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Dongjun Lee

446.328 Mechanical System Analysis

기계시스템해석

Dongjun Lee (이동준)

Department of Mechanical & Aerospace EngineeringSeoul National University

Dongjun Lee

Today

‐ RLC circuit

‐mathematical analogy

‐ impedance

‐ op‐amp

‐ op‐amp circuits

‐ dc motors

‐ step motors

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Dongjun Lee

RLC Components

variable resistance: potentiometer

voltage source: can provide any amount 

of current (cf. constant pressure pump)

resistor: dissipate energy similar to  damping [Ω

Ohm’s law

position sensing!

Dongjun Lee

RLC Components

charge stored in capacitor

capacitor [F]: two conductors separated by 

non‐conducting medium

V can’t change instantaneously

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inductor [H]: coupling between current‐flow and induced magnetic field

magnetic flux due to coiling

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,1

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Dongjun Lee

Kirchhoff’s Laws

1. voltage law: 

sum of voltage drops around 

a closed‐loop is zero

Δ Δ Δ Δ 0

2. current law: 

sum of currents at a junction is zero

0ex) equivalent capacitance

1 1 parallel    

series    

equivalent resistance

series   

parallel    

equivalent inductance

series   

parallel    

impedance

Dongjun Lee

Impedance

impedance

ex) impedance as a transfer function

force

velocity

admittance

1

force

velocity

passive 

circuit

system

+

‐

passive 

circuit

system

incident wave

reflected wave

scatteringoperator

ex) series and parallel connections

series    parallel    

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Dongjun Lee

Mathematical Analogy

analogous! spring

1

dampingmassposition

1

velocity

mathematically equivalent can use same tool/analysis across different domains!

(e.g., transfer function, state‐space approach, …)

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Dongjun Lee

2ndOrder Response of RLC Circuit

1→

1

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‐ natural frequency  

‐ damping ratio 

‐ response of  with unit‐step  1?‐ for circuits, usually over‐damped (e.g., L = 1mH, R = 200 ‐> C < 100nF)

‐ state‐space   0 11/ / , ⋯ → ,

small

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Dongjun Lee

Input and Output Impedance

input impedance output impedance* we want:

1. very large input impedance  

no effect on the source circuit w/o change in 

2. very low output impedance 

no voltage drop in  and no effect by the loading circuit

* if  ≫ , we can consider 

two systems decoupled from each other 

can analyze them separately!

system I system II

Dongjun Lee

Input/Output Impedance ‐ Example

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11

1

1

series   

parallel    

11 1

if    ≫

* input impedance: circuit impedance seen from input source to ground

* output impedance: circuit impedance seen from output source with input short‐circuited

measurement system

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Dongjun Lee

Operational Amplifier

‐ ideal op‐amp

1) infinite input impedance       ∞ → 02) infinite open‐loop gain          ∞ → 3) zero output impedance          0 →output = 

‐ difference amplifier  large open‐loop gain: 10 ~10

‐ : ~ Ω,    0Ω

Dongjun Lee

OP‐Amp Circuits ‐ I

virtual ground

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Dongjun Lee

voltage follower

OP‐Amp Circuits ‐ II

high input impedance w/ low output impedance!

Dongjun Lee

V1

V2

R4

adjustable gain

if  ,   1 2

instrumentation amplifier

OP‐Amp Circuits ‐ III

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Dongjun Lee

OP‐Amp Circuits ‐ IV

differentiator

integrator1

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time constant   

Dongjun Lee

first‐order LPF

OP‐Amp Circuits ‐V

second‐order LPF

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11

1 1

1

v1

1

i2

i1

i

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Dongjun Lee

‐ input impedance     2 Ω‐ input offset voltage  for  0 2‐ input offset current for  0 20‐ output short circuit current = max. current from  25‐ common mode rejection ratio:   

10 10 ← /2

‐ slew‐rate .

‐ bandwidth:  (w/ fb, 

to avoid instability in high‐frequency w/ ‐180deg phase‐lag

‐maximum output     _

Real OP‐Amp (LM741C)

Dongjun Lee

Magneto‐Mechanical Coupling

i [x]

conductor

1) magnetic‐field induces force 2) motion induces voltage drop

f = BLi [z]

f: magnetic‐field induced forcei: currentB: magnetic field flux densityL: conductor length

conductor

v: motion [x]

E: back emf (electromotive force)v: velocityB: magnetic field flux density L: conductor length

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Dongjun Lee

Armature‐Controlled Motor

torque

modeling

# of colis armatureradius

back emf

linearvelocity

angularvelocity

armaturelength

unwind armature coil

Dongjun Lee

dc‐Motor Modeling

mechanical side electrical side

externaltorque

‐ shaft torque proportional to current

‐ no‐load max. shaft velocity proportional to voltage (w/   0‐ 1(if  in  ] and  in  / ])

kT: torque constant ke = 1/kn: speed 

constant

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Dongjun Lee

dc‐Motor System TF

mechanical side electrical side

often small

: torque constant 1/ : speed 

constant

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transfer matrix

Dongjun Lee

Example: Tachometer

back emf

governingequation

(tachometer)   

i

Vo

1‐st order with  /if  1[rad/s]  ‐>  →63% steady‐state value at 

0

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Dongjun Lee

Example: Speaker

systemmodeling

m

k

c

x,f

speaker coil +diaphragm

electrical amplifier

diaphragm & coilmove together

in B

Dongjun Lee

Real dc‐Motor

2: rotor – permanent magnet 5: stator – winding8: graphite brush or metal brush 7: commutator

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Dongjun Lee

Commutator and Brush

‐ commutator: sequentially alternate current w.r.t. motor rotation

‐ brush (graphite/metal): maintain contact with rotating rotor ‐> friction 

‐ brushless motor: proximity sensor + circuitry w/ NO mechanical contact

‐> less friction/ higher performance

* armature motor

brush

commutator

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dc‐motor data sheet

motor diagram

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Gearbox

[shaft w] = [motor w] / N

[shaft T] = N x [motor T] x η

‐ dc motor: designed for high speed and low torque

‐ real task: low speed and high force

‐ gear ratio = N : 1

motor wm output shaft wo

speed reduction

torqueamplification

efficiency

Dongjun Lee

Gearbox

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Dongjun Lee

Encoder

- incremental quadrature encoder

three tracks: A, B, Index (+ A-, B-, Index- for differential)index pulse: zero position, number of turns

resolution = counts per turn * 4

current input position output?

Dongjun Lee

Encoder

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Dongjun Lee

Resolver

‐ analog sensor (i.e., infinite‐resolution), absolute over a single turn

‐ started from military applications, in harsht industrial environments

‐ hot, humid, dusty, oily, or mechanically demanding environment

‐ AC voltage (6 VAC to 60 VAC, 400 Hz to 10,000 Hz) applied to the rotating coil 

(rotor) induces a voltage across the gap in the stationary coil (stator)

Dongjun Lee

Encoder vs Resolver

Four unique states of the A and B signals known as quadrature output: A high –B low, A high – B high, A low – B high, and A low – B low.Since these four detectable states occur for every line on the codewheel, the resolution of the graduated lines can be multiplied by four.The smallest resolvable angle is then ¼ the angle between the coded lines ( = 360 / 4 * n). A 10 bit encoder (1024 lines) can be used to resolve 12 bits (4096 counts).

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Dongjun Lee

PWM

input?

PWM (pulse width modulation)

- torque command signal is converted into pulses- higher duty cycle (i.e. 100*t/T[%]) -> higher current - much less power consumption than linear amplifier

T: period

t: width

linear amplifier:larger V*I dissipation

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Dongjun Lee

Motor Driver (Amplifier)

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Dongjun Lee

Stepper Motors

variable reluctance

iron tooth core

permanent magnet hybrid

obsoletebetter 

larger torque

24‐48 cpt

cheaper

best

largest torque

100‐400 cpt

expensive

permanentmagnet

polarity‐changingstator 

tooth‐attracting

stator

tootheadpermanentmagnet:

focused magneticflux

Dongjun Lee

Stepper Motor Principle

‐ digital pulse to change stator polarity ‐> angle increment

‐ pulse rate ‐> angular rate

‐ simple to control + low power 

‐ no feedback: may be slipped or exceeding from desired angle!

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Dongjun Lee

Next Lecture

‐ frequency response