Datasheet

AD2S1205
Rev. A | Page 17 of 20
CIRCUIT DYNAMICS
LOOP RESPONSE MODEL
0
6339-011
ERROR
(ACCELERATION)
–
θ
IN
θ
OUT
VELOCITY
k1 × k2
1 – z
–1
1 – bz
–1
1 – z
–1
c 1 – az
–1
c
Sin/Cos LOOKUP
Figure 11. RDC System Response Block Diagram
The RDC is a mixed-signal device that uses two ADCs to
digitize signals from the resolver and a Type II tracking loop
to convert these to digital position and velocity words.
The first gain stage consists of the ADC gain on the Sin/Cos
inputs and the gain of the error signal into the first integrator.
The first integrator generates a signal proportional to velocity.
The compensation filter contains a pole and a zero that are used
to provide phase margin and reduce high frequency noise gain.
The second integrator is the same as the first and generates the
position output from the velocity signal. The Sin/Cos lookup has
unity gain. The values for each section are as follows:
ADC gain parameter (k1
NOM
= 1.8/2.5)
)V(
)V(
p
REF
IN
V
V
k2 = (12)
Error gain parameter
π××= 21018
6
k2 (13)
Compensator zero coefficient
4096
4095
=a
(14)
Compensator pole coefficient
4096
4085
=b
(15)
Integrator gain parameter
000,096,4
1
=c (16)
INT1 and INT2 transfer function
1
1
)(
−
−
=
z
c
zI (17)
Compensation filter transfer function
1
1
1
1
)(
−
−
−
−
=
bz
az
zC (18)
R2D open-loop transfer function
)()()(
2
zCzIk2k1zG ×××= (19)
R2D closed-loop transfer function
)(1
)(
)(
zG
zG
zH
+
=
(20)
The closed-loop magnitude and phase responses are that of a
second-order low-pass filter (see Figure 12 and Figure 13).
To convert G(z) into the s-plane, an inverse bilinear transfor-
mation is performed by substituting the following equation
for z:
s
t
s
t
z
−
+
=
2
2
(21)
where t is the sampling period (1/4.096 MHz ≈ 244 ns).
Substitution yields the open-loop transfer function G(s).
)1(2
)1(
1
)1(2
)1(
1
4
1
)1(
)(
2
22
b
bt
s
a
at
s
s
ts
st
ba
ak2k1
sG
−
+
×+
−
+
×+
×
++
×
−
−×
= (22)
This transformation produces the best matching at low frequencies
(f < f
SAMPLE
). At such frequencies (within the closed-loop bandwidth
of the AD2S1205), the transfer function can be simplified to
2
1
2
1
1
)(
st
st
s
K
sG
a
+
+
×≅ (23)
where:
ba
ak2k1
K
b
bt
t
a
at
t
a
−
−×
=
−
+
=
−
+
=
)1(
)1(2
)1(
)1(2
)1(
2
1
Solving for each value gives t
1
= 1 ms, t
2
= 90 μs, and K
a
≈ 7.4 ×
10
6
s
−2
. Note that the closed-loop response is described as
)(1
)(
)(
sG
sG
sH
+
=
(24)
By converting the calculation to the s-domain, it is possible to
quantify the open-loop dc gain (K
a
). This value is useful to
calculate the acceleration error of the loop (see the Sources of
Error section).