Transcript Electrical circuits wyklad 6x

Dr inż. Agnieszka Wardzińska
Room: 105 Polanka
[email protected]
cygnus.et.put.poznan.pl/~award
Advisor hours:
Monday: 9.30-10.15
Wednesday: 10.15-11.00
Coupling coils
The coupling occurs when two coils are
placed near each other (See Fig.3.1). The first
coil current I1 gives magnetic field B1. When
the distance between two coils are small,
some of the magnetic field will pass through
coil 2. Variation of I1 with time, induce
electromagnetic field associated with the
changing magnetic flux in the second coil:
Mutual inductance
Dot convention
v2  M
v1   M
di2
dt
v1  M
di2
dt
di1
dt
v2   M
di1
dt
If the current ENTERS the dotted terminal of one coil, the reference polarity of the
mutual voltage in the second coil is POSITIVE at the dotted terminal of the second coil.
If the current LEAVES the dotted terminal of one coil, the reference polarity of the
mutual voltage in the second coil is NEGATIVE at the dotted terminal of the second coil.
Series connection
The effect of mutual inductance for inductors
connected together in series so that the magnetic
field of one links with the other, changes total
inductance. The increase or decrease of the
inductance depends on their orientation to each
other.
The coils are said to be Cumulatively Coupled
(Fig.3.2) if the magnetic flux produced by the
current flows through the coils in the same
direction.
The coils are said to be Differentially Coupled if the current flows through the
coils in opposite directions.
Parallel connection
The mutual inductance of the coils
connected in parallel, when the currents
goes through them in the same way
(parallel aiding inductors, see Fig3.4) can If one of the two coils was reversed (see
be calculated as:
Fig.3.5) the mutual inductance, M will have
a cancelling effect on each coil instead of an
aiding effect (parallel opposing
inductors).
Time-domain and Frequency-domain
Analysis
jM
V1
I1
jL1
jL2
I2
b) Frequency-domain circuit
a) Time-domain circuit
Time Domain
di1
di2
v1  i1 R1  L1
M
dt
dt
di
di
v2  i2 R2  L2 2  M 1
dt
dt
Frequency Domain
V1  ( R1  j L1 ) I1  j MI 2
V2  j MI1  ( R2  j L2 ) I 2
V2
Energy in a Coupled Circuit
 The total energy w stored in a mutually coupled inductor is:
 Positive sign is selected if both currents ENTER or LEAVE the dotted terminals.
 Otherwise we use Negative sign.
1 2 1
w  L1i1  L2i2 2  Mi1i2
2
2
Methods of analysing the coupligs
 Kirchoffs laws
 The rules for positive or negative couplings work for current in
branches
 Mehs current method
 The rules for positive or negative couplings work for current in
loops
 Uncupling
 The rules for positive or negative couplings work for the node
joining the branches with coupled coils.
Uncoupling
 Sometimes it could be useful replace mutual coupled
inductors by ordinary uncoupled inductors. If coupled
inductors are connected into same node, then the
replacement is
EXAMPLES
Transformers
 On the mutual inductance bases the transformer operation. The transformer is
constructed of two coils, the flux generated in one of the coils induced voltage
across the second coil. The source coil is called primary coil and the coil to
which the load is applied is called secondary.
 The basic types of transformers:
 the iron-core transformer
 the air-core transformer
 the variable-core transformer

Three basic operations of a transformer are:
 Step up/down
 Impedance matching
 Isolation
The symbol used for the transformer in
circuit theory
Linear Transformers
 A transformer is generally a four-terminal device comprising two or more
magnetically coupled coils.
 The transformer is called LINEAR if the coils are wound on magnetically linear
material.
 For a LINEAR TRANSFORMER flux is proportional to current in the windings.
 Resistances R1 and R2 account for losses in the coils.
 The coils are named as PRIMARY and SECONDARY.
Reflected Impedance for Linear Transformers
 Let us obtain the input impedance as seen from the source,
ZR
V  ( R1  j L1 ) I1  j MI 2
0   j MI1  ( R2  j L2  Z L ) I 2
V
2M 2
Zin   R1  j L1 
 R1  j L1  Z R
I1
R2  j L2  Z L
 2M 2
ZR 
R2  j L2  Z L
REFLECTED IMPEDANCE
• Secondary impedance seen from the primary side is the Reflected
Impedance.
Equivalent T Circuit for Linear Transformers
 The coupled transformer can equivalently be represented by an EQUIVALENT
T circuit using UNCOUPED INDUCTORS.
a) Transformer circuit
transformer
b) Equivalent T circuit of the
La  L1  M , Lb  L2  M , Lc  M
Equivalent П Circuit for Linear Transformers
 The coupled transformer can equivalently be represented by an EQUIVALENT
П circuit using uncoupled inductors.
a) Transformer circuit
b) Equivalent Π circuit of the transformer
L1L2  M 2
L1L2  M 2
L1L2  M 2
LA 
, LB 
, LC 
L2  M
L1  M
M
Power – DC circuit
 The electric power in watts associated with an electric circuit or a circuit
component represents the rate at which energy is converted from the electrical
energy of the moving charges to some other form, e.g., heat, mechanical
energy, or energy stored in electric fields or magnetic fields.
 For a resistor in a DC Circuit the power is given by the product of applied
voltage and the electric current.
P U I
[W ]  [V ]  [ A]
 When calculating the power dissipation of resistive components, we can also
use one of the two other power equations (they are conversions of the above
using Ohm’s law):
2
2
U
P U I  RI 
R
power is is additive for any configuration of circuit: series, parallel,
series/parallel, or otherwise.
Maximum Power Transfer Theorem
 Maximum Power Transfer Theorem states that the
maximum amount of power will be dissipated by a
load if its total resistance Rl is equal to the source total
resistance Rs of the network supplying power.
For maximum power:
The Maximum Power Transfer Theorem
does not assume maximum or even high
efficiency, what is more important for AC
power distribution.
Example
 Calculate the total power of the load. Check the
additivity rule. Calculate Rw to get the maximum power
transfer.
Power in AC circuits
 Instantaneous electric power
The time varying value of the amplitude of the sinusoidally oscillating
magnitude S and doubling the frequency around the mean value P. It is
measured in voltampere (VA).
 Active power or Real power
where ' is an phase shift between current and voltage.
The average value of power (for the period) actually consumed by the
device, able to be processed into another form (eg. mechanical, thermal),
this power is always non-negative. It is measured in watt (W).
 Reactive power
The value a purely contractual linked to periodic changes in the energy
stored in the reactive components (coil, capacitor), this power can be
positive (induction, where ' > 0) or negative (capacitive, when ' < 0). It
is measured in volt-ampere reactive (var).
 Complex power
It is proportional to the RMS values of current and voltage, and marked with the
letter S. Complex power is formally defined as a complex number in the form of
a complex product of the RMS voltage U and coupled current I. It is measured
in volt-ampere (VA). The complex power is a complex sum of real and reactive
power:
 Apparent power
 The power resulting from the amplitude of voltage and current,
including both the active power and reactive power. The apparent
power can be also calculated as the magnitude of complex power S. It is
measured in volt-ampere (VA). We can easy calculate the apparent
power: reactive (var).
or
power triangle
 We can define the power triangle the trigonometric
form showing the relation appearant power to true
power and reactive power. It is presented below:
 The angle between the real and complex power ' is a phase
of voltage relative to current. It mean the angle of
difference (in degrees) between current and voltage. The
ratio between real power and apparent power in a circuit is
called the power factor. It’s a measure of the efficiency of a
power distribution. The power factor is the cosine of the
phase angle ' between the current and voltage cos':
The power factor is by definition a dimensionless and its value is between -1 and 1.
When power factor is equal to 0, the energy flow is entirely reactive. When the
power factor is 1, all the energy supplied by the source is consumed by the load.