mode in all three portions of the function ILOAD (VOUT).
In the first stage of the transient,
the current is IINIT and the output
voltage rises linearly from zero to
VINIT during the time t1. The capacitive load CLOAD1 can be expressed as:
IINIT t 1
VINIT
The output voltage as a function of time is:
dVOUT (t) IINIT + • VOUT (t)
=
dt
1
[IINIT +
• VOUT (t)] dt
(7)
which leads to the first order
differential equation
dVOUT (t)
IINIT
• VOUT (t)
=0
dt
IINIT t 1
(8)
The solution for Equation (8) is:
VOUT (t) =
IINIT
+ VINIT e
(
IINIT
(10)
The output voltage at time t2 is VFIX
VOUT (t 2 ) = VFIX =
=
IINIT
+ VINIT e
(
IINIT
(11)
The duration of interval t2 is
(5)
t2 =
(6)
(9)
Equation (8) describes VOUT(t) from
the start of stage 2, t = 0, to time
t2 , when the current limit reaches
its maximum value, ILIMIT.
t1
Substituting the expression (2)
in equation (5) produces:
VOUT (t) =
or
VOUT (0) = VINIT =
(4)
1
VOPER
with initial condition
In the second stage of the transient, the
current increases linearly from IINIT to
ILIMIT according to (2) as the output voltage increases from VINIT to VFIX . Time
t2 represents the duration of this stage,
completed when the output voltage
reaches VFIX . VFIX is usually set to (0.5 to
0.9)VOPER (it must be lower than VOPER) by
proper selection of the resistive divider.
VOUT (t) =
VFIX
VOUT
• ln
VFIX +
VINIT +
IINIT
IINIT
(12)
The third equation should describe how
CLOAD1 is charged with current ILIMIT during the interval t3 from VFIX to some
intermediate-level VINTERIM where the
There are two common problems that require solution when charging a load
with a Hot Swap circuit: (1) the maximum pure capacitive load for a successful
power-up transient; and (2) the maximum capacitive load, which could be added
in parallel to a resistive load, RL, for a successful power-up transient.
1A/DIV
10mA/DIV
VC(TIMER)
500mV/DIV
VOUT
2V/DIV
VOUT
2V/DIV
1A/DIV
VOUT
2V/DIV
VC(TIMER)
500mV/DIV
2ms/DIV
VC(TIMER)
500mV/DIV
2ms/DIV
2ms/DIV
Figure 2. (Case 1) Pure capacitive load. Operating
point leaves current limit mode in the third area,
where current limit is ILIMIT.
Figure 3. (Case 2) Pure capacitive load with a limited
SO (natural output voltage slew rate). Operating
point leaves current limit mode in the first area,
where current limit is IINIT.
Figure 4. (Case 4) Pure capacitive load with a limited
SO (natural output voltage slew rate). Operating
point leaves current limit mode in the second area,
where current limit rises linearly.
operation point leaves a current limit
mode because the MOSFET transconductance drops off in the triode region. When
this region is entered, successful start-up is
assured, but the point at which this region
begins is difficult to solve for because it
involves MOSFET parameters that may
not be available. This region is usually
small, so it makes sense to simplify the
description of this region with the assumption that CLOAD1 is charged from VFIX to
VOPER with ILIMIT. In this case the time:
Case 2: S O Limits Current at Point A
C
(V
V )
t 3 = LOAD1 OPER FIX
ILIMIT
(13)
according to (4)
t1 =
IINIT
(14)
Taking into account T = t1 + t2 + t3 , the
(15)
T
IINIT
=
VFIX +
VINIT 1
V
V
+ ln
+ OPER FIX
IINIT
IINIT
I
LIMIT
VINIT +
The minimum value of CLOAD1 from
equation (15) is obtained with TMIN ,
VINIT_MAX , IINIT_MIN , and VFIX_MAX . The
same parametric limits must be used
for expression of CNO_FLT that follows.
For the case where the slew rate
limit, SO , causes the operating point
to leave a current limit mode exactly
at point A in Figure 1, the capacitive
IINIT T
VINIT
(16)
IINIT_MIN , TMIN and VINIT_MAX .
Figure 2 shows this type of start-up
transient.
It should be noted that the maximum
output voltage slew rate in this case is
The output voltage slew rate range
for this case falls in the range:
IINIT
(16a)
IINIT
(15a)
It is constant while the operating point
resides in the current limit mode.
ILIMIT
(15b)
Figure 3 shows this transient.
design ideas
during the time t2_4 it becomes equal to SO.
The following method is recommended.
The output voltage rises from zero
to the VFIX during the time t1_4 , which
can be expressed from (16) as
VOUT
2V/DIV
1A/DIV
VOUT
2V/DIV
1A/DIV
VC(TIMER)
500mV/DIV
VC(TIMER)
500mV/DIV
2ms/DIV
Figure 5. R-C load without limitation for SO.
Operating point leaves current limit mode in the third
area, where current limit is ILIMIT.
Case 3: S O Limits Current at Point B
To produce an analytical expression of
CLOAD3 for the case when the operating point leaves a current limit mode at
point B, assume that it happens exactly
as the output voltage reaches VFIX .
The full duration of operation in current limit mode, T, includes two intervals t1 and t2 . According to (14) t1 is:
t 1_ 3 =
IINIT
(17)
In a second part of this transient, the output voltage changes according to expression (10) and the duration of interval t2 is:
t2 =
ln
VFIX +
VINIT +
IINIT
IINIT
and from (17) and (18)
(18)
t 1_ 4 =
The output voltage VOUT(t2_4) should be
equal to the voltage on the second stage
of the characteristic Figure 1 (equation 2):
2ms/DIV
Figure 6. R-C load with limited SO. Operating point
leaves current limit mode in the first area, where
current limit is IINIT.
T
IINIT
(19)
VFIX +
VINIT 1
+ ln
I
IINIT
VINIT + INIT
The output voltage slew rate range
for this case can be defined as:
IINIT
ILIMIT
VOUT (t 2_ 4 ) =
(19a)
(19b)
Case 4: S O Limits Current Between
Points A and B
(20)
Substituting ILOAD (t2_4) in (14) with
SO CLOAD4 , taking into an account
that t2_4 = T − t1_4 , and placing the
equal sign between (14) and (11)
forms the following equation:
From equation (19), the minimum
value of CLOAD3 is produced using TMIN ,
VINIT_MAX , IINIT_MIN , and VFIX_MAX .
IINIT
=
IINIT
+ VINIT e
(
(21)
This transcendental equation (21) with
unknown CLOAD4 can be solved with the
MATLAB, Mathematica) or LTspice.
Figure 4 demonstrates a successful
power-up transient, where the current is limited for a time less than the
timer period at maximum slew rate.
As a result, the transient begins in current limit mode and finishes the inrush
in slew rate limited operation.
If the output voltage slew rate SO is in the
range SLOAD1_MIN ≤ SO ≤ SLOAD1_MAX , current limiting stops before the output voltage reaches VFIX (before point B), meaning
that during the time t1_4 the output voltage
slew rate is a constant (due to IINIT) and
to the numerical solutions can serve as a basis for
a detailed optimization of Hot Swap solutions.
FOR SUCCESSFUL POWERUP TRANSIENT WITHOUT THE
LIMITATION OF THE OUTPUT
VOLTAGE SLEW RATE AND A
The solution to equation (23) describes
the output voltage from the beginning
of this stage to the time t2 , when
the output voltage reaches VFIX .
VOUT (t) =
If the passive load can be defined as a
resistive load RL and all Hot Swap circuit
parameters (VOPER , VFIX , IINIT, ILIMIT, T)
are known, then the maximum capacitive load should be found to ensure a
successful power-up transient. A successful power-up transient is completed only
after the current reaches ILIMIT, because
the slew rate is fast enough to stay in a
current limit for the entire transient.
The differential equation
for the first stage is:
CLR1
dVOUT (t) VOUT (t)
+
= IINIT
dt
RL
(22)
The equation (22) solution is:
(
VOUT (t) = IINIT • RL 1 e
t CLR1RL
)
(23)
At the end of the first stage (t1) the
output voltage is equal to VINIT and
IINIT • RL
t 1 = CLR1 • RLOAD • ln
IINIT • RL VINIT
(24)
dVOUT (t) VOUT (t)
+
= IINIT + VOUT (t) (25)
dt
RL
The first component on the left
side of equation (25) is a current
charging the capacitor; the second
one is a resistive current.
RL 1
t
CLR1RL
IINITRL
1 (26)
RL 1
IINITRL
+
IINITRL VINIT
I R
VFIX + INIT L
RL
RL 1
CLR1 = T ÷ +
ln
+ (31)
RL 1 V + IINITRL
INIT
RL 1
VFIX ILIMITRL
VOPER ILIMITRL
The time interval t2 could be expressed
as a function of CLOAD from (26) as:
Figure 5 shows measured results
for this case.
I R
VFIX + INIT L
RLCLR1
RL 1
t2 =
ln
RL 1 V + IINITRL
INIT
RL 1
FOR SUCCESSFUL POWER-UP
TRANSIENT WHEN CURRENT IS
LIMITED BY THE OUTPUT VOLTAGE
SLEW RATE AND A DEFINED
(27)
The differential equation
for the third stage is:
dV (t) V (t)
CLR1 OUT + OUT = ILIMIT
dt
RL
(28)
With the assumption that the output voltage slew rate, SO , does not affect the transient, it is possible to say that the output
voltage is changing according to (28) up to
VOPER . The solution for equation (28) is:
VOUT (t) =
The differential equation for the second
stage is:
CLR1
I R
= VINIT + INIT L e
RL 1
RL ln
( VFIX
ILIMITRL ) e
(
t CLR1RL )
+ILIMITRL
(29)
At the end of this stage:
The transient in Figure 6 illustrates
this case, when the operating point
leaves the current limit mode in the
first stage of current limit area.
VOUT (t 3 ) = VOPER
And time t3 equals:
t 3 = RLCLR1ln
VFIX ILIMITRL
VOPER ILIMITRL
The value of the output voltage slew
rate, SO , defines how long a Hot Swap
circuit should operate in the current
limit mode. This event (leaving the
current limit mode) can happen at any
moment of the three stages of the transient. For defined points A and B, during the third stage it is possible to use
equations from the previous section. For
any intermediate points, it is possible
to use the approach demonstrated in
“Case 4” with a transcendental equation.
(30)
Since T = t1 + t2 + t3 , CLOAD for this case is:
CONCLUSION