Download Easy Calculation Yields Load Transient Response

Easy Calculation Yields
Load Transient Response
By John Betten, Application Engineer and Member of Group
Technical Staff, and Robert Kollman, Distinguished Member
of Technical Staff, Texas Instruments, Dallas
A simple method can accurately calculate the
necessary amount of output capacitance and
bandwidth of a power converter to provide the
required voltage transient performance.
presents a simplified model of the open-loop power supply power stage and the resulting output impedance of a
typical filter. The buck power switching elements have been
replaced by a voltage source with zero output impedance.
The output impedance is determined by driving the output with a 1-A ac source and measuring the output voltage. Consequently, the output impedance at low frequency
is set by the output inductor’s equivalent series resistance
(ESR) and increases with the inductor impedance. The
impedance rises until the output inductor and capacitor
resonate. At this frequency, the magnitude of the impedance is determined by the Q of the circuit and can be many
times the impedance of the filter elements. Once past resonance, the output impedance drops with increasing frequency until the capacitor’s ESR is reached. Finally, at
higher frequencies, there is a parasitic or equivalent series
inductance (ESL) in the capacitor that causes the impedance to increase. From Fig. 1, the output-voltage variation
to a rapid load step can be quickly estimated, neglecting
any electronic circuits can rapidly shift
between a sleep state and full load operation, placing a heavy requirement
on the power converter to stabilize its
output voltage. An excessive voltage
variation could cause the electronics to malfunction, reset,
latch up or fail. It is typically the function of the power
converter’s output capacitance to provide output voltage
holdup during any heavy load steps. The amount of capacitance placed on the converter’s output often is miscalculated, increasing overall cost if too much capacitance is
used and compromising system performance if too little is
used. In addition, the effect of the output capacitance is
frequently overlooked, and too much effort is spent trying
to increase the converter’s bandwidth to improve the load
transient response.
Quite simply, the power supply’s voltage variation to a
load transient is its peak output impedance (in the frequency domain) times the change in load current. Fig. 1
M
E1
++
- -
LOUT
ResrL
4.7 uH
0.015
0
10
COUT
0
100 uF
-10
0.005
1 Aac +
0 Adc -
I1
dB Ohms
ResrC
Lesl
-20
LOUT
-30
COUT
2 nH
-40
0
0
Lesl
ResrL
-50
100 Hz
VDB (R8:2)
1.0 KHz
10 KHz
Frequency
100 KHz
ResrC
1.0 MHz
Fig. 1. Simplified voltage-mode buck power stage model and output impedance.
Power Electronics Technology
February 2005
40
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G1
10
COUT
100 uF
ResrC
0.005
1 Aac
0 Adc
+
-
0
I1
-10
dB Ohms
0
++
- -
TRANSIENT PERFORMANCE
0
-20
Cout
-30
Lesl
Lesl
-40
2 nH
ResrC
-50
100 Hz
1.0 KHz
0
10 KHz
100 KHz
Frequency
1.0 MHz
Fig. 2. Simplified current-mode buck power stage model and output impedance.
the effect of any feedback network. If the load step has a
rate of rise much greater than one divided by the filter resonant frequency, the filter characteristic impedance sets the
maximum impedance and the voltage variation in the open
circuit case is just Zout × Istep, where:
The step-load repetition rate may play an important role
and needs to be identified. If the pulse-load repetition rate
is equal to the resonance frequency of the output filter, then
the Q of the filter will make the output impedance much
greater than Zout defined in Equation 1 and a rather large
voltage variation can result. If the repetition rate is at a
rapid rate and is above the bandwidth of the power supply,
the ESL sets the maximum output impedance. However in
Zout = Lout/C out
(1)
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TRANSIENT PERFORMANCE
7
60
6
5
Peaking
dB Ohms and dB gain
40
Loop gain
0
4
3
2
1
Z OUT open loop
0
-40
0
20
40
60
80
100
Phase Margin
Z OUT closed loop
Fig. 4. Good phase margin reduces peaking at crossover.
-80
100 Hz
1.0 KHz
10 KHz
Frequency
100 KHz
have as a constant current source feeding the output filter
capacitor, negating its effect in the ac model. At low frequencies, the current-mode output impedance is much
higher than with voltage-mode control; however, at higher
frequencies, the impedances are identical.
Closing a feedback loop around the output voltage helps
lower the output impedance. From classical feedback
theory, the closed-loop output impedance is equal to the
open- loop output impedance divided by one plus the loop
gain. Fig. 3 shows this graphically. The output impedance
of a voltage-mode power supply is presented and has a
shape similar to Fig. 1. The closed-loop gain of the power
supply is shown as an integrator-type response and, in gen-
1.0 MHz
Fig. 3. Closing the loop significantly reduces output impedance.
a properly bypassed system, where distributed capacitance
usually supplies pulse currents to the load, this is usually
not the case.
Current-mode Control
Fig. 2 presents a simplified current-mode buck power
stage model and corresponding output impedance, assuming that the voltage loop is open. Note that in current-mode
control, the control loop makes the output inductor be-
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TRANSIENT PERFORMANCE
-20
5.05 V
-30
5.00 V
Voltage mode
dB Ohms
Current
mode
-40
Current mode
4.95 V
-50
4.90 V
Voltage mode
-60
4.85 V
-70
4.80 V
-80
100 Hz
1.0 KHz
10 KHz
100 KHz
4.75 V
19.9 ms
1.0 MHz
20.0 ms
20.1 ms
eral, has a -1 slope. The closed-loop output impedance also
is graphed. As expected, at low frequencies, substantial reductions in output impedance are seen. However, near the
crossover frequency, the closed-loop impedance approaches
the open-loop impedance and at much higher frequencies
they are identical.
An interesting effect can be seen in the closed-loop output impedance at the loop gain crossover frequency (where
loop gain crosses 0 dB or where loop gain equals 1). Again,
the closed-loop output impedance is set by the open-loop
impedance divided by the loop gain of 1 at an angle related
to the phase margin or simply:
Zoutclosedloop = Zoutopenloop /(
1-1(coss m + j*sins m ))
(2)
While not really apparent in Fig. 3 due to the logarithmic scale, there is a 1.15 scale factor in peaking in the closedloop-output impedance near the loop-gain crossover frequency due to a 55-degree phase margin. Fig. 4 presents
this graphically. Peaking at the crossover frequency can be
as little as 0.7 with 90 degree of phase margin or approaching infinity with 0-degree phase margin (but in reality, the
system goes unstable with 0-degree phase margin).
20.4 ms
Finally, at high frequency, the two output impedances converge as the improvement made by closing the loop wanes.
Fig. 6 shows the transient response of these two power
supplies to a 5-A load step. There are two distinct differences between the two. First, the current-mode supply has
a smaller voltage deviation than the voltage-mode supply.
This is due to the voltage-mode converter’s lower phase
margin and corresponding higher output impedance near
its crossover frequency. This impedance sets the initial transient voltage spike during a load step; hence, the voltage
spike is a little larger than seen with current-mode control.
The second difference is the voltage-mode supply recovers
much quicker than the current-mode supply. This is due
to the lower output impedance of the voltage mode at the
lower frequencies, which allows an output-voltage recovery time that is faster when compared to a similar design
using current-mode control.
Output Filter Design
Some simplifications can be made designing the output filter. If one neglects the peaking caused by less than
60 degrees of phase, the closed-loop output impedance is
approximately equal to the open-loop output impedance
at the crossover frequency. The transient response to a load
step can be calculated by multiplying the load step by the
filter output impedance at the crossover frequency. In both
voltage mode and current mode, this impedance usually is
set by the output capacitor. Given this fact, Fig. 7 provides
a graphical method of establishing loop bandwidth and
output capacitance requirements. In this figure, the capacitance is assumed to have no ESR. First the required closedloop output impedance is calculated from the equation
below for a given transient load. For example, if a 50-mV
voltage transient is desired for a 1-A pulse load, then the
required closed-loop output impedance is 50 mΩ.
Current- vs. Voltage-mode Control
Fig. 5 presents a comparison between current-mode control and voltage-mode control on the power supply’s closed
loop output impedance. In both cases, the loops cross 0 dB
at a frequency approximately equal to the loop gain shown
in Fig. 3. However, the current-mode loop has a better phase
margin (90 degrees versus 45 degrees). Notice that at lower
frequencies, the output impedance using voltage-mode
control is lower. This is due to the lower impedance of the
output inductor in voltage mode versus the current source
equivalent of current-mode control. As the frequency nears
the overall loop-gain crossover or bandwidth of 30 kHz,
the voltage-mode loop exhibits higher output impedance
due to the peaking caused by the lower phase margin.
February 2005
20.3 ms
Fig. 6. Comparison of voltage-mode and current-mode control to a
5-A step load.
Fig. 5. Comparison of closed-loop output impedance with currentmode control and voltage-mode control.
Power Electronics Technology
20.2 ms
Time
Frequency
Zout_ req = DVp /DIstep
46
(3)
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Closed-Loop Impedance (Ohms)
TRANSIENT PERFORMANCE
100
10 uF
20 uF
50 uF
100 uF
10
200 uF
500 uF
1000 uF
10
10
10
1•10 -3
1•10 -3
1•10 -4
1•10 -5
Crossover Frequency (Hz)
Fig. 7. Graphical method determines output capacitance
requirements.
Next, select the loop-gain crossover frequency for the
converter. If the loop crossover is not known, then a typical estimate would be to use one-fifth of the switching frequency. Find the intersection between the required closedloop impedance and the loop-gain crossover frequency to
find the output capacitance needed to meet the voltage
transient requirement. This also can be worked from the
opposite direction by starting with the loop-gain crossover
frequency, finding the intersection with the desired output capacitor value (a known output capacitor) and then
working back to find the capacitor impedance. This allows
for the transient voltage to be calculated for any given step
current and output capacitor used.
In reality, many capacitors have an ESR that cannot be
neglected and that needs to be accounted for in the output
filter design. In fact, many times, ESL needs to be accounted
for as well. But looking at the effects of the ESR and output
capacitance only, the output voltage variation is equal to
the load-current step multiplied by the output impedance
of the filter at crossover. For large value capacitors (polymer, tantalum, aluminum electrolytic) and reasonably high
crossover frequencies, the combined output impedance is
usually dominated by the ESR of the output capacitor. This
is because the impedance due to the capacitance itself at
the crossover frequency is typically much lower than the
ESR. To design the output filter, just determine the impedance requirement, choose a capacitor with that ESR, and
guarantee that the crossover frequency is above the zero
formed by the output capacitor and its ESR. Note that this
crossover frequency can be relatively low at the expense of
response time.
To summarize the important points, a power supply’s
response to a load step is approximately equal to its output
impedance at its crossover frequency multiplied by the
magnitude of the load step. The type of control strategy
can impact this impedance. Typically, a current-mode
power supply will have smaller output impedance and a
smaller transient response; however, it will be slower to
recover from a load step. Finally, it is a simple calculation
to determine output filter capacitance or ESR from a transient load-step requirement and an estimation of the loopPETech
crossover frequency.
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