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Parasitic resistance current sensing topology for coupled inductors
Shangyang Xiaoa; Weihong Qiua; Jun Liua; Thomas X. Wub; Issa Batarsehb
a
Intersil Corporation, Milpitas, USA b School of Electrical and Computer Engineering, University of
Central Florida, Orlando, FL, USA
To cite this Article Xiao, Shangyang , Qiu, Weihong , Liu, Jun , Wu, Thomas X. and Batarseh, Issa(2009) 'Parasitic
resistance current sensing topology for coupled inductors', International Journal of Electronics, 96: 1, 51 — 61
To link to this Article: DOI: 10.1080/00207210802492336
URL: http://dx.doi.org/10.1080/00207210802492336
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International Journal of Electronics
Vol. 96, No. 1, January 2009, 51–61
Parasitic resistance current sensing topology for coupled inductors
Shangyang Xiaoa*, Weihong Qiua, Jun Liua, Thomas X. Wub and Issa Batarsehb
a
Intersil Corporation, Milpitas, USA; bSchool of Electrical and Computer Engineering, University of
Central Florida, Orlando, FL, USA
Downloaded By: [University of Central Florida] At: 03:54 26 August 2010
(Received 10 July 2008; final version received 13 September 2008)
Traditional current sensing topology based on inductor equivalent series resistance fails
to extract phase currents for coupled inductors due to the presence of the magnetising
inductance. This article proposes a new direct-current resistance current sensing
topology for coupled inductors. By implementation of a simple resistor-capacitor
network, the proposed topology can preserve the coupling effect between phases. As a
result, real phase inductor currents and total current can be sensed. Detailed
mathematical analysis and design equations are presented in this article. Sensitivity
and mismatch issues are addressed. Experimental results show that the proposed
topologies are able to extract phase current as well as total current with acceptable
accuracy.
Keywords: power electronics; DC–DC converter; voltage regulator; coupled inductors;
current sensing; inductor parasitic resistance
1.
Introduction
The ever-stringent transient and efficiency requirements of recent microprocessors have
posed great challenges on voltage regulator (VR) design (Voltage Regulator-Down http://
www.intel.com). Coupled inductors are an emerging topology for power supplies. In
recent years, multiphase VRs with coupled inductors have drawn more and more attention
from both industry and academia because of their better performance than traditional
discrete-inductor multiphase VRs. The work by Schultz and Sullivan (2001), Jieli et al.
(2002), Ledenev et al. (2004), Herbert (2005), Xu et al. (2007), was involved in various
coupled-inductor topologies and modelling in which the number of coupled phases is
larger than two; whereas in the study by Pietkiewicz and Tollik (1998), Chen et al. (1999),
Wong et al. (2000), Wu et al. (2006), Zhou et al. (2005), two–two coupled-inductor VRs
were explored. Since the linear structure of multiphase coupled inductors is inherently
asymmetrical, with coupling phase number more than two, the non-identical magnetic
characteristics for different phases may lead to sub-harmonic output ripple (Wu et al.
2006). As a result, two–two coupled-inductor VR is usually preferred in industry. This
article will limit the scope to discussion of two–two coupled inductors only.
As technologies advance, current mode control, droop control (also referred as
adaptive voltage positioning), over-current-protection, phase-current limit and other
advanced power management features are becoming common industry practice.
*Corresponding author. Email: sxiao@intersil.com
ISSN 0020-7217 print/ISSN 1362-3060 online
Ó 2009 Taylor & Francis
DOI: 10.1080/00207210802492336
http://www.informaworld.com
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52
S. Xiao et al.
To implement these technologies, the phase currents need to be sensed and fed back to the
PWM (pulse width modulation) controller. Therefore, a proper current sensing technique
is desired. Conventionally, a dedicated sense resistor is placed after each phase inductor to
achieve high-accuracy current sensing. As next generation microprocessors demand more
power and higher efficiency, this method becomes undesirable since it introduces
significant conduction loss. Recently, the metal–oxide–semiconductor field-effect transistor (MOSFET) on-resistance (Rdson) current sensing has been utilised in industrial
products. Unfortunately, the tolerance of MOSFET on-resistance usually falls in the 30–
40% range (Hua and Luo 2006). Poor current sensing accuracy leads to phase-to-phase
current unbalance. For high-current multiphase VRs, current unbalance leads to stability
and thermal issues which could be disastrous. The current-sensing schemes presented by
Ma and Lee (1994), Zhou et al. (1999), Dallago et al. (2000), Forghani-zadeh and RinconMora (2002, 2005) show advantages in terms of current sensing accuracy. However,
complexity limits their utilisation in VR application.
2.
Limitations of conventional DCR current sensing topology
In recent years, inductor parasitic-resistance current sensing (also referred as DCR
current sensing) has been prevailing in the VR industry (LinFinity 1998; Walters et al.
1998; Dong et al. 2006; ISL6327), although the concept can date back to Maxwell bridge
decades ago (http://en.wikipedia.org/wiki/Maxwell_bridge). The inductor parasiticresistance (DCR) current sensing method employs a resistor and a capacitor across
the inductor to extract the voltage drop on the output inductor parasitic resistance. If
the resistor-capacitor (RC) network components are selected such that the RC time
constant matches the inductor time constant (L/DCR), the voltage across the capacitor
is equal to the voltage drop across the DCR, i.e., the phase current is replicated
(ISL6327). Because the tolerance of inductor copper trace can be controlled within up to
5% with current process (Hua and Luo 2006; LC1740-R30R10A), the accuracy of DCR
current sensing can be significantly improved compared to the MOSFET Rdson sensing
method. In addition, despite that the RC sensing network has some AC losses, the DC
loss on it is almost zero. Furthermore, the overall cost of the RC network may be even
cheaper than a sense resistor. Thus, as a cost-effective practice, DCR sensing is widely
implemented in VR applications. However, even though DCR current sensing has been
well investigated for discrete inductors, little work has been done on current sensing for
coupled inductors. In most coupled-inductor applications, the traditional DCR sensing
method is simply applied to coupled inductors (Dong et al. 2006; Xu et al. 2007; Wu
et al. 2006). The RC network is designed based on the leakage inductance while the
magnetising inductance is neglected. Since the phase currents of coupled inductors are
different to those of discrete inductors due to the presence of magnetising inductance
and coupling effect, the application of conventional DCR current sensing to coupled
inductors is questionable (Xiao et al. 2008). Figure 1 shows a two-phase coupledinductor VR with traditional DCR current sensing topology, where VP1 and VP2
represent the phase voltages (referring to VO); R and C are the current-sensing network
components for each phase. A widely utilised model (Pietkiewicz and Tollik 1998; Chen
et al. 1999; Wong et al. 2000; Schultz and Sullivan 2001; Jieli et al. 2002; Ledenev, et al.
2004; Herbert 2005; Xu et al. 2007; Zhou et al. 2005; Wu et al. 2006) of coupled
inductors is employed. This circuit model consists of a leakage inductance (Lk) in series
with the inductor DCR for each winding, and an ideal transformer with a magnetising
inductance (LM) on one of the windings.
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International Journal of Electronics
Figure 1.
53
Coupled inductors with traditional DCR current sensing topology.
Assuming that the coupled inductors are symmetric for the purpose of simplicity, the
phase voltage VP1 is written in s-domain as:
VP1 ¼ sðLk þ LM Þ iL1 sLM iL2 þ DCR iL1
ð1Þ
where s is a complex argument in s-domain. Since VP1 is also the voltage across R and C,
the voltage across C for phase 1 can be expressed as:
VC ¼
sðLk þ LM Þ iL1 sLM iL2 þ DCR iL1
1 þ sR C
ð2Þ
With the RC time constant matching the inductor Lk/DCR time constant, Equation 2
is simplified to:
VC ¼ DCR iL1 þ
sLM ðiL1 iL2 Þ
1 þ sR C
ð3Þ
The first term represents the DC current across DCR whereas the second term cannot
be cancelled out due to the magnetising inductance. Therefore, Equation 3 suggests that
traditional DCR current sensing method cannot extract the correct phase current
information. It is the superposition of the phase DC current and additional AC
components.
Figure 2 shows experimental results obtained from a built prototype. The top one
shows the sensed phase current by traditional DCR sensing and the bottom waveform
represents the real phase current. With the RC time constant matching the time constant
of leakage inductance and DCR, it is found that the sensed current is distorted.
Consequently, wrong phase current information is fed back to the controller which leads
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54
S. Xiao et al.
Figure 2. Experimental current waveforms for coupled inductors with traditional DCR current
sensing (X-axis: 5 mV/div for channel 1 and channel 2; Y-axis: 2 mS/div).
to improper phase over-current-protection as well as inferior current and voltage
regulation in return.
Motivated by the demand of a cost-effective current sensing technique for
coupled inductors, a new DCR current-sensing topology is proposed in this article. By
implementation of a simple RC network, the phase currents as well as total current
can be extracted. The design equations for the RC sensing network are derived by stepby-step mathematical analysis. These equations are given in terms of leakage
inductance and magnetising inductance since these coefficients are usually provided in
the coupled-inductor datasheets by vendors (LC1740-R30R10A). Sensitivity and
mismatch issues are addressed. A prototype has been built to verify the theoretical
results.
3. Proposed inductor parasitic resistance current sensing topology
Figure 3 shows the proposed parasitic resistance current sensing topology for two–two
coupled inductors. This topology consists of a capacitor C and two resistors for each
phase, where R1, R2 are for phase one and R3, R4 are for phase two, respectively. For the
purpose of simplicity, two phases can be assumed to be symmetric. Then, R1 is equal to R4
and R2 is equal to R3. VP1 and VP2 are denoted as phase voltages (referring to VO). Under
some circumstances, it is necessary to use a resistor divider to scale down the sensed
current or to compensate the temperature variation of DCR. This can be accomplished by
placing a thermal resistor RC in parallel with the capacitor C for each phase.
By properly sizing the resistor and capacitor values in the topology, the phase current
can be represented by the voltage across C, i.e. VC. VC is then sent to a current amplifier
inside the PWM controller for current and voltage regulation as shown in Figure 1.
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International Journal of Electronics
Figure 3.
Proposed DCR Current Sensing topology for two-phase coupled inductors.
The phase voltage VP1 given in Equation 1 is valid here and similarly VP2 can be given in
the s-domain by:
VP2 ¼ sðLk þ LM Þ iL2 sLM iL1 þ DCR iL2
ð4Þ
Solving Equation 4 for iL2 and then substitution of iL2 into Equation 1 leads to the
solution for iL1:
iL1 ¼
VP1 ½s ðLk þ LM Þ þ DCR þ s LM VP2
½s ðLk þ LM Þ þ DCR2 s2 L2M
ð5Þ
In the case that RC is not stuffed, the voltage across current sensing capacitor C for
phase one can be expressed as:
VC ¼
R1
1þsR1 C1
R1
1þsR1 C þ R2
VP1 þ
R2
1þsR2 C1
R2
1þsR2 C þ R1
VP2
ð6Þ
If the RC network is designed properly, VC should be able to represent the voltage
drop across DCR of each phase. For phase one, the following equation will be satisfied:
iL1 DCR ¼ VC
ð7Þ
56
S. Xiao et al.
Inserting Equation 5 into Equation 7 and substitution of VC in Equation 6 lead to:
R1
R2
VP1 þ
VP2
R1 þ R2 þ sR1 R2 C
R1 þ R2 þ sR1 R2 C
VP1 ½s ðLk þ LM Þ þ DCR þ s LM VP2
DCR
¼
½s ðLk þ LM Þ þ sLM þ DCRðsLk þ DCRÞ
ð8Þ
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Since the normal switching frequency of VR falls in the hundreds of kilohertz
range, the term s(Lk þ LM) is usually much greater than DCR. Equation 8 can be
decoupled into two equations by dropping off VP1 and VP2. The two equations are
then rearranged as:
Lk
R1 1 þ s DCR
Lk þ LM
Lk þ 2LM
ð9Þ
Lk
R2 1 þ s DCR
LM
¼
R1 R2
L
þ
2LM
k
ðR1 þ R2 Þ 1 þ s R1 þR2 C
ð10Þ
ðR1 þ R2 Þ ð1 þ
R2
s RR11þR
2
CÞ
¼
Simplifying Equations 9 and 10 yields:
R1
Lk þ LM Lk
R1 R2
¼
¼
C
R1 þ R2 Lk þ 2LM DCR R1 þ R2
ð11Þ
The above derivation suggests that if the resistors and capacitors in the proposed
topology are designed such that Equation 11 is satisfied, then by reversing the derivation
procedure, Equation 7 can be obtained, which means that the voltage across C is a replica
of the voltage across the inductor DCR, i.e., the phase current is duplicated. Therefore,
Equation 11 can serve as the design equation for the proposed current sensing topology.
For some applications in which the DCR voltage drop is too high or temperature
compensation is desired, a scale-down resistor or thermal resistor RC is usually placed in
parallel with C. In this case, VC is a scaled version of the DCR voltage. Letting the sensing
gain be K, the following equations can be obtained:
Lk
R1 R2 RC
R1
Lk þ LM
¼
C
¼
DCR R1 RC þ R2 RC þ R1 R2
R1 þ R2 Lk þ 2 LM
ðR1 þ R2 Þ RC
K¼
R1 RC þ R2 RC þ R1 R2
4.
ð12Þ
Sensitivity and mismatch analysis
The above analysis has assumed ideal coupled inductors with known, matched
coefficients. In practice, it is difficult to obtain accurate values of the parasitic
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International Journal of Electronics
resistance, which varies with frequency, proximity effects (and therefore current),
temperature, etc. And it would not be expected to be balanced among windings in a
given inductor. The other parameters such as leakage inductance and magnetising
inductance have tolerance up to 30% and can also change with time as magnetic
materials cycle and age (LC1740-R30R10A). With the tolerance and mismatch issues,
the sensed current signal may be distorted. Therefore, it is necessary to evaluate the
sensitivity and mismatch effects on sensed currents. Since the tolerance of the RC
sensing components is typically as low as 1%, the discussion will be focused on the
coupled inductor coefficients only.
Letting the variations of the leakage inductance, magnetising inductance, DCR to be
dLk, dLM, dDCR and the resulted voltage deviation on sensing capacitor C to be dVC,
respectively, then the effect of each specific coefficient on the sensed current can be
analysed. By adding the variations into Equations 1 and 4, insertion of these equations
into Equation 6 leads to the equation for VC with deviation. dVC can be solved by
subtraction of VC from this equation. If the current sensing sensitivity ratio is defined as
the sensing deviation dVC over the correct voltage VC on the sensing capacitor, then for
leakage inductance variation dLk:
dVC dLk
1
¼
VC
Lk 1 þ R1 þR2
ð13Þ
s R1 R2 C
For magnetising inductance, the sensitivity ratio is obtained as:
dVC dLM
R2
1
¼
VC
LM R1 R2 1 þ R1 þR2
ð14Þ
s R1 R2 C
The sensitivity ratio for DCR can be given as:
dVC
dDCR
1
¼
DCR þ dDCR 1 þ sR1 R2 C
VC
ð15Þ
R1 þR2
Equations 13, 14 and 15 suggest that coefficient variations result in current sensing
errors and the errors vary with frequency. To demonstrate sensitivity of each coefficient,
the coupled inductor LC1740-R30R10A from NEC/Tokin is employed as an example
(LC1740-R30R10A). According to design equations in Equation 11, the current sensing
values as well as the inductor coefficient are given in Table 1.
Table 1.
Coefficients of LC1740-R30R10A and current sensing values.
Leakage
inductance (nH)
110% + 30%
Magnetising
inductance (nH)
DCR (mO)
R1 (kO)
R2 (kO)
C (mF)
200% + 30%
0.4% + 5%
3.1
2
0.22
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58
Figure 4.
S. Xiao et al.
Normalised sensitivity ratio for leakage inductance, magnetising inductance and DCR.
Figure 5. Experimental results for proposed topology. System parameters: Vin ¼ 12 V,
VO ¼ 1.2 V, LO ¼ LC1740-R30R10A from NEC/Tokin, CO ¼ 36 6 22 mF MLCC. Controller:
ISL6266 from Intersil (X-axis: 20 V/div for channel 1 and channel 4, 10A O/div for channel 3, and
5 mV O/div for channel 2; Y-axis: 1 mS/div).
International Journal of Electronics
59
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Based on Equations 13, 14 and 15, normalised sensitivity ratio (sensitivity ratio over
tolerance) of the coefficients can be plotted versus frequency in Mathcad as shown in
Figure 4.
It can be concluded from Figure 4 that:
(1) The tolerance of magnetising inductance is amplified by around 1.8 times at high
frequencies, indicating the AC portion of the sensed current waveform will be
distorted if the magnetising inductance data is not accurate enough. However, it
has little effect on the DC level of sensed current.
(2) The deviation of sensed current is almost proportional to the leakage inductance
tolerance at high frequencies. Similar to the magnetising inductance, it has limited
impact on the DC current level.
(3) This topology is sensitive to DCR tolerance at DC (low frequencies), and the
sensed current deviation is nearly proportional to the DCR tolerance. At high
frequencies, DCR tolerance has negligible effect on sensed current.
As a common industry practice, a thermal resistor RC as shown in Figure 3 is usually
placed close to the inductor to compensate the DCR temperature variation. Therefore, the
DC current sensing error can be controlled within acceptable range if the topology is
designed properly. The AC portion of sensed current may have errors that stem from
leakage and magnetising inductance tolerance.
Figure 6. Comparison of real phase current and sensed current by proposed topology (X-axis:
20 V/div for channel 1 and channel 4, 10A O/div for channel 3, and 5 mV O/div for channel 2;
Y-axis: 1 mS/div).
60
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5.
S. Xiao et al.
Experimental verification
A 400 kHz two-phase VR with coupled inductors has been developed to verify the
proposed topology. The component values and coefficients of the topology are listed in
Table 1, where RC is not stuffed. The experimental results are shown in Figure 5. From top
to bottom, the first waveform represents the sensed current by the proposed topology and
the second waveform is the phase inductor current captured by a current probe,
respectively. The two bottom waveforms are the PWM switching voltages.
To investigate the current sensing error, the sensed current and real current in Figure 5
are overlapped as shown in Figure 6. From Figures 5 and 6, it is found that the sensed
current agrees with the real phase current in terms of DC level. The AC portion (valley
current) is distorted a little because of the tolerance of leakage and magnetising
inductance. The high-frequency jitters on the sensed waveforms are random noise resulting
from measurement.
6.
Conclusion
A current sensing topology for coupled inductors has been proposed in this article. By
implementation of simple RC networks, the phase coupling information can be
preserved. Mathematical derivation and design formula were presented in detail.
Sensitivity and mismatch issues are addressed in terms of mathematical experimental
results show that the proposed topology is able to sense real phase current with
acceptable accuracy.
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