Texas Instruments 1 AAJ 4Q 2016

AutomotiveAnalog Applications Journal

System modeling in MATLAB Simulink® for PLL-based resolver-to-digital converters

IntroductionA previous article in the Analog Applications Journal described the fundamental architecture of a resolver-to-digital converter (RDC).[1] This article addresses how to simulate the performance of a RDC in the powertrain system and how to analyze real events such as hard braking and sudden acceleration in the automobile. Different automotive subsystems have diverse require-ments. For example, the belt-driven alternator/starter system can have an acceleration of 50,000 revolutions per minute per second (RPM/s); whereas for crank-driven systems, acceleration can be in the order of 20,000 RPM/s. Similarly, industrial applications may have a highly dynamic servo motor that can accelerate from 0 to 5,000 RPM within as little as two to three milliseconds.

Some applications involve hard braking and instant acceler ation where the motor stops because of an obstacle. For this application, acceleration or deceleration can be as

high as 200,000 rad/s2. In this case, higher position errors are temporarily tolerated, however, returning to true posi-tion values is required within a few milliseconds. The delay from reading the input position to outputting the angle has to be known accurately for proper motor control.

Resolver-to-digital conversion architectureFigure 1 shows the RDC with PLL/VCO-based architecture that converts analog resolver signals into digital angle and velocity outputs. The angle information is amplitude modulated onto the signals from the stator windings . These signals must be demodulated and post-processed to extract angle and velocity information.

The main objective of the RDC architecture is to calcu-late the rotation angle, Θ, and the velocity of the resolver shaft. Figure 1 shows how the angular position information is extracted from the envelope or voltage peaks of the input sine and cosine signals.[1]

By Ankur VermaAutomotive Solutions

Figure 1. Simplified resolver-to-digital converter (RDC) architecture

Angle ( )ϕandVelocityOutput

DAC

DAC

COS

Analog Digital

SIN

10

16

10

DifferentialComparator

K × sin × sin tΘ ω

Reference Signal = sin tω

V = K × sinϕERR ( – )Θ ϕΘ ϕ=

sin ϕ

Rectify/CarrierDemodulator

PI Control Loop

Sine/CosineROM

K × cos × sin tΘ ω

cos ϕ

Sine

Analog Front End (AFE)

Cosine

Fro

m R

eso

lver

+

+

+

–

–

–

εPULSE =

t ( – )K × sin × sinω Θ ϕ

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AutomotiveAnalog Applications Journal

System modelingVarious use cases at the system level can be simulated with the help of the MATLAB Simulink-based architecture model, such as has been done for the PGA411-Q1 in Figure 2. There are two major blocks in the system model for use-case analysis of different configurations.

Resolver blockThe input resolver block shown in Figure 3 supports various real-system scenarios, such as:

1. Static angle testing

2. Step response in input angle

3. Custom acceleration/deceleration

4. Sudden acceleration/hard braking

5. Effects of coupled noise in the system

The outputs from the resolver blocks are simulated signals of the resolver sensor:

Sine signal from resolver sensor = sin Θ × sin wt (1)

Cosine signal from resolver sensor = cos Θ × sin wt (2)

where Θ = resolver angle and w = the exciter or excitation frequency applied at the R1-R2 rotor winding.

The amplitude-modulated signals at the sine and cosine winding of the resolver are created inside the resolver block. There are multiple ways to give inputs to this block.

Figure 2. MATLAB Simulink model for real system analysis

Figure 3. Input resolver block diagram

DifferentialComparator

PGA411-Q1

ε_pulse

threshold

Loop error

Looperrorcomp

z1

Unit Delay4

[A]

[D]

[C]

Goto 2

[B]

[E]

Goto 4

[A]

From 8

[B]

From 7

0.3

faultthreshold

Z-d

Z-d

Delay 1

-C-

GainCos

-C-

GainSin

[H]

Goto 6

-K-

Gain_Noise

phi_hys

ToWorkspace

[M]

Goto 8

IEx pins

sin input

cos input

exciter out

Exciter Phase Configuration

csin ϕ

cos ϕ

Sine/Cosine ROM DAC

ε_pulse

exciter

VϕERR

CarrierDemodulator

Vϕ_ERR

exciter

velocity

ϕ

+

–

output Θ

PI ControlLoop

K Θ sin ωtsin

K Θ sin ωtcos

K Θ sin ωtcos

K Θ sin ωtsin cos_gain

sine_gain

sin ϕ

cos ϕ

Out 1

Out 2

Mixercosin

sinin

Noise

AFE

Manual Switch 2 Manual Switch

angle

Step

Step 1

Accelerationand Braking

Delay

Goto 3

Goto 1

Goto

sin input

exciter

cos input

input anglenoise

Sine/CosineCreation

ManualSwitch 3 Manual

Switch 1

SineWave

RandomNumber

CosWave

Exciter

InputStep

FromWorkspace 1

FromWorkspace

IdealSignals

Resolver

-K-

Gain_Noise

K Θ sin ωtcos

K Θ sin ωtsin

cosin

sinin

Noise

Manual Switch 2 Manual Switch

Step

Step 1

Accelerationand Braking

sin input

exciter

cos input

input anglenoise

Sine/CosineCreation

ManualSwitch 3 Manual

Switch 1

SineWave

RandomNumber

CosWave

Exciter

InputStep

FromWorkspace 1

FromWorkspace

IdealSignals

Resolver

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The configurable parameters are shown in Table 1.

Table 1. MATLAB Simulink® parameters

Input parameter Function

sim_time Simulation time

RESOLVER BLOCK INPUTS

Static angle / constant RPM

Angle Static angle or initial angle: specify angle here for static tests

Speed Rotation speed (RPM) "enter speed in RPM" - rotor rotation, "0" for static testing

Direction Rotation direction: 0 = CW ; 1 = CCW

Step input

step_time Enter the step time for the angle change

angle1 Angle1 is the initial value

angle2 Angle2 is the final value

Custom inputs - for acceleration and deceleration

cosin Enter custom parameters from .m file

sinin Enter custom parameters from .m file

Phase delay w.r.t. exciter

signal_delay Delay = signal_delay × 100 ns (simulation step size)

Noise

InNoise Noise on Sin/Cos inputs

sin_phase Sin signal phase delay in degrees

cos_phase Cos signal phase delay in degrees

PGA411-Q1 BLOCK INPUTS

SelfExt_Sel Excitation frequency select. Range = 10000 Hz to 20000 Hz in seven steps

Analog front-end

GainCos Cos input gain. Range = 2.8 to 4.2

GainSin Sin input gain. Range = 2.8 to 4.2

Tracking loop

DKp_Sel Range = 0 to 5%, KPGain base value (DKP)

MKp_Sel Range = 0 to 3%, KPGain multiplier for AMODE = 1 (DKP × MKP)

DKi_Sel Range = 0 to 7%, Number of bits for PI gain loop (DKI)

OHYS Hysteresis selection: 0 - Off; 1 - 1 bit; 2 - 2 bits; 3 - 3 bits

AMODE Accelerated mode: 0 - Off; 1 - On

BMODE Resolution mode: 0 - 10 bit; 1 - 12 bit

1. Sine and cosineSin (frequency × t + phase)Frequency: 2p × fR, where fR = speed / 60Phase: Direction × p + (sin_phase × 2p / 360) + (angle

× 2p / 360)

Cosine (Frequency × t + phase)Frequency: 2p × fR, where fR = speed / 60 Phase: p / 2 + direction × p + (cos_phase × 2p / 360) + (angle × 2p / 360)where fR = speed / 60

2. Acceleration and brakingCustom sinin and cosin signals (shown in Figure 3) can be fed directly into the sine/cosine creation block. These input signals are multiplied with the exciter signal per Equations 1 and 2. The objective is to generate an array of data for the cosin and the sinin signals.

The sin [2p × (fstart × t + 0.5 × accel × t2)] and cos [2p × (fstart × t + 0.5 × accel × t2)] commands can be used to create custom signals.

3. Step response Sine input step

• Initial value: sin (angle1 × 2p / 360)

• Final value: sin (angle2 × 2p / 360)

Cosine input step

• Initial value: sin [p / 2 + (angle1 × 2p / 360)]

• Final value: sin [p / 2 + (angle2 × 2p / 360)]

Note that step_time specifies the time of the step.

4. Noise analysisThe impact of noise on the performance can be analyzed as well. The system’s mechanical vibration may result in some noise that causes change in the output angle. The variation has to be limited in a certain bounded condition, for example ±05 degrees. Random noise is generated and multiplied with a factor K, as shown in Figure 3.

Analog front-endThe analog front-end (AFE) shown in Figure 1 consists of programmable gain amplifiers and a comparator. The AFE block conditions the output signals of the resolver by removing noise, sets correct input DC bias, and appropri-ately gains up the AC signal to be used by the subsequent blocks. GainSin and GainCos in Table 1, and shown in Figure 2, are the parameters that control the gain of AFE.

Tracking loopThe proportional-integral (PI) control is implemented in the tracking loop as shown in Figures 1 and 2. It helps to bring the steady-state error to zero and improves transient response. Because of the added benefit of proportional control, it does not cause any offset and leads to faster response than integral control alone. The phase-locked loop (PLL) is a feedback system that forces the voltage-controlled oscillator (VCO) to replicate the input angle. The phase detector generates an error signal proportional to the phase difference between the two inputs. The VCO performs integration of the control voltage and provides the integral factor in the loop transfer function.

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AutomotiveAnalog Applications Journal

The PI control-loop filter shown in Figure 1 and high-lighted in Figure 4 can be analyzed with Equations 3 through 9, where the values are defined as follows:

a(t) = Error pulse post demodulation in Figure 4,y(t) = Output of the PI controller shown in Figure 4,f(t) = Output of the integrator as shown in Figure 4, DKP = Proportional gain term shown in Figures 4 and 5, MKP = Proportional gain term shown in Figure 5 when

acceleration block is used,AMODE = Acceleration mode shown in Figure 5,KI = integral gain term shown in Figure 4, KP = MKP × DKP × Gain from mixer when AMODE is ON, and “z” indicates the z-transform.

y t PI K a t K f tOUT P I( ) = × −( ) + × −( ) at 1 1

(3)

f t K a t f tP( ) = × ( ) + −( )1

(4)

y t K a t K K a t K f tP I P I( ) = × −( ) + × × −( ) + × −( )1 1 2

(5)

Also,

K f t y t K a tI P× −( ) = −( ) − × −( )2 1 2

(6)

y t K a t K K a t

t K a t

P I P

P

( ) = × −( ) + × × −( )+ −( ) − × −( )

1 1

1 2 y

(7)

y t y t K a t

K K a t K a t

P

I P P

( ) − −( ) = × −( )+ × × −( ) − × −( )

1 1

1 2

(8)

y z

a zK z

K z

zP

I( )( ) = × ×

+( ) −

−−

−

−1

1

1

1

1

(9)

For high acceleration rates, define KP = MKP × DKP × gain from mixer when AMODE is ON and KI = 1/2DKI.

In order to follow the rotation angle under rapidly changing conditions, another loop-acceleration block is added in Figure 5 that can change the control-loop feed-back gain. The higher-gain option helps the control loop to

Figure 5. PI control with acceleration block

Loop Acceleration

PI Controller (Type II)

MKP

DKP

Acceleration Mode (AMODE)

A = 1

ON

OFF

Σ AngleOutput

VelocityOutput

KIKI

KP

Scale

∫

∫∫

SampleandHold

Error-Pulse PostDemodulation

DelayedReference Pulse

16

16

Integrator

Integrator

Integrator

1/DKI

y(t)a(t)

f(t)

Figure 4. Example of a PI control loop

PI Controller (Type II)

Σ AngleOutput

VelocityOutput

KIKI

KP y(t)a(t)

f(t)

Scale

∫∫

Error Pulse PostDemodulation(Synchronous

Detection)16

16

Integrator

Integrator

1/DKI

DKP

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AutomotiveAnalog Applications Journal

track a fast rotation angle much easier. In the acceleration mode, the proportional gain is increased by several times (DKP × MKP) compared to the normal mode (DKP).

VCOIntegrator =− −1

1 1z (10)

The transfer function of the PI control loop is the multi-plication of Equations 9 and 10.

Transfer Functionz

z =

−× × ×

+( ) −

−

−

−−

−

−

1

11

1

11

1

1K z

K z

zP

I

(11)

ResultsThree stringent real-system scenarios (amongst several tested) are shown in Figures 6 through 8. Figure 6 depicts an example event of hard braking when a vehicle stops due to an obstacle, where the acceleration can be as high as –200,000 rad/s2 and velocity decreases rapidly from 19,100 RPM to 1 RPM. The output angle tracks the angle calculated at the input with less than 2 LSBs of error, indi-cating that the tracking loop successfully never falls out of lock. Figure 7 shows a braking event at –50,000 rad/s2 deceleration when the PGA411-Q1 is operated under 12-bit output mode. The tracking loop never falls out of

Figure 6. Braking event under 10-bit mode at –200,000 rad/s2

15001000500

0

20100

–10

20

–2–4

1

0

6

4

2

0

300020001000

0–1000

Time (ms)

0 5 10 15 20 25 30

Output andExpected

Angles

Velocity (Hz)

Error Flag(1 or 0)

Input Sineand Cosine

(rad)

ErrorAmplitude

(LSBs)

Output/ExpectedAngle Error

(LSBs)

Figure 7. Braking event under 12-bit mode at –50,000 rad/s2

6000

4000

2000

0

420

–2–4

20

–2–4

1

0

6

4

2

0

600

400

200

0

Output andExpected

Angles

Velocity (Hz)

Error Flag(1 or 0)

Input Sineand Cosine

(rad)

ErrorAmplitude

(LSBs)

Output/ExpectedAngle Error

(LSBs)

Time (ms)

0 5 10 15 20 25 30

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AutomotiveAnalog Applications Journal

lock here again. Figure 8 indicates an acceleration for applications such as a belt-driven alternator/starter system. With a starting velocity of 0 RPM, it may reach an acceleration of 50,000 rad/s2 in 30 ms. The velocity may increase to 14,324 RPM and the loop successfully tracks to the input angle again. Similar use cases for a real system can be analyzed with the help of the previously described model.

ConclusionsReal use cases for a system can be analyzed with the help of the Figure 2 model. Some differences may be observed between real system performance and the model-based performance. Hence, it is recommend that the simulation results only be used as a starting point and system-parameter adjust-ments are to be based on real system performance.

References1. Ankur Verma and Anand Chellamuthu, “Design consid-

erations for resolver-to-digital converters in electric vehicles,” Texas Instruments Analog Applications Journal (SLYT661), 1Q 2016.

2. Ankur Verma and Amanda Weise, “Rotary position sensing for electric vehicles,” Part 1 and Part 2, EDN, October 2, 2015.

3. Ankur Verma and J. Panacek. “PGA411-Q1 PCB Design Guidelines,” Texas Instruments Application Report (SLAA697), March 2016.

4. Ankur Verma and F. Xu. “Troubleshooting Guide for PGA411-Q1,” Texas Instruments Application Report (SLAA687), February 2016.

5. Ankur Verma and Brian Rodriguez, “Electrical design considerations for industrial resolver sensing applica-tions,” TI White Paper (SLYY100), June 2016.

Figure 8. Acceleration event under 12-bit mode at 50,000 rad/s2

600040002000

0

5

0

–5420

–2–4

1

0

–1

6420

200015001000500

0

Time (ms)

0 5 10 15 20 353025 40

Output andExpected

Angles

Velocity (Hz)

Error Flag(1 or 0)

Input Sineand Cosine

(rad)

ErrorAmplitude

(LSBs)

Output/ExpectedAngle Error

(LSBs)

6. Ankur Verma, “Software Developer’s Guide for PGA411-Q1,” Application Report (SLAA708) July 2016.

7. Ankur Verma, Fei Xu and Jiri Panacek, “PGA411-Q1 Step-by-Step Initialization with Any Host System,” TI Application Report (SLAA688), March 2016.

8. Ankur Verma, “Accuracy? Resolution? Arc minutes? How to take charge of your motor control design,” TI E2E™ Motor Drive & Control blog, August 17, 2016.

Related Web sitesProduct information:PGA411-Q1

AcknowledgementsThe author would like to acknowledge Srinath Hosur (systems architect), Arun Vemuri (systems architect), Ashish Khandelwal (design manager), Ted Lekan (digital design engineer from mixed signal automotive), and Charles Sestok (systems architect, Kilby Lab) from Texas Instruments for their support and guidance in system architecture and modeling.

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