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Exam Notes - WEC

WEC

Module

Electromechanical System Design (ES2C6)

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The University of Warwick

Academic year: 2022/2023

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Control Systems

Open loop control system = input transducer converts input into form that controller will understand, controller

drives a process. It cannot compensate for any disturbances (no feedback process)

Closed loop control system = output signal fed back and subtracted from reference to obtain error signal, if error is

detected the controller will make correction to reach desired output

Modelling approaches

Physical modelling approach = using first principles and physics to explain a model

System identification approach = using a data-driven modelling approach where the relationship between the

Control Systems
Modelling approaches
Transfer functions
- Process used:
Forms of functions
Poles + stability
Response definitions
Steady state errors
PID (Proportional integral derivative controller) controllers
Zeiger-Nichols approach
- First method:
- Second method:
Representing Model in MATLAB and Simulink
Inertia and torque
- Torque and angular velocity
- Inertia of shapes............................................................................................................................................................
Transmissions
- General gears equations
- Worm + wheel gear.......................................................................................................................................................
- Planetary gearbox
- Belt and pulley
- Lead and screw
- Conveyor
- Rack and pinion
Motion profiles
Motor Equations
Op Amps
- Inverting
- Non-Inverting
- Low pass filters..............................................................................................................................................................
Sampling
Wheatstone bridge
Load cells
Encoders + current sensors
Magnetism
- Hand rules
- Equations
PMDC Motors..................................................................................................................................................................
- PWM
- Transistors and MOSFETS
H bridge
Magnetic force and work
- Maxwell pulling force
Sensors
Machines
Capacitors and inductors
Waves
Complex Impedance
- General equations
- Resonant circuits
  - Process:
- Inductors and capacitors.............................................................................................................................................
Types of power................................................................................................................................................................
Power factor....................................................................................................................................................................
- Process used:
3 Phase AC.......................................................................................................................................................................
- Phase sequences
  - Star connected
  - Delta connected
  - Converting between connection types
- 3 phase power.............................................................................................................................................................
Transformers
- Ideal vs non-ideal
- Process:

system unstable if natural response grows without limit as time approaches infinity

system marginally stable if natural response remains constant or oscillates

order of system = highest order of polynomial s in denominator of transfer function so in first order system there will

only be 1 pole

Pole of input function = forced response

Pole of transfer function = natural response

Pole of real axis = exponential response in form of e-at, more negative pole = faster decay to 0

for transfer function

𝑎

𝑠+𝑎 with unit step input, 𝑎 =

1 𝑡𝑖𝑚𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, therefore larger a = faster time constant = faster

transient response

Response definitions

Note that TF = transfer function

Term Definition

Rise time (tr) the time for the response to go from 0 to 0 of its final value ≈ 2/a

Settling time (ts) time for the response to reach and stay within 2% 1 of its final value ≈ 4/a, 4

𝜁𝜔𝑛

2 nd order TF 𝐺(𝑠) = 𝜔𝑛 2

𝑠 2 +2𝜁𝜔𝑛𝑠+𝜔𝑛 2

Damping ratio (ζ)

ratio of the exponential decay frequency to the natural frequency, −

ln(%𝑂𝑆 100 )

√𝜋 2 +(ln(%𝑂𝑆 100 )

2 Natural frequency

(ωn)

the frequency oscillation of the system without damping

Peak time (tp) time required to reach the first or maximum peak, 𝜋

𝜔𝑛√1−𝜁 2

Percentage

overshoot (%OS)

amount of the waveform that over- shoots the steady state or final value occurring at

peak time and expressed as a percentage of the steady state value, %𝑂𝑆 = 𝑒

− 𝜁𝜋

√1−𝜁

Overdamped 2 poles on real axis, 𝑐(𝑡) = 𝐾 1 𝑒𝜎 1 𝑡 + 𝐾 2 𝑒𝜎 2 𝑡

Underdamped 2 complex poles where real = exponential response and imaginary = sinusoidal response,

𝑐(𝑡) = 𝐴𝑒−𝜎𝑑𝑡 cos(𝜔𝑑𝑡 − 𝜑)

Critically damped 2 repeating poles so exponential response and exponential response x time, 𝑐(𝑡) =

𝐾 1 𝑒−𝜎 1 𝑡 + 𝐾 2 𝑡𝑒−𝜎 1 𝑡

Undamped 2 imaginary only poles so sinusoidal response, 𝑐(𝑡) = 𝐴 cos(𝜔 1 𝑡 − 𝜑)

Steady state errors

Steady state error = difference between input and output for prescribed test input as t → ∞

Finite steady state error = when steady state is fixed amount away from ideal steady state

Zero steady state error: input = output

Infinite steady state error = steady state is increasing at a lower rate than the input is increasing by, typically used

with ramp waveforms

Input name What’s constant Time function Laplace transform Steady state error

Step Position 1

1 𝑠

1 1 + 𝐾𝑝

=

𝑠𝑅(𝑠)

1 + lim

𝑠→

𝐺(𝑠)

Ramp Velocity t

1 𝑠 2

1 𝐾𝑣

Parabola Acceleration 0 2

1 𝑠 3

1 𝐾𝑎

Kp, Kv, Ka = static error constants associated with their respective input types

For 0 steady state error, TF must be in form

(𝑠+𝑧 1 )(𝑠+𝑧 2 )(𝑠+𝑧 3 )...(𝑠+𝑧𝑛)

𝑠𝑛(𝑠+𝑝 1 )(𝑠+𝑝 2 )(𝑠+𝑝 3 )...(𝑠+𝑝𝑛) where n ≥ 1 for step, ≥ 2 for ramp etc

PID (Proportional integral derivative controller) controllers

proportional handles feedback, integral removes bias offset (steady state error) and derivative improves dynamic

response + stability

written as 𝑢(𝑡) = 𝐾𝑝𝑒(𝑡) + 𝐾𝑖 ∫ 𝑒(𝑡) 𝑑𝑡 + 𝐾𝑑

𝑑𝑒(𝑡)

𝑑𝑡

→

𝑈(𝑠)

𝐸(𝑠)

= 𝐾𝑝 +

𝐾𝑖

𝑠

+ 𝐾𝑑𝑠 where:

Closed loop response Rise time Overshoot Settling time Steady state error Effect of higher value

𝑲𝒑 Decrease Increase Small change Decrease More oscillations

𝑲𝒊 Decrease Increase Increase Decrease Fewer oscillations

𝑲𝒅 Small change Decrease Decrease No change Larger amplitude

Controller tuning = process of selecting the controller parameters to meet given performance specifications

Zeiger-Nichols approach

Symbol Meaning Symbol Meaning

𝑲𝒑 Proportional gain 𝑻𝒊 Integral time

𝑲𝒊 Integral gain, 𝐾𝑃

𝑇𝑖

𝑻𝑫 Derivative time

𝑲𝑫 Derivative gain, 𝐾𝑃𝑇𝐷 𝑳 Delay time

𝑲𝒄𝒓 Critical value 𝑻 Time constant

𝑷𝒄𝒓 Critical time period

Only use this if plant has no built in integrator e., transfer function has no 1/s term

First method:

Experimental method of finding response of plant to unit step input

S-shaped curve mostly due to slow response time

Transfer function =

𝐾𝑒−𝐿𝑠

𝑇𝑠+

Second method:

Set Ti = ∞, TD = 0. Then using only proportional action control,

increase Kp to critical value Kcr where output first shows

oscillation. Critical time period known as Pcr which is

determined experimentally

Representing Model in MATLAB and Simulink

Using MATLAB to represent model:

• Sys = tf(num,dem)  num = coefficients of polynomials in numerator and dem is the same but for

denominator

• Step(sys)  unit step response of transfer function (if you right click on graph and click

characteristics you can see various characteristics of the curve)

• lsim(sys,u,t)  where sys = transfer function, u = input, t = time

𝑁 = 𝑁 2 𝑁 1 𝑁 4 𝑁 3 𝑁 6 𝑁 5 ... (−1)𝑛

Worm + wheel gear.......................................................................................................................................................

Lead = distance the worm moves forward in 1 revolution = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑒𝑡ℎ 𝑜𝑛 𝑤𝑜𝑟𝑚 × 𝑎𝑥𝑖𝑎𝑙 𝑝𝑖𝑡𝑐ℎ, 𝐿 = 𝑁 1 𝑝𝑎 where lead and axial pitch are both measured in metres Single start = 1 tooth, double start = 2 teeth, triple start = 3 teeth For wheel to drive the worm, coefficient of friction between gears > angle between lead and worm diameter (dw), tan 𝜆 < 𝜇) but in most cases tan 𝜆 ≫ 𝜇

Planetary gearbox

suitable for high torque, low speed applications. Cheap, compact, and efficient Fixed sun (solar) Fixed ring (planetary) Fixed carrier (star) Carrier torque 𝑇𝑟𝑖𝑛𝑔𝑁𝑟𝑖𝑛𝑔+𝑁𝑠𝑢𝑛 𝑁𝑟𝑖𝑛𝑔 𝑇𝑠𝑢𝑛 𝑁𝑟𝑖𝑛𝑔+𝑁𝑠𝑢𝑛 𝑁𝑠𝑢𝑛 −𝑇𝑠𝑢𝑛 𝑁𝑟𝑖𝑛𝑔 𝑁𝑠𝑢𝑛 Angular velocity → gear ratio 1 − 𝜔𝑟𝑖𝑛𝑔 𝜔𝑐𝑎𝑟𝑟𝑖𝑒𝑟 = − 𝑁𝑠𝑢𝑛 𝑁𝑟𝑖𝑛𝑔 1 − 𝜔𝑠𝑢𝑛 𝜔𝑐𝑎𝑟𝑟𝑖𝑒𝑟 = − 𝑁𝑟𝑖𝑛𝑔 𝑁𝑠𝑢𝑛 𝜔𝑠𝑢𝑛 𝜔𝑟𝑖𝑛𝑔 = − 𝑁𝑟𝑖𝑛𝑔 𝑁𝑠𝑢𝑛 If component is fixed, its angular velocity = 0

Belt and pulley

Symbol Explanation Symbol Explanation 𝑱𝑷(𝑴,𝑳) Inertia of motor and load pulleys (kg m 2 ) 𝝎𝑴 Velocity of motor shaft (rad/s) 𝑫𝑷(𝑴,𝑳) Diameter of motor and load pulleys (m) 𝜽𝑳 Position of the load 𝑴𝑩 Mass of belt (kg) 𝝎𝑳 Velocity of load (rad/s) 𝜽𝑴 Position of motor shaft (rad) 𝒃𝑴,𝑳 Motor and load side damping (Nm/(rad/s)) 𝜃𝑀 = 𝑁𝜃𝐿 𝐽𝑃𝐿→𝑀 = 𝐽𝑃𝐿 ( 1 𝑁 ) 2 𝑁 = 𝐷𝑃𝐿 𝐷𝑃𝑀 𝜔𝑀 = 𝑁𝑎𝐿 𝐽𝐵→𝑀 = 𝑀𝐵 (𝐷𝑃𝑀 2 ) 2 𝑏𝑇 = 𝑏𝑀 + 𝑏𝐿→𝑀 𝑎𝑀 = 𝑁𝑎𝐿 𝐽𝐿→𝑀 = 𝐽𝐿 ( 1 𝑁) 2 𝑏𝐿→𝑀 = 𝑏𝐿 ( 1 𝑁) 2 𝐽𝑇𝑜𝑡 = 𝐽𝑀 + 𝐽𝑃𝑀 + 𝐽𝑃𝐿→𝑀 + 𝐽𝐵→𝑀 + 𝐽𝐿→𝑀 𝑇𝑀 = 𝐽𝑇𝑜𝑡𝑎𝑀 + 𝑏𝑇𝜔𝑀 + 𝑇𝐿 ( 1 𝑁 )

Lead and screw

Lead (L) = distance that the nut moves for 1 rotation of the screw = number of starts(teeth) x pitch, measured in metres per revolution or metres per radian, , 𝐿 = 𝑁 1 𝑝𝑎 Nut linear velocity = rotational velocity of screw x lead, 𝑉𝑁 = 𝜔𝑠𝑐𝑟𝑒𝑤𝐿 𝐽𝑠𝑐𝑟𝑒𝑤 = 𝑀𝑠𝑐𝑟𝑒𝑤𝑟 2 2 𝐽𝐿→𝑀 = 𝑀𝐿 ( 𝐿 2𝜋 ) 2 𝑖𝑛 𝑚 𝑟𝑒𝑣 = 𝑀𝐿𝐿 2 𝑖𝑛 𝑚 𝑟𝑎𝑑 Gravity forces = mass of load x gravity x sin (angle between system and ground), 𝐹𝑔 = 𝑀𝐿𝑔 sin 𝛾 Frictional force (Ff) = 𝜇𝑀𝐿𝑔 cos 𝛾(𝑠𝑔𝑛(𝑉𝐿)) where sgn(VL) ensures that friction acts against direction of motion 𝑇𝐿→𝑀 = 𝐿(𝐹𝑝 + 𝐹𝑔 + 𝐹𝑓) = 𝐿(𝐹𝑝 + 𝑀𝐿𝑔 sin 𝛾 + 𝜇𝑀𝐿𝑔 cos 𝛾(𝑠𝑔𝑛(𝑉𝐿))) 𝑤ℎ𝑒𝑟𝑒 𝐿 𝑖𝑛 𝑚 𝑟𝑎𝑑

𝑇𝐿→𝑀 =

𝐿

2𝜋

(𝐹𝑝 + 𝐹𝑔 + 𝐹𝑓) =

𝐿

2𝜋

(𝐹𝑝 + 𝑀𝐿𝑔 sin 𝛾 + 𝜇𝑀𝐿𝑔 cos 𝛾(𝑠𝑔𝑛(𝑉𝐿))) 𝑤ℎ𝑒𝑟𝑒 𝐿 𝑖𝑛

𝑚

𝑟𝑒𝑣

𝑇𝑀 = (𝐽𝑀 + 𝐽𝑆𝑐𝑟𝑒𝑤 + 𝑀𝐿 (

𝐿

2𝜋

)

2 ) 𝑎 +

𝐿

2𝜋

(𝐹𝑝 + 𝐹𝑔 + 𝐹𝑓) 𝑤ℎ𝑒𝑟𝑒 𝐿 𝑖𝑛

𝑚

𝑟𝑒𝑣

Conveyor

Quantity Equation Quantity Equation

𝜽𝑴

2𝜃𝐿

𝐷𝑃1 𝜽̇𝑴

2𝜃̇𝐿

𝐷𝑃

𝜽̈𝑴

2𝜃̈𝐿

𝐷𝑃1 𝑭𝒕𝒐𝒕𝒂𝒍

𝐹𝑔 + 𝐹𝑓 + 𝐹𝑝

𝑭𝒈 (𝑀𝐵 + 𝑀𝐿)𝑔 sin 𝛾 𝑭𝒇 𝜇𝐿(𝑀𝐵 + 𝑀𝐿)𝑔 cos 𝛾

𝑻𝑳→𝑴

𝐷𝑃

2 𝐹𝑡𝑜𝑡𝑎𝑙 𝑻𝑴 See below

Rack and pinion

Quantity Equation Quantity Equation

𝜽𝑴

2𝜃𝐿

𝐷𝑃 𝜽̇𝑴

2𝜃̇𝐿

𝐷𝑃

𝜽̈𝑴

2𝜃̈𝐿

𝐷𝑃

𝑱𝑳→𝑴 (𝑀𝑅 + 𝑀𝐿) (𝐷𝑃

2 )

2 𝑭𝒈 (𝑀𝑅 + 𝑀𝐿)𝑔 sin 𝛾 𝑭𝒇 𝜇𝐿(𝑀𝑅 + 𝑀𝐿)𝑔 cos 𝛾

𝑱𝑻 𝐽𝑀 + 𝐽𝑃 + 𝐽𝐿→𝑀 𝑻𝑳→𝑴 (𝐹𝑝 + 𝐹𝑔 + 𝐹𝑓) (

𝐷𝑃

2 )

Motion profiles

Motoring (total) time = Acceleration time + constant velocity (slew) time + deceleration time, 𝑡𝑀 = 𝑡𝐴𝐶𝐶 + 𝑡𝑠 + 𝑡𝐷𝐸𝐶

K value = fraction of the total run time for which the velocity is constant where constant velocity time = k x total

time, 𝑡𝑠 = 𝑘𝑡𝑀 → 𝑘 =

𝑡𝑀−𝑡𝐴𝐶𝐶−𝑡𝐷𝐸𝐶

𝑡𝑀

Trapezoidal Cosine Polynomial

K for

given

𝜃̇𝑚𝑎𝑥

2𝐿

𝑡𝑀𝜃̇𝑚𝑎𝑥

− 1

2𝐿

𝑡𝑀𝜃̇𝑚𝑎𝑥

− 1

3𝐿

𝑡𝑀𝜃̇𝑚𝑎𝑥

− 2

K for

given

𝜃̈𝑚𝑎𝑥

√1 −

4𝐿

𝑡𝑀 2 𝜃̈𝑚𝑎𝑥

√1 −

2𝜋𝐿

𝑡𝑀 2 𝜃̈𝑚𝑎𝑥

1

2 (√

√3𝑎𝑚𝑎𝑥 𝑡𝑀 2 − 16𝐿

𝑡𝑀√𝑎𝑚𝑎𝑥

− 1)

K for

given 𝜃̇ {

𝑎𝑡 𝑡 ∈ [𝑡𝑎, 𝑡𝑏]

𝑎𝑡𝑏 𝑡 ∈ [𝑡𝑏, 𝑡𝑐]

𝑎𝑡𝑏 − 𝑎(𝑡 − 𝑡𝑐) 𝑡 ∈ [𝑡𝑐, 𝑡𝑑]

} {

0𝜃̇𝑚𝑎𝑥(1 − cos(𝜃 1 ̇ 𝑡)) 𝑡 ∈ [𝑡𝑎, 𝑡𝑏]

𝑎𝑡𝑏 𝑡 ∈ [𝑡𝑏, 𝑡𝑐]

𝑎𝑡𝑏 − 𝑎(𝑡 − 𝑡𝑐) 𝑡 ∈ [𝑡𝑐, 𝑡𝑑]

}

{

2 𝜃̇

𝑚𝑎𝑥 𝑡

𝑡𝑎𝑐𝑐

− 𝜃̇

𝑚𝑎𝑥𝑡 2

𝑡𝑎𝑐𝑐 2

𝑡 ∈ [𝑡𝑎, 𝑡𝑏]

𝜃̇𝑚𝑎𝑥 𝑡 ∈ [𝑡𝑏, 𝑡𝑐]

𝜃̇𝑚𝑎𝑥 − 𝜃̇

𝑚𝑎𝑥(𝑡 − 𝑡𝑐)

2 𝑡𝑑𝑒𝑐 2

𝑡 ∈ [𝑡𝑐, 𝑡𝑑]

}

K for

given 𝜃̈ {

𝑎 𝑡 ∈ [𝑡𝑎, 𝑡𝑏]

0 𝑡 ∈ [𝑡𝑏, 𝑡𝑐]

−𝑎 𝑡 ∈ [𝑡𝑐, 𝑡𝑑]

} {

0𝜃̇𝑚𝑎𝑥𝜃 1 sin(𝜃̇ ̇ 1 𝑡) 𝑡 ∈ [𝑡𝑎, 𝑡𝑏]

0 𝑡 ∈ [𝑡𝑏, 𝑡𝑐]

−0𝜃̇𝑚𝑎𝑥𝜃 3 sin(𝜃̇ ̇ 3 𝑡) 𝑡 ∈ [𝑡𝑐, 𝑡𝑑 ]

}

{

2 𝜃̇

𝑚𝑎𝑥

𝑡𝑎𝑐𝑐

− 2 𝜃̇

𝑚𝑎𝑥𝑡

𝑡𝑎𝑐𝑐 2

𝑡 ∈ [𝑡𝑎, 𝑡𝑏]

0 𝑡 ∈ [𝑡𝑏, 𝑡𝑐]

−2 𝜃̇

𝑚𝑎𝑥 (𝑡 − 𝑡𝑐)

𝑡𝑎𝑐𝑐

𝑡 ∈ [𝑡𝑐, 𝑡𝑑]

}

Term Explanation

Capacitor impedance 𝑋𝑐 = 1

2𝜋𝑓𝐶

𝑽𝒐𝒖𝒕 𝑉𝑖𝑛 𝑋𝑐

√𝑅 2 +𝑋𝑐 2

Corner frequency = cut off

frequency = cut-off frequency

Frequency at which the output is -3dB smaller than the input (about 70% of the

input size), 𝑓𝑐 =

1 2𝜋𝑅𝐶

Gain −20 log (𝑉𝑜𝑢𝑡

𝑉𝑖𝑛

)

Pass band Range of frequencies that pass the filter (below corner frequency)

Stop band Range of frequencies that do not pass the filter

Sampling

Nyquist Frequency = highest frequency in the measured analogue signal

sample + hold = find value at specified time then hold that value until next sample time

quantization level = number of values that a digital scan can take, unfortunately normally sample falls between

quantization levels, leading to signal distortion, determined by number of bits as needs to convert signal points to

binary

Wheatstone bridge

Term Equation Term Equation

𝑽𝑪 𝑉𝑠

𝑅 2

𝑅 1 +𝑅 2 𝑽𝑫

𝑉𝑠

𝑅 4

𝑅 3 +𝑅 4

Balanced Bridge

𝑅 1

𝑅 2 =

𝑅 3

𝑅 4

𝑉𝑜𝑢𝑡 = 0

Unbalanced Bridge

𝑅 1

𝑅 2

+ 1 =

𝑉𝑠(𝑅 3 +𝑅 4 )

𝑉𝑜𝑢𝑡(𝑅 3 +𝑅 4 )+𝑉𝑠𝑅 4

𝑉𝑜𝑢𝑡 ≠ 0

𝑽𝒐𝒖𝒕 𝑉𝐶 − 𝑉𝐷

Load cells

Term Equation Term Equation

Gauge

factor

𝜀∆𝑅

𝑅

= 𝐺 𝑽𝒐𝒖𝒕 Rated output x excitation voltage

Load cell

voltage

Excitation voltage (VS, see

wheatstone bridge) normally

significantly larger than output

voltage (see wheatstone bridge) so

rated output units is normally mV/V

Application

point

How can the load cell be used? Does the load need

to be applied at a particular point? Does it work in

compression and/or tension?

Capacity

Its maximum load, given in weight

or force

Rotary

torque

sensor

sensor coupled to rotating shaft to measure

deformation when torque is applied

Rated

Output

How does its output change with

respect to applied load?

output mass (Output voltage / rated output) * rated capacity

Encoders + current sensors

Incremental Encoder resolution = 360° / number of holes per layer, also known as pulses per revolution where

resolution = 360° / pulses per revolution

Absolute encoder resolution = = 360° / 2number of bits

𝑉 = 𝐾𝐻

𝐵𝐼

𝑡 used in hall effect current sensors

Magnetism

Hand rules

Right hand grip rule: curl fingers in right hand and point thumb in direction of current, fingers = direction of field

Flemings left hand rule: thumb = force, magnetic field = pointer, current = middle at 90⁰

Equations

Term Equation Term Equation

Magnetic flux density 𝐵 =

𝜇 0 𝐼

2𝜋𝑟 (T, Tesla) Magnetic field intensity 𝐻 = 𝑁𝐼/𝑙 (amp turns per meter)

Magnetic flux ∅ = 𝐵𝐴 (Weber’s (Wb)) Total flux available = flux linkage  = N, (Wbt (Weber turns))

Useful magnetic field B = μH (T, Tesla) Material permeability 𝜇𝑚 = 𝑟 0 (Henry m-1)

Force due to field 𝐵𝐼𝑙 sin 𝜃 (N, Newtons) Induced EMF (Wire) 𝑒 = −

𝑑∅

𝑑𝑡

𝑉 (V, volts)

Induced EMF (Coil) 𝑒 = −

𝑑

𝑑𝑡 (𝑁∅) = −

𝑑

𝑑𝑡 𝑉

PMDC Motors

𝑇𝑀 = 𝐵𝑁𝐼𝑙2𝑟 sin 𝜃 but sin 𝜃 approximately 1 so abbreviated into 𝑇𝑀 = 𝐾𝑇𝐼 where 𝐾𝑇 = 2𝐵𝑁𝑙𝑟 = torque constant

therefore 𝑇𝑀 = 𝐵𝑁𝐼𝑙2𝑟 sin 𝜃 = 𝐾𝑇𝐼 = 𝐽𝑀𝑎𝑀 + 𝑏𝑀𝜔𝑀 + 𝑇𝐿

Term Equation Term Equation

𝒌𝒆 torque constant, back EMF constant Steady state current

𝑏𝜔 + 𝑇𝐿

𝐾

𝒌𝑻 Torque constant Induced EMF −𝑁

𝑑∅

𝑑𝑡

= 𝑁𝑙2𝑟𝐵

𝑑𝜃

𝑑𝑡

= 𝑘𝑒𝜔𝑅

L Armature inductance PMDC Motor voltage 𝐿

𝑑𝐼

𝑑𝑡 + 𝑅𝐼 + 𝐾𝑒𝜔𝑀

K 𝑘𝑇 = 𝑘𝑒 Output power 𝑇𝐿𝜔

Velocity

𝑉𝑘𝑇 − 𝑅𝑇𝐿

𝑅𝑏 + 𝑇𝐿 2

Motor efficency

𝑇𝐿𝜔

𝑖𝑉

=

𝑇𝐿(𝐾 2 𝑉 − 𝐾𝑅𝑇𝐿)

𝑉(𝑏𝜔 + 𝑇𝐿)(𝑅𝑏 + 𝐾 2 )

R Armature Resistance Generated torque JMaM + bM𝜔𝑀 + useful torque

V Applied voltage Electrical losses 𝐼 2 𝑅

Angular Velocity

𝐾𝑉−𝑅𝑇𝐿

𝑅𝑏+𝐾 2

 L6 slide 4, from lecture slides

PWM

Inductor current decay: 𝑖 = 𝐼(1 − 𝑒

−𝑅𝐿𝑡

).

Duty ratio = ratio of on/off time for a PWM

Transistors and MOSFETS

N type MOSFET means transistor switched off at 0V, P type MOSFET means switched on at 0V

Sensors

Max angular velocity =

2 1+𝑘

(

𝐿𝑒𝑎𝑑

𝑡𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒

)

Acceleration =

4 (1+𝑘) 2 (

𝐿𝑒𝑎𝑑

𝑡𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 2 )

Trapezoidal motion trajectory

𝑇𝑃𝑒𝑎𝑘 = 𝐽𝑇𝑎 + 𝑏𝑠𝜔𝑚𝑎𝑥 + 𝑇𝐿

Accuracy: How closely related is the output of the sensor to the true value of the input?

Drift: Changes in the output of the sensor not due to changes to the input. Generally, temperature related.

Hysteresis: The difference between the output when related to an increasing input, and the output when related to

a decreasing input. →

Linearity: The consistency of the input/output ratio over the range of the sensor. If a sensor is linear, it simplifies

processing

Precision: How consistent are the outputs or the sensor for the same input, taken at different instances. Also known

as repeatability. If you have high precision but low accuracy you have a systematic error

Range: The maximum and minimum values that a sensor can confidently measure

Resolution: The smallest change in the measured input that can be detected by the sensor.

Sensing system: sensor → signal conditioning (e., filter, amplification,

attenuation) → microcontroller

Sensitivity (gain): The amount the output changes with changes to the input, i., the gain of the sensor. High

sensitivity = low noise tolerance + large output changes, low sensitivity = cant detect small changes

Smart sensor = sensor that also incorporates signal conditioning but expensive + complex

Threshold crossing = determining when sensor has found input greater than threshold level

Machines

Basic components of any machine = ability to create a magnetic field on demand, something to channel the magnetic

field, something to be usefully acted upon by the field

Electromagnet: Uses coil of wire + current to make magnetic field, high relative permeability material directs field

downwards, will lift/hold/drop scrap metal

Solenoid: Use of a coil of wire and a current to create a magnetic field, high relative permeability material is used to

direct the magnetic field to flow across an airgap, metal slug in the solenoid is actuated

Relay: Use of a coil of wire and a current to create a magnetic field, high relative permeability material is used to

direct the magnetic field to flow across an airgap, arm of the relay is moved to make or break an electrical switch

Capacitors and inductors

Property Capacitor Inductor

Capacitance

Inductance

𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑝𝑒𝑟𝑚𝑖𝑡𝑡𝑖𝑣𝑖𝑡𝑦 × 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑙𝑎𝑡𝑒𝑠

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑝𝑙𝑎𝑡𝑒𝑠

, 𝑄

𝑉

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑢𝑟𝑛𝑠 2 × 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦 × 𝑎𝑟𝑒𝑎

𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑐𝑜𝑖𝑙

, 𝑁

2 𝜇𝐴

𝑙

Current

𝐶 𝑑𝑣

𝑑𝑡

1 𝐿

∫ 𝑉𝐿(𝑡) 𝑑𝑡

𝑡

𝑡 0

+ 𝐼 0

Voltage 𝑄

𝐶

= 1

𝑐

∫ 𝐼 𝑑𝑡 + 𝑉 0 𝐿 𝑑𝑖

𝑑𝑡

Series

𝐶𝑡𝑜𝑡𝑎𝑙 =

1 𝐶 1

+ 1𝐶

2 + 1𝐶

3 𝑒𝑡𝑐

𝐿𝑡𝑜𝑡𝑎𝑙 = 𝐿 1 + 𝐿 2 + 𝐿 3 𝑒𝑡𝑐

Parallel

𝐶𝑡𝑜𝑡𝑎𝑙 = 𝐶 1 + 𝐶 2 + 𝐶 3 𝑒𝑡𝑐

𝐿𝑡𝑜𝑡𝑎𝑙 =

1 𝐿 1

+ 1

𝐿 2

𝑒𝑡𝑐

Energy 0𝐶𝑉 2 0𝐿𝐼 2

Step

Response 𝑉𝐶(𝑡) = 𝑉𝑖𝑛 + 𝑒

− 𝑡𝑅𝐶(𝑉

0 − 𝑉𝑖𝑛) 𝐼𝐿(𝑡) = 𝑉

𝑖𝑛

𝑅

+ 𝑒

− 𝑡𝑅𝐶

(𝐼 0 − 𝑉

𝑖𝑛

𝑅

)

Waves

Term Equation Term Equation

A Amplitude 𝝋 Phase shift,

∆𝑡

𝑇 × 2𝜋 𝑜𝑟 360 for deg or rad

𝝎 Radian frequency, 2𝜋𝑓

Periodic

signal

𝑥(𝑡) = 𝑥(𝑡 + 𝑛𝑇)

𝒇 Natural frequency (Hz)

Sine

wave

𝑥(𝑡) = 𝐴𝑐𝑜𝑠(𝜔𝑡 + 𝜑) where positive phase shift = shifted left

𝒙𝑹𝑴𝑺 𝐴 ÷ √2 = √

𝑇

∫ 𝑥 2 (𝑡)𝑑𝑡

𝑇

0 Average

value

< 𝑥(𝑡) > = 1

𝑇

∫ 𝑥(𝑡)𝑑𝑡

𝑇

0

Complex Impedance

General gears equations

𝐴𝑒𝑖𝜃 = Acos(𝜃) + 𝐴𝑗𝑠𝑖𝑛(𝜃) = 𝐴∠𝜃

𝑌∠𝑎

𝑋∠𝑏

=

𝑌

𝑋

∠(𝑎 − 𝑏)

In phasor diagram: positive real = right, positive imagined = up, DRAW PHASOR DIAGRAM

Current through resistor: 𝑖(𝑡) =

𝐴

𝑅 cos(2𝜋𝑓𝑡) =

𝐴

𝑅 cos(𝜔𝑡), phasor form: 𝐼(𝑗𝜔) =

𝐴

𝑅 ∠

𝑉 = 𝑉𝑝 cos(𝜔𝑡)

𝐼 = 𝐼𝑝 cos(𝜔𝑡)

Resonant circuits

Resonant frequency = the electrical frequency at which the circuit draws only active power i. reactive power is zero

Applies to RLC circuits

Process:

1. Find total impedance of circuit (hardest step):

a. 𝑍𝐿 = 𝑗𝜔𝐿, 𝑍𝐶 =

−𝑗

𝜔𝐶

b.

1 𝑍𝑇

=

𝑍 1 𝑍 2

𝑍 1 +𝑍 2

c. Attempt to get all components to have same denominator then add the components

together

d. Don’t be afraid to use brackets or

𝑎

𝑏+𝑐

×

𝑏−𝑐

2. Split impedance into complex and imaginary parts

Causes current to lag voltage = decrease power factor so more complex power needed for same active power =

higher energy costs + lower efficiency

Separate reactive power systems connected to power system = more active power per unit complex power

Resistance Capacitor reactance Inductor reactance

Equation 𝑉𝑟𝑚𝑠 2

𝑃

𝑉𝑟𝑚𝑠 2

𝑄𝐶

𝑉𝑟𝑚𝑠 2

𝑄𝐿

Process used:

1. 𝑍𝐶 =

−𝑗

𝜔𝐶

=

−𝑗

2𝜋𝑓𝐶

, 𝑍𝐿 = 𝑗𝜔𝐿 = 𝑗2𝜋𝑓𝐿

2. Find impedance of each of the branches  INCLUDE RESISTORS

3. Create equation of total impedance using parallel equation 𝑍𝑇 =

𝑍 1 𝑍 2

𝑍 1 +𝑍 2

4. Convert to Z∠𝜃 form to simplify easier as

𝑌∠𝑎

𝑋∠𝑏

=

𝑌

𝑋

∠(𝑎 − 𝑏)

5. Once you have total impedance, find 𝐼 using 𝐼 =

𝑉𝑖𝑛

𝑍𝑇

remember 𝑉𝑟𝑚𝑠 =

𝑉𝑝𝑒𝑎𝑘

√

6. 𝑃 = 𝐼𝑅𝑀𝑆 2 𝑅 → 𝑄 = 𝐼𝑅𝑀𝑆 2 𝑋 → |𝑆| = √𝑃 2 + 𝑄 2

7. Power factor =

𝑅𝑒𝑎𝑙 𝑝𝑜𝑤𝑒𝑟

𝐶𝑜𝑚𝑝𝑙𝑒𝑥 𝑝𝑜𝑤𝑒𝑟

=

𝑃

𝑆

 can stop at this point if question asks

8. To build circuit with specific power factor:

a. Find load angle using 𝜑 = cos−1(𝑃𝑜𝑤𝑒𝑟 𝑓𝑎𝑐𝑡𝑜𝑟)

b. Find required reactive power using 𝑃 tan(𝑙𝑜𝑎𝑑 𝑎𝑛𝑔𝑙𝑒)

c. Find required reactive power to reach load angle using 𝑄𝐶 = 𝑄 − 𝑃 tan(𝑙𝑜𝑎𝑑 𝑎𝑛𝑔𝑙𝑒) if

required power > reactive power or 𝑄𝐿 = 𝑄 + 𝑃 tan(𝑙𝑜𝑎𝑑 𝑎𝑛𝑔𝑙𝑒) if required power <

reactive power

d. Find required reactance using 𝑋𝐶 =

𝑉𝑟𝑚𝑠 2

𝑄𝐶

=

1 𝜔𝐶

=

1 2𝜋𝑓𝐶

if required reactance is negative

or

𝑉𝑟𝑚𝑠 2

𝑄𝐿

= 𝜔𝐿 if required reactance is positive  struggle to explain b and c

e. Solve for required capacitance or inductance

f. The required capacitance or inductance should be in parallel with prior total impedance

g. Find new reactive power using 𝑄 = 𝑃 tan(𝑙𝑜𝑎𝑑 𝑎𝑛𝑔𝑙𝑒)

h. Find new complex power by using |𝑆| = √𝑃 2 + 𝑄 2

3 Phase AC

Ideal 3 phase power source comprises of voltages/currents with the same amplitude, frequency, impedance and are

out of phase by 120° with each other. The 3 phase loads have the same impedance on each phase

Balanced 3 phase system = all phase voltages, phase impedances and phase currents are equal and assumes each

phase is connected to an identical load of impedance Z

Phase sequences

𝑉𝑝ℎ = phase voltage

Phase Positive Negative

𝑽𝑨 𝑉𝑝ℎ cos(𝜔𝑡 + 𝜑) 𝑉𝑝ℎ cos(𝜔𝑡 + 𝜑)

𝑽𝑩 𝑉𝑝ℎ cos(𝜔𝑡 + 𝜑 − 120) 𝑉𝑝ℎ cos(𝜔𝑡 + 𝜑 − 240)

𝑽𝑪 𝑉𝑝ℎ cos(𝜔𝑡 + 𝜑 − 240) 𝑉𝑝ℎ cos(𝜔𝑡 + 𝜑 − 120)

Each source has a positive and side Star (Wye) connected system = the negative of each phase is connected to the ground – or Neutral Delta connected system = the negative of each phase is connected to another phase L9 slide 9 from lecture slides Term Definition Term Definition Phase voltages Voltage across a Phase, e., 𝑉𝑎𝑛 Phase current Current flowing in each phase, e., 𝐼𝑎𝑏 Line voltages Voltage across two lines, e., 𝑉𝑎𝑏 Line Current Current flowing in each line, e., 𝐼𝑎 Phase and line quantities will always either be the same, of have a difference in magnitude of √3 and a phase shift of ±∠30° 𝑉𝐿 = 𝑉𝑎𝑏 = line voltage, 𝑉𝑝ℎ = phase voltage Star connected Voltage Positive Negative 𝑽𝒂𝒏 𝑉𝑝ℎ cos(𝜔𝑡) 𝑉𝑝ℎ cos(𝜔𝑡) 𝑽𝒃𝒏 𝑉𝑝ℎ cos(𝜔𝑡 − 120) 𝑉𝑝ℎ cos(𝜔𝑡 − 240) Line voltage (𝑽𝒂𝒃) 𝑉𝑎𝑛 − 𝑉𝑏𝑛 = √ 3 𝑉𝑝ℎ cos(𝜔𝑡 + 30) 𝑉𝑎𝑛 − 𝑉𝑏𝑛 = √ 3 𝑉𝑝ℎ cos(𝜔𝑡 − 30) 𝑽𝑳 √3|𝑉𝑝ℎ|∠(𝑉𝑝ℎ + 30) √3|𝑉𝑝ℎ|∠(𝑉𝑝ℎ − 30) 𝑽𝒑𝒉 |𝑉𝐿| √ ∠(𝑉𝐿 − 30) |𝑉𝐿| √ ∠(𝑉𝐿 + 30) Phase current = line current Delta connected Current Positive Negative 𝑰𝒂𝒃 𝐼𝑝ℎ cos(𝜔𝑡) 𝐼𝑝ℎ cos(𝜔𝑡) 𝑰𝒄𝒂 𝐼𝑝ℎ cos(𝜔𝑡 − 240) 𝐼𝑝ℎ cos(𝜔𝑡 − 120) Line current (𝑰𝒂) 𝐼𝑎𝑏 − 𝐼𝑐𝑎 = √3𝐼𝑝ℎ cos(𝜔𝑡 − 30) 𝐼𝑎𝑏 − 𝐼𝑐𝑎 = √3𝐼𝑝ℎ cos(𝜔𝑡 + 30) 𝑰𝑳 √3|𝐼𝑝ℎ|∠(𝐼𝑝ℎ − 30) √3|𝐼𝑝ℎ|∠(𝐼𝑝ℎ + 30) 𝑰𝒑𝒉 |𝐼𝐿| √ ∠(𝐼𝐿 + 30) |𝐼𝐿| √ ∠(𝐼𝐿 − 30) Phase voltage = line voltage

Ideal vs non-ideal

Assumption made about ideal transformer Reality Windings have 0 resistance Windings have resistance Core permeability = ∞ so 0 reluctance Reluctance > 0 so finite Core permeability No leakage flux (all flux remains in core) There is leakage No core losses Real + reactive power loss due to Hysteresis and eddy currents Windings resistance modelled as resistor in series with windings (curls) on both primary and secondary coils Magnetic flux leakage modelled as reactance in series with windings (curls) on both primary and secondary coils When modelling the transformer, the secondary resistors and reactance’s are transferred over to the primary winding to simplify the model For very large transformers, winding resistance negligible relative to leakage reactance so omitted Simplified transformer model from L9 slide 14 →

Process:

This is very question specific so these steps may not be in the correct order / relevant to each question - Find N using 𝑁𝑃 𝑁𝑆 = 𝑉𝑖𝑛 𝑉𝑜𝑢𝑡 = 𝐼𝑜𝑢𝑡 𝐼𝑖𝑛 - Find impedance of what you can (page 15 useful for capacitors and inductors) - Find relevant impedance using 𝑍𝑃 = 𝑍𝑆𝑁 2 and convert to phasor (𝑎∠𝑏 form) - Remember that V = IR = IZ, P = I 2 R and other simple equations if they’re needed - 𝑁𝑃 𝑁𝑆 = 𝑉𝑖𝑛 𝑉𝑜𝑢𝑡 = 𝐼𝑜𝑢𝑡 𝐼𝑖𝑛

Multiple Choice

Multiple Choice

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Exam Notes - WEC

Module: Electromechanical System Design (ES2C6)

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University: The University of Warwick

Multiple Choice

Multiple Choice

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Table of Contents

Control Systems ................................................................................................................................................................ 2

Modelling approaches ...................................................................................................................................................... 2

Transfer functions ............................................................................................................................................................. 3

Process used: ................................................................................................................................................................ 3

Forms of functions ............................................................................................................................................................ 3

Poles + stability ................................................................................................................................................................. 3

Response definitions ......................................................................................................................................................... 4

Steady state errors ............................................................................................................................................................ 4

PID (Proportional integral derivative controller) controllers ............................................................................................ 5

Zeiger-Nichols approach ................................................................................................................................................... 5

First method: ................................................................................................................................................................. 5

Second method: ............................................................................................................................................................ 5

Representing Model in MATLAB and Simulink ................................................................................................................. 5

Inertia and torque ............................................................................................................................................................. 6

Torque and angular velocity ......................................................................................................................................... 6

Inertia of shapes............................................................................................................................................................ 6

Transmissions .................................................................................................................................................................... 6

General gears equations ............................................................................................................................................... 6

Worm + wheel gear ....................................................................................................................................................... 7

Planetary gearbox ......................................................................................................................................................... 7

Belt and pulley .............................................................................................................................................................. 7

Lead and screw ............................................................................................................................................................. 7

Conveyor ....................................................................................................................................................................... 8

Rack and pinion ............................................................................................................................................................. 8

Motion profiles ................................................................................................................................................................. 8

Motor Equations ............................................................................................................................................................... 9

Op Amps ............................................................................................................................................................................ 9

Inverting ........................................................................................................................................................................ 9

Non-Inverting ................................................................................................................................................................ 9

Low pass filters .............................................................................................................................................................. 9

Sampling .......................................................................................................................................................................... 10

Wheatstone bridge ......................................................................................................................................................... 10

Load cells ......................................................................................................................................................................... 10

Encoders + current sensors ............................................................................................................................................. 10

Magnetism ...................................................................................................................................................................... 11

Hand rules ................................................................................................................................................................... 11