EE533 – Machines & Drives

Chapter 2: Modeling of DC Motors

Fall 2016 Dr. KHELDOUN A.

Chapter 2: Modeling of DC motor

Objectives

I. Introduction

II. Operation principle: review

III. Equivalent circuit

IV. Dynamic model

V. Transfer functions of DC motor

VI. Measurement of DC motor parameters

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 Chapter 2: Modeling of DC motor

I. Introduction

DC motor in service for more than a century

Dominated variable speed applications before Power

Electronics was introduced

Main advantage:
Precise torque and speed control without sophisticated
electronics

For low power applications the cost of DC motor plus drives

is still economical.

Chapter 2: Modeling of DC motor

II. Principle of Operation

Field winding - on stator pole

if produces f
Armature winding –on rotor
ia produces a
f and a mutually perpendicular
maximum torque
Using left hand rule
Rotor rotates clockwise
For unidirectional torque and
rotation
ia must be same polarity
under each field pole
achieved using commutators
and brushes

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 Chapter 2: Modeling of DC motor

II. Principle of Operation

Since this mechanism is called commutator, DC machinery is also

known as commutating machinery
Different DC motors are available and mainly differ in the
excitation circuitry
The type of excitation depends on connections of field winding
with respect to armature winding
This results in 5 different DC machines:
Separately Excited
Shunt Excited
Series Excited
Compounded
Permanent Magnet

Chapter 2: Modeling of DC motor

II. Principle of Operation

1) Separately Excited DC machine

Field winding separated from armature winding
Independent control of if ( f ) and ia (T)

Subsequently torque and flux can be controlled

independently and precisely.

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 Chapter 2: Modeling of DC motor

II. Principle of Operation

2) Shunt Excited DC machine
Field winding parallel to armature
winding
Good for constant voltage
T vs characteristic almost
constant
AR = armature reaction
Circuit connection
(as T , ia , armature flux
weakens main flux f ,
)
Troubling consequences for
variable-voltage operation
Coupling of f (if ) and T (ia)
production
Mechanical characteristic

Chapter 2: Modeling of DC motor

II. Principle of Operation

3) Series Excited DC machine

Field winding is connected in
series with armature winding

There is no independence of
Circuit connection
control between torque and flux

T ia 2 since if = ia

T resembles a hyperbola

High starting torque

No load operation must be

avoided (T = 0, ) Mechanical characteristic

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 Chapter 2: Modeling of DC motor

II. Principle of Operation L- Long-shunt connection

4) DC Compound machine S- Short-shunt connection

Combines best feature of series

and shunt
Series – high starting torque
Shunt – no load operation
Cumulative compounding
shunt and series field strengthens
each other.
Differential compounding
shunt and series field opposes each
other.

Chapter 2: Modeling of DC motor

II. Principle of Operation

5) PM DC machine
Field provided by magnets
Less heat
No field winding resistive losses
Compact
Armature similar to separately excited
machine
Disadvantages:
Can’t increase flux
Risk of de-magnetisation
due to armature reaction
Permanent Magnet DC machine

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 Chapter 2: Modeling of DC motor

III. Equivalent circuit - Separately excited DC machine -

Ra La Lf Rf

+ Ia + If +

Va ea Vf
wm
_ _ Tem _
TL

Armature circuit (rotor) Field circuit (stator)

P .Z
Induced voltage ea v x B .l ea .Wm . KWm .
2 .a
P: number of poles
P .Z 2 Wave winding
K Z: armature conductors a
2 .a P Lap winding
a: parallel paths

Chapter 2: Modeling of DC motor

III. Equivalent circuit - Separately excited DC machine -

Induced electromagnetic torque

From the previous equivalent circuit and using 2nd Kirchhoff's law, the terminal
relationship is written as

Va ea R a .I a Multiplying the equation by Ia

2 2
V a .I a e a .I a R a .I a Pin Pa R a .I a
RaIa2 represents losses in the armature. This means that Pa is the effective power that has
been converted to mechanical power. P a: air-gap power
P .Z
Pa e a .I a T em .W m .W m . .I a T em .W m
2 .a
P .Z
T em . .I a K .I a
2 .a

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 Chapter 2: Modeling of DC motor

IV. Dynamic model - Separately excited DC machine -

The dynamic model of the motor takes into account the instantaneous
variation of electrical and mechanical quantities such as: current,
voltage, speed, etc. Therefore previous steady state equation would be
as follows: ia ,I a

di ………….(1)
va ea R a .i a La a La
dt
Ra
va ,Va
Where T em ea
m
ea K m . Kg m

is constant

Chapter 2: Modeling of DC motor

IV. Dynamic model - Separately excited DC machine -

The motor is driving a mechanical load which must be represented by

model that relates the speed to torque. For simplicity, we assume that
the load has a moment of inertia J (kg.m2/s2), a viscous friction B
(N.m/rad/s) and a net torque TL . Therefore, the motion equation can be
obtained as below
d m ia , I a
Te J B m TL ………….(2)
dt
La
Where
Ra
Tem K .ia . K g ia v a ,V a
T em
is constant m ea

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 Chapter 2: Modeling of DC motor

II. Dynamic model - Separately excited DC machine -

Equations (1) and (2) can be cast under the following form

dia Ra 1 Kg
ia va m ….............(3)
dt La La La
d Kg B 1
m
ia m TL ……………..(4)
dt J J J
From which the state-space model of the DC motor can be derived

Ra Kg 1
0
si a La La ia La va
s m Kg B m 1 TL
0
J J J
Where s is differential operator with respect to time

Chapter 2: Modeling of DC motor

II. Dynamic model - Separately excited DC machine -

Ra Kg 1
0
si a La La ia La va
s m Kg B m 1 TL
0
J J J
Under the compact form ,we obtain X AX BU

where Ra Kg
La La ia
A System matrix X State variable vector
Kg B m

J J
1
0 va
B La Control matrix U Input vector
1 TL
0
J

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 Chapter 2: Modeling of DC motor

II. Dynamic model - Separately excited DC machine -

The roots of the system are the eigenvalues of matrix A are obtained
by solving the following equation

1 0
det( I A) 0 where I A 2X2 identity matrix
0 1
The solution of this equation are :
2 2
1 Ra B 1 Ra B Ra B Kg
1 4
2 La J 2 La J JLa JLa
2 2
1 Ra B 1 Ra B Ra B Kg
2 4
2 La J 2 La J JLa JLa

Chapter 3: Modeling of DC motor

IV. Dynamic model - Separately excited DC machine -

2 2
1 Ra B 1 Ra B Ra B Kg
1 ,2 4
2 La J 2 La J JLa JLa

The computation of these roots are very important from the control
viewpoint.
The real part of the eigenvalues indicates whether the system
is stable or not. In our case, it is negative :
1 Ra B
Real 1,2 0
2 La J

Therefore, SE-DC motor is stable on open-loop operation

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 Chapter 3: Modeling of DC motor

V. Block diagram and transfer functions

Taking Laplace transform of (1) and (2) and neglecting initial

conditions:
di a
va ea R a .i a La Va ( s ) K g m (s) Ra La s .I a ( s )
dt
d m
Te J B m TL Kg Ia( s ) Js B m ( s ) TL ( s )
dt

Va ( s ) K g m (s)
Ia( s ) ….............(5)
Ra La s
K g I a ( s ) TL ( s )
m (s) ……………..(6)
Js B

The latter relationships are used to construct the bloc diagram of the
DC motor.

Chapter 3: Modeling of DC motor

V. Block diagram and transfer functions

T L(s)

+ Ia(s) Te(s) -
1 1
Va (s) Kg + m (s)
- Ra sLa B sJ

Kg
From either bloc diagram or eqs. (5 and 6) the transfer function
relating speed to the input voltage can be derived
Taking TL(s)=0
Va ( s ) K g m (s)
Ia( s )
Ra La s m(s) Kg
G V(s) 2
Kg Ia( s ) Va ( s ) Js B La s Ra Kg
m( s )
Js B

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 Chapter 3: Modeling of DC motor

V. Block diagram and transfer functions

With the same procedure, the transfer function relating speed to load
torque is derived
Taking Va(s)=0

Kg m (s)
Ia( s ) (s) La s Ra
Ra La s G T(s) m
2
TL ( s ) Js B La s Ra Kg
K g I a ( s ) TL ( s )
m (s)
Js B
The DC motor is linear system, hence the speed response due
simultaneously to the input voltage and load torque disturbance can be
obtained upon applying the superposition theorem

m (s) G V ( s )Va ( s ) G T ( s )TL ( s )

The Laplace inverse of m(s) gives the speed time response (t).

Chapter 3: Modeling of DC motor

VI. Measurement of DC motor parameters

To obtain time response , analyze motor behavior or design it control system,

one needs the model parameters. Here below are given the methods used to
measure the dc motor parameters Ra, La and Kg.

A. Measurement of armature resistance Ra

DC voltage applied at armature terminals such that rated ia circulates

Vdc
Ra
ia , rated

This gives the dc value for Ra (it includes armature resistance and brush
resistance)
Better measurement can be obtained if this value is corrected by taking into
account the effect of temperature at which motor is expected to operate at
steady state
Similar procedure can be applied to find Rf of field circuit

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 Chapter 3: Modeling of DC motor

VI. Measurement of DC motor parameters

B. Measurement of armature Inductance La

Apply low AC voltage through an autotransformer at armature terminals
Measure ia
Motor must be at standstill (i.e. m = 0 and ea = 0)

Chapter 3: Modeling of DC motor

VI. Measurement of DC motor parameters

The following equation is used to compute La

2
V a ,rms 2
Ra
I a ,rms
La
2 .f

Where

Va,rms, Ia,rms are applied rms voltage and measured rms current respectively

f is frequency in Hz of the applied voltage

Ra is the ac armature resistance which is different from the dc resistance

because of the skin effect produced by the alternating current.

Similar procedure can be applied to find Lf of field circuit

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 Chapter 3: Modeling of DC motor

VI. Measurement of DC motor parameters

C. Measurement of EMF constant

Rated voltage applied across the field circuit and kept constant
The dc motor is connected to another motor (prime mover) which drives it up to
rated speed (given in the nameplate)
Voltmeter connected to armature terminals reads value of ea
Get values of ea at different shaft speeds ( m)
Plot ea vs. m
The obtained curve looks like the one ea (V)
shown in the figure
Slope of this curve gives the emf constant
Kg ea

ea
Kg
m

(rad/s)
Units of Kg = [V-s/rad]

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