(Tan) High-frequency pole (from the Tan averaged model (4))
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Transcript of (Tan) High-frequency pole (from the Tan averaged model (4))
(Tan)
High-frequency pole(from the Tan averaged model (4))
][ˆ)1(]1[ˆ ][ˆ ninini cLL αα −+−=
Discrete-time dynamics: )(ˆ)(ˆ zizi Lc →
Z-transform: )(ˆ)1()(ˆ )(ˆ 1 zizzizi cLL αα −+= −
1 1
1
)(ˆ)(ˆ
−−−
=zzi
zi
c
L
ααDiscrete-time (z-domain) control-to-
inductor current transfer function:
ss TjsT ee ωαα
αα
−− −−
→−−
11
11
Difference equation:
• Pole at z = α • Stability condition: pole inside the unit circle, |α| < 1
• Frequency response (note that z1 corresponds to a delay of Ts in time domain):
Equivalent hold:
ic[n]
m1m2
ic + ic
iL[n]
d[n]TsiL[n-1]
ma(t)
)(ˆ)(ˆ ),(ˆ][ˆ sizitini LLLL →→
iL(t) iL[n]
Ts
Equivalent hold
• The response from the samples iL[n] of the inductor current to the inductor current perturbation iL(t) is a pulse of amplitude iL[n] and length Ts
• Hence, in frequency domain, the equivalent hold has the transfer function previously derived for the zero-order hold:
s
e ssT−−1
Complete sampled-data “transfer function”
s
sT
sTc
L
sT
e
esi
si s
s
−
−
−−−
=1
11
)(ˆ)(ˆ
αα
2
2
1
2
'
1
mm
DD
mm
mm
mm
a
a
a
a
+
−−=
+−
−=α
Control-to-inductor current small-signal response:
Example
• CPM buck converter: Vg = 10V, L = 5 H, C = 75 F, D = 0.5, V = 5 V,
I = 20 A, R = V/I = 0.25 , fs = 100 kHz
• Inductor current slopes:m1 = (Vg – V)/L = 1 A/s
m2 = V/L = 1 A/s
2
2
2
2
1
2
1
1
'
1
mmmm
mm
DD
mm
mm
mm
a
a
a
a
a
a
+
−−=
+
−−=
+−
−=α
D = 0.5: CPM controller is stable for any compensation ramp, ma/m2 > 0
s
sT
sTc
L
sT
e
esi
si s
s
−
−
−−−
=1
11
)(ˆ)(ˆ
αα
Control-to-inductor current responses for several compensation ramps (ma/m2 is a parameter)
102 103 104 105-40
-30
-20
-10
0
10
20
magnitude [db] iL/ic magnitude and phase responses
102 103 104 105
-150
-100
-50
0
frequency [Hz]
phase [deg]
ma/m2=0.1
ma/m2=0.5
ma/m2=1
ma/m2=5
5
10.5
0.1
MATLAB file: CPMfr.m
First-order approximation
hfs
s
sT
sTc
L
sssT
e
esi
si s
s
ωπωααα
α
+=
−++
≈−
−−
=−
−
1
1
)/(111
11 1
1)(ˆ)(ˆ
)/(1
)/(1
πω
πω
s
ssT
s
s
e s
+
−≈−
ππαα s
a
shf
f
mm
DD
ff
2
221
1
1
1
+−=
+
−=
Control-to-inductor current response behaves approximately as a single-pole transfer function with a high-frequency pole at
Same prediction as HF pole in basic model (4) (Tan)
Control-to-inductor current responses for several compensation ramps (ma/m2 = 0.1, 0.5, 1, 5)
102 103 104 105-40
-30
-20
-10
0
10
20
magnitude [db] iL/ic magnitude and phase responses
102 103 104 105
-150
-100
-50
0
frequency [Hz]
phase [deg]
1st-order transfer-function approximation
Second-order approximation
2
2/)2/(11
21
11
1
1
)(ˆ)(ˆ
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛
−++
≈−
−−
=−
−
ss
s
sT
sTc
L
sssTe
esisi s
s
ωωααπα
α
2
2
2/)2/(21
2/)2/(21
⎟⎟⎠
⎞⎜⎜⎝
⎛++
⎟⎟⎠
⎞⎜⎜⎝
⎛+−
≈−
ss
sssT
ss
ss
e s
ωωπ
ωωπ
2
221
12
1
12
m
mDD
Qa+−
=+−
=πα
απ
Control-to-inductor current response behaves approximately as a second-order transfer function with corner frequency fs/2 and Q-factor given by
102 103 104 105-40
-30
-20
-10
0
10
20
magnitude [db] iL/ic magnitude and phase responses
102 103 104 105
-150
-100
-50
0
frequency [Hz]
phase [deg]
Control-to-inductor current responses for several compensation ramps (ma/m2 = 0.1, 0.5, 1, 5)
2nd-order transfer-function approximation
2nd-order approximation in the small-signal averaged model
DC gain of line-to-output Gvg-cpm(based on model (4))
Example
• CPM buck converter: Vg = 10V, L = 5 H, C = 75 F, D = 0.5, V = 5 V,
I = 20 A, R = V/I = 0.25 , fs = 100 kHz
• Inductor current slopes:m1 = (Vg – V)/L = 1 A/s
m2 = V/L = 1 A/sD = 0.5: CPM controller is stable for any compensation ramp, ma/m2 > 0
Select: ma/m2 = Ma/M2 = 1, Ma = 1 A/s
A/V 25.02
'==
LTDD
F sg
1/A 1.01
2
1
21
=−+
=s
a
m TMMMF
Example (cont.)
kHz 2.81
2
1==
LCfo π
1==LC
RQ
47.047.01
1==
+
+= Q
LVRCFRVF
QQgm
gm
ckHz 3.1851 ==+= ogm
oc fRVF
ff
kHz 4.81 =≈ ccp fQf
kHz 39/2 =≈= cchfp Qfff
Duty-cycle control
Peak current-mode control (CPM)
Compare to first-order approximation of the high-frequency sampled-data control-to-current model
hfs
s
sT
sTc
L
sssT
e
esi
si s
s
ωπωααα
α
+=
−++
≈−
−−
=−
−
1
1
)/(111
11 1
1)(ˆ)(ˆ
)/(1
)/(1
πω
πω
s
ssT
s
s
e s
+
−≈−
kHz 32221
1
1
1
2
==+−
=+−
=πππα
α ss
a
shf
ff
mm
DD
ff
Control-to-inductor current response behaves approximately as a single-pole transfer function with a high-frequency pole at