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ECE 4951. Lecture 2: Power Switches and PID Control of Motorized Processes. Power SemiConductors. High Voltage (100’s of Volts) High Current (10’s of Amps) High Power Transistors, SCR’s Power Diode Power BJT, IGBT Power MOSFET Thyristor (Power SCR), GTO. High Power DC Switch.

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ece 4951

ECE 4951

Lecture 2:

Power Switches and PID Control of Motorized Processes

power semiconductors
Power SemiConductors
  • High Voltage (100’s of Volts)
  • High Current (10’s of Amps)
  • High Power Transistors, SCR’s
    • Power Diode
    • Power BJT, IGBT
    • Power MOSFET
    • Thyristor (Power SCR), GTO
high power dc switch
High Power DC Switch
  • Use Power Transistor as a Switch (On/Off) on a Power Circuit
  • Small Signal (Low power) Controls Large Signal (Like a Relay)
  • Combine with Inductors and Capacitors for Wave-Shaping
power mosfets
Power MOSFETs
  • Simple to Bias
  • Hundreds of Volts
  • Tens of Amps
  • Low Gate Voltages
    • Vgs < +/- 20 Volts (DO NOT EXCEED)
  • Fairly Fast Switching times (200 nS)
dc dc chopper
DC-DC Chopper
  • Power Transistor “Chops” High Voltage DC into Low Voltage DC (DC to DC Transformation)
chopper output waveforms
Chopper Output Waveforms
  • Transistor Chops Voltage into Square Wave
  • Inductor Smoothes Current
biasing circuit for p mosfet switch
Biasing Circuit for P-MOSFET Switch
  • Design Goals:
    • 5V Logic to turn on/off switch
    • Want MOSFET lightly in saturation when on (Vgs=10-15V) [Avoid approaching Vgs=+/-20V]
    • Want to control a 24V circuit
    • Want to protect Logic Source from Transients
motor types
Motor Types
  • AC, DC or Universal
  • DC Motors:
    • Wound Stator
      • Series
      • Shunt
      • Compound (both)
    • Permanent Magnet Stator
    • Brushless (Permanent Magnet Rotor)
speed torque characteristics of dc motors
Speed – Torque Characteristicsof DC Motors
  • Shunt – Constant Speed
  • Series – ‘Traction Motor’
  • Compound – Anywhere in between
  • PM Motor – Constant Speed like a Shunt Motor
gear heads
Gear Heads
  • Common to take a High-Speed, Low Torque motor (Permanent Magnet) and match it to a Gear Head.

T = P/ω

  • Produces a Low-Speed, High Torque Motor where speed can be varied with applied voltage.
servomotor
ServoMotor
  • Designed for Classic Feedback Control
  • Shaft position is encoded and available to control system as an electronic signal
  • Shaft encoder usually matched with a gearhead PM motor (or equivalent)
feedback control
Feedback Control
  • Reference Signal Calls for Specific Shaft Position, θ
  • Controller responds, with actual shaft position fed back to achieve swift and accurate outcome
stepper motor
Stepper Motor
  • Designed for Pulsed Input(s)
  • Each Pulse advances (or retards) shaft by a fixed angle (No feedback needed to know θ)
pid control
PID Control
  • A closed loop (feedback) control system, generally with Single Input-Single Output (SISO)
  • A portion of the signal being fed back is:
    • Proportional to the signal (P)
    • Proportional to integral of the signal (I)
    • Proportional to the derivative of the signal (D)
when pid control is used
When PID Control is Used
  • PID control works well on SISO systems that can are or can be approximated to 2nd Order, where a desired Set Point can be supplied to the system control input
  • PID control handles step changes to the Set Point especially well:
    • Fast Rise Times
    • Little or No Overshoot
    • Fast settling Times
    • Zero Steady State Error
  • PID controllers are often fine tuned on-site, using established guidelines
control theory
Control Theory
  • Consider a DC Motor turning a Load:
    • Shaft Position, Theta, is proportional to the input voltage
looking at the motor
Looking at the Motor:
  • Electrically (for Permanent Magnet DC):
combining elect mech
Combining Elect/Mech

Torque is Conserved: Tm = Te

1 and 2 above are a basis for the state-space description

this motor system is 2 nd order
This Motor System is 2nd Order
  • So, the “plant”,G(s) = K / (s2 + 2as + b2)
    • Where a = damping factor, b = undamped freq.
  • And a Feedback Control System would look like:
physically we want
Physically, We Want:
  • A 2nd Order SISO System with Input to Control Shaft Position:
adding the pid
Adding the PID:
  • Consider the block diagram shown:
  • C(s) could also be second order….(PID)
pid mathematically
PID Mathematically:
  • Consider the input error variable, e(t):
    • Let p(t) = Kp*e(t) {p proportional to e (mag)}
    • Let i(t) = Ki*∫e(t)dt {i integral of e (area)}
    • Let d(t) = Kd* de(t)/dt {d derivative of e (slope)}

AND let Vdc(t) = p(t) + i(t) + d(t)

Then in Laplace Domain:

Vdc(s) = [Kp + 1/s Ki + s Kd] E(s)

pid implemented
PID Implemented:

Let C(s) = Vdc(s) / E(s) (transfer function)

C(s) = [Kp + 1/s Ki + s Kd]

= [Kp s + Ki + Kd s2] / s (2nd Order)

THEN

C(s)G(s) = K [Kd s2 + Kp s + Ki]

s(s2 + 2as + b2)

AND

Y/R = Kd s2 + Kp s + Ki

s3 + (2a+Kd)s2 + (b2+Kp) s + Ki

implications
Implications:
  • Kd has direct impact on damping
  • Kp has direct impact on resonant frequency

In General the effects of increasing parameters is:

Parameter: Rise Time Overshoot Settling Time S.S.Error

Kp Decrease Increase Small Change Decrease

Ki Decrease Increase Increase Eliminate

Kd Small Change Decrease Decrease None

tuning a pid
Tuning a PID:
  • There is a fairly standard procedure for tuning PID controllers
  • A good first stop for tuning information is Wikipedia:
  • http://en.wikipedia.org/wiki/PID_controller
deadband
Deadband
  • In noisy environments or with energy intensive processes it may be desirable to make the controller unresponsive to small changes in input or feedback signals
  • A deadband is an area around the input signal set point, wherein no control action will occur
time step implementation of control algorithms digital controllers
Time Step Implementation of Control Algorithms (digital controllers)
  • Given a continuous, linear time domain description of a process, it is possible to approximate the process with Difference Equations and implement in software
  • Time Step size (and/or Sampling Rate) is/are critical to the accuracy of the approximation
from differential equation to difference equation
From Differential Equation to Difference Equation:
  • Definition of Derivative:

dU = lim U(t + Δt) – U(t)

dt Δt0Δt

  • Algebraically Manipulate to Difference Eq:

U(t + Δt) = U(t) + Δt*dU

dt

(for sufficiently small Δt)

  • Apply this to Iteratively Solve First Order Linear Differential Equations (hold for applause)
implementing difference eqs
Implementing Difference Eqs:
  • Consider the following RC Circuit, with 5 Volts of initial charge on the capacitor:
  • KVL around the loop:

-Vs + Ic*R + Vc = 0, Ic = C*dVc/dt

OR dVc/dt = (Vs –Vc)/RC

differential to difference with time step t
Differential to Difference with Time-Step, T:
  • Differential Equation:

dVc/dt = (Vs –Vc)/RC

  • Difference Equation by Definition:

Vc(kT+T) = Vc(kT) + T*dVc/dt

  • Substituting:

Vc(kT+T) = Vc(kT) + T*(Vs –Vc(kT))/RC

coding in scilab
Coding in SciLab:

R=1000

C=1e-4

Vs=10

Vo=5

//Initial Value of Difference Equation (same as Vo)

Vx(1)=5

//Time Step

dt=.01

//Initialize counter and time variable

i=1

t=0

//While loop to calculate exact solution and difference equation

while i<101, Vc(i)=Vs+(Vo-Vs)*exp(-t/(R*C)),

Vx(i+1)=Vx(i)+dt*(Vs-Vx(i))/(R*C),

t=t+dt,

i=i+1,

end

integration by trapezoidal approximation
Integration by Trapezoidal Approximation:
  • Definition of Integration (area under curve):
  • Approximation by Trapezoidal Areas
trapezoidal approximate integration in scilab
Trapezoidal Approximate Integration in SciLab:

//Calculate and plot X=5t and integrate it with a Trapezoidal approx.

//Time Step

dt=.01

//Initialize time and counter

t=0

i=2

//Initialize function and its trapezoidal integration function

X(1)=0

Y(1)=0

//Perform time step calculation of function and trapezoidal integral

while i<101,X(i)=5*t,Y(i)=Y(i-1)+dt*X(i-1)+0.5*dt*(X(i)-X(i-1)),

t=t+dt,

i=i+1,

end

//Plot the results

plot(X)

plot(Y)

coding the pid
Coding the PID
  • Using Difference Equations, it is possible now to code the PID algorithm in a high level language

p(t) = Kp*e(t)  P(kT) = Kp*E(kT)

i(t) = Ki*∫e(t)dt 

I(kT+T) = Ki*[I(kT)+T*E(kT+T)+.5(E(kT+T)-E(kT))]

d(t) = Kd* de(t)/dt  D(kT+T) = Kd*[E(kT+T)-E(kT)]/T

example permanent magnet dc motor
Example: Permanent Magnet DC Motor

State-Space Description of the DC Motor:

0. θ’ = ω (angular frequency)

1. Jθ’’ + Bθ’ = KtIa  ω’ = -Bω/J + KtIa/J

  • LaIa’ + RaIa = Vdc - Kaθ’ 

Ia’ = -Kaω/La –RaIa/La +Vdc/La

In Matrix Form:

dc motor with pid control
DC Motor with PID control

//PID position control of permanent magnet DC motor

//Constants

Ra=1.2;La=1.4e-3;Ka=.055;Kt=Ka;J=.0005;B=.01*J;Ref=0;Kp=5;Ki=1;Kd=1

//Initial Conditions

Vdc(1)=0;Theta(1)=0;Omega(1)=0;Ia(1)=0;P(1)=0;I(1)=0;D(1)=0;E(1)=0

//Time Step (Seconds)

dt=.001

//Initialize Counter and time

i=1;t(1)=0

//While loop to simulate motor and PID difference equation approximation

while i<1500, Theta(i+1)=Theta(i)+dt*Omega(i),

Omega(i+1)=Omega(i)+dt*(-B*Omega(i)+Kt*Ia(i))/J,

Ia(i+1)=Ia(i)+dt*(-Ka*Omega(i)-Ra*Ia(i)+Vdc(i))/La,

E(i+1)=Ref-Theta(i+1),

P(i+1)=Kp*E(i+1),

I(i+1)=Ki*(I(i)+dt*E(i)+0.5*dt*(E(i+1)-E(i))),

D(i+1)=Kd*(E(i+1)-E(i))/dt,

Vdc(i+1)=P(i+1)+I(i+1)+D(i+1),

//Check to see if Vdc has hit power supply limit

if Vdc(i+1)>12 then Vdc(i+1)=12

end

t(i+1)=t(i)+dt,

i=i+1,

//Call for a new shaft position

if i>5 then Ref=10

end

end

references
References:
  • Phillips and Nagle, Digital Control System Analysis and Design, Prentice Hall, 1995, ISBN-10: 013309832X

WAKE UP!!

team assignment
Team Assignment:
  • Build a Power Switch using an N-MOS transistor (IRF 830) and an opto-isolator (PS-2501). Use your programmable device to turn a 12 V circuit on and off.
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