Chapter 9 Analysis and Calculation of Sinusoidal Circuit

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Chapter 9 Analysis and Calculation of Sinusoidal
Circuit
Main content
1 resistance and conductivity
2 Resistance Series-parallel Connection
Key point
3 Phase Graph
4.analysis of Sinusoidal Steady State Circuit
5 power of Sinusoidal Steady State Circuit
nodus
6 Complex power
7 The transmission of maximal power
Key point
8 Series connection resonance
Key point
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9 .1 resistance and conductivity
The magnitude of complex impedance
Ohms Law
complex impedance
The angle of complex impedance
conclusionthe magnitude ofZ is the result which
the rms value of total voltage divide the rms
value total current,and the angle of Zis the
result which total voltages angle minus total
currents angle ?
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impedance
Impedance triangle
Reactance Xgt0,Z is inductive Xlt0,Z is
capacitive X0,Z is resistant
N0
admittance
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Phasor model
The relation between total voltage and total
current
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Note
In sinusoidal AC circuit,if the physics is
expressed by phasor and component s parameter
is expressed by complex impedance ,the equations
is similar to that of DC circuit.
6

R
L
C
N0
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Assume R?L?C is identified, Whether is Circuit
character identified? (impedance?inductance?capaci
tance?)
Forbidden!
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9 .2 Resistance Series-parallel Connection
(?)simple series connection circuit
ZZ1Z2
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(?)Simple Parallel Connection Circuit
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Example 9-2
solution
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9.3 Phase Graph of Circuit
In series current is reference phasor,KVL In
parallel voltage is reference phasor ,KCL
First draw reference phasor
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example 9-4 draw the phasor graph of example
R1
L
-
-
C
R2
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9.4 Analysis of Sinusoidal Steady State Circuit
Two Solution
1 Phase Operation By Virtue of Compound.
2.use phasor graph to solve
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solution 1Phase Operation By Virtue of Compound

If I110A? UAB 100V,
solution the number of A?UO
Assume
As reference phasor,
namely
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If I110A? UAB 100V,
Solve the number of A?UO
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Solution 2use phasor graph to solve
If I110A? UAB 100V,
solvethe number of A?UO
assume
I10 A? UO 141V
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Complemented example
solvecurrent of every branch
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Original circuit
Phasor model
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solution?nodes electric potential method
A
Nodes equation
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Solve the current of every branch from nodes
electric potential
A
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Solution ? Thevenins theorem
A
A

Z
B
B
solvecurrent of every branch
Solve
?
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Conclusion the solution steps of sinusoidal AC
circuit
1?Draw phasor model graph according to initial
circuit graph(stay circuits structure is
changeless)
2?write equations or draw phasor graph according
to phasor model
3?use complex number or phasor graph to solve
4?transform the result to required form
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9.5 Power of Sinusoidal Steady Circuit
?. resistive circuit
i
1 .instantaneous equation
2. Rms value equation
3. Phase equation
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Power of Resistive Circuit

• Instantaneous power pinstantaneous voltage
  multiply instantaneous current

minuscule
Conclusion
1. (consumed energy component)
3. Is proportional to
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2. average power( (active power) P the
average of the instantaneous power over one period
capital
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?.inductive circuit
2 Rms value equation
3.
3. Phase equation
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Power of Inductance Circuit
1. Instantaneous power p
Transformation of energy is reversible
Store Energy
Extract energy
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2. average power P (real power)
Instantaneous power
Conclusion in a purely inductive circuit,the
average power is zone. It only exchange energy
with source?
3. Reactive Power Q
Q definition A measure of the power associated
with purely inductive circuit the maximal
value of instantaneous power?
unit (var)
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?.capacitive circuit
1? Instantaneous equation
2 Rms value
3. Phase equation
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Power of Conductance Circuit
1.Instantaneous power p
i
u
?t
p
P gt 0
deliver
absorb
absorb
Store Energy
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2 average power P
3. Reactive Power Q
the maximal value of instantaneous power
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?? Calculation of Power of No Circuit

N0
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The relation among P?U?? I

Total voltage
The angle between u and i
Total current
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3. Reactive Power Q
In R?L?C series connection circuit ,storing
energy component L?C do not consume energy but
they can swallow and disgorge energy ,whose scale
is expressed by reactive power Q ?
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4Apparent Power S
In circuit the rms value of total voltage
multiply the rms value of total current ?
unitVA?KVA
notice SU I is used to measure to the maximal
power provided by generator(rated voltagerated
current)
5. Power triangle
Real power P
Reactive power Q
Apparent Power S
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The relation of power in circuit
if i leading u ,(capacitive circuit)
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Judge Right Or Wrong
in R-L-C series circuit
Neglect the phase of U
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Judge Right Or Wrong
?
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Judge Right Or Wrong
in R-L-C sinusoidal AC circuit
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9. 6 Compound Power

• Compound Power
• power is describer as complex number

Reactive Q unit var
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Real power is conservation reactive power is
conservation complex power is conservation
Complex power is conservation ,but apparent power
is not conservation
Complex power is expressed
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Examplecircuit shown as follows,solve the
complex power of every branch?
Solution ?
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Solution ?
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Increasing of Alternative Current Power and Power
Factor
Total current
Power factor
Total voltage
The angle between u and i
The meaning of Increasing of Alternative Current
Power and Power Factor (1)make the best of the
capacity of electric of source
SUI If S1000KVA,and
0.4,P400KW
0.9,P900KW
(2)reduce the energy loss and voltage
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Put out questions
Usually many
loads is inductive and its Equivalent circuit and
the phasor relation are as follows
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The relation between power factor cosF and
circuit parameter
Note
is defined by the character of loads? It relates
to circuit parameter, frequency but do not
relate to circuit voltage and current
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Example
40W incandescent lamp
40W daylight lamp
Require the capacity of generating and
supplying-electric facility is large
power station usually ask the user
or the users will be punished?
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The power factor of usual circuit
Purely resistive circuit
Purely inductive circuit or purely capacitive
circuits
R-L-C series circuit
electromotor none load full load
daylight lamp (R-L-C series circuit)
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Principals of Increasing Power Factor
Make sure initial loads state is constant ,
Namely load voltage and real power is constant
the measurement of increasing power factor
Parallel capacitive
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Calculation of Parallel- Connection Capacity
if initial circuits power factor is cos?
L, require to compensate cos? L to cos? , solve
the parallel connection C ? (if U?P is certain)

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Analysis Basis in all course P?U are constant?
From the phasor diagram
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Node (1)after paralleling connection capacity
,initial loads state is constant , namely ,I1
and cos?1 are constant (2)total current I
reduce,because IX reduce but IR is constant
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Which is better for compensating power factor to
inductive or capacitive?
?
Lack of compensate
Excessive compensate
inductive( smaller)
Capacitive ( larger)
Conclusion when is the same ,if compensate
to capacitive ,it requires the capacity of
capacitance is large ,and it is not economical
.so circuits usual work in lack of compensation
state
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By paralleling capacitive to compensate,the real
power of circuit is whether change?
Through computing we know total power is
constant?
qualitative illustrationthe resistances in
circuits do not change ,so consumed powers do not
change?
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what else ways to increase power factor except
paralleling capacitance ?
After compensate
C
R
L
If external voltage is constant ,load can not get
rated work voltage and current?
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Example.
Iff50Hz, U380V, P20kW, cosj10.6(lagging)?Requ
ire increase power factor is 0.9 ,solve parallel
capacitance C?
solution
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Supplement Power Measurement
Power instrument
Average torque
measurethe measure of P Us measure? Is
measure?cos?
When measure,P?U?I must not exceed each measure?
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9-7 Maximum Power Transmission
Zi Ri jXi, ZL RL jXL
RL and XLcan change at will

The Conditions of making ZL get maximal power
Best Match
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supplement example
If
Hope R to get the maximal power ,solve C?
Use thevenins theorem equivalent circuit
Solution 1
1/(j2C) -j1
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Supplement Work Principal of Lamp
Double metal electrode
Fixed electrode
filament
Light tube
Ballast L
link
-

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Generate splendent light and release electricity

• Impose Line voltage on two electrode

Double metal electrode is heated and get through
circuit(L?filament?started electrode)
Current is large
Electrode eject a large of electron
Temperature rise 800 ?C --1000 ?C
Start up implement get through
Release electricity end
Double metal electrode get cold and cut off
Ballast generate high voltage pulse
Electrode shoot a large of electron,because of
voltage pulse filament breakdown and release
electricity
Ultraviolet ray irradiate fluorescent powder and
generate visible light
Mercury atom impact with high energy electron
and generate ultraviolet ray
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9-8 Resonance of Series Connection Circuit
Sometimes use
Sometimes avoid
In L?C circuit ,if the port voltage and current
of circuit are in phase ,it is called resonance ?
Define resonance
?? Resonance of Series Connection Circuit
1?the conditions of resonance
When
Circuit arise resonance
resonant angular frequency
Fixed Frequency
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the condition of RLC series circuit arising
resonance
(1). L C constant ,change w ?
w0 is decided by circuit parameter a RLC
series connection circuit only have a
correspondingw0 when frequency equal the
resonance frequency ,circuit arise resonance?
(2). The voltage frequency is constant ,change ,L
or C ( usually change C )?
Usually radios choose broadcasting station ,in
essence choose different frequent single through
changing c to make circuit arise resonance.
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2?the feature of Resonance of RLC Series
Connection Circuit
Are in the same phase
(2). Input port Z is purely resistance ? In
circuit the value of Z is minimal?
(3). current I have the maximal value I0U/R (U
is certain)?
(4). LC series connection total voltage is zone

Electric source voltage is imposed on resistance
URU0
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In Series resonance circuit ,inductive voltage
and capacitive voltage is same in value ,but the
direction is reverse .So series connection
resonance is called Voltage Resonance
The phasor diagram in resonance circuit
When w0L1/(w0C )gtgtR, UL UC gtgtU ?
(5). power
PRI02U2/R,resistance power is up to the
maximal value?
Land C exchange energy ,loads and electric source
do not exchange energy?
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??Characteristic Impedance and Quality Factor
1. characteristic impedance ?
When arise resonance ,inductive reactance equal
capacitive reactance
units?
? do not relative to response frequency ,only is
defined by circuit parameter?
2. quality factor Q
None units
It is one of targets which illustration
performance of circuit ,and it is decided by
circuit parameter?
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The meanings of quality factor Q
(a) The relation of voltage
In resonance circuit Q is equal to the result
that inductive voltage UL0(or capacitive
voltageUC0) divide electric source voltage ?It
show the multiple that voltage is magnified.
UL0andUC0 are the external voltage Q multiples,if
w0L1/(w0C )gtgtR ,Q is high,L and C arise high
voltage ,sometimes it can be useful ,but
sometimes it should be avoided?
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Example a radio C150pF,L250mH,R20?
If a single voltage is 10mV , inductance voltage
is 650mV ,what we need.
In power system ,because electric source voltage
is high ,and when arise resonance the overwhelm
voltage breakdown insulater and destroy
equipments .what should be avoid.
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Thought
How do the common radios choose broadcasting
station? Whats that we choose broadcasting
station in essence ?

• Why some radios noise is large ,while some is
  small?
• Why we can hear several broadcast of
  broadcasting stations?
• What is the interruption that power system affect
  communication ?

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?. Selectivity and general resonance curve
(a) selectivity
The current is large to resonance single ,while
the current is small to other single .the
capacity of choosing different single is called
selectivity.
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Supplement example.
The circuit parameter of a radios
L250mH, R20W, C150ph, U1U2 U3 10mV, w
05.5?106 rad/s, f0820 kHz.
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(Very small)
?receive the broadcast of beijing820kHz?
Choose the single of w 0 from several frequent
single , namely selectivity?
The good or bad of selectivity is relative to
resonance curve, the more pointed of resonance
,the better of selectivity ?
If LC id constant ,R is large,the curve is
smooth, the selectivity is bad?
The affection which Q to selectivity the
change of R affects selectivity in essence Q
affect selectivity?
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(b) general resonance curve
In order to compare with different resonance loop
easily, we usually make current resonance curves
abscissa and ordinate divide w0 and I(w0)
respectively.
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General Resonance Curve
The larger Q,resonance curve is more pointed ?If
leave the resonance point a little ,the curve
will descend abruptly ,circuit has strong
capability of restraining to non-resonance
frequency current , so its selectivity is good?
So ,Q is a important target which illuminate the
character of resonance circuit ?
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Called BW (Band Width)
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The frequent character of UL(? ) and UC(?)
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U(? )
UL(w )
w 0, UL(w )0 0ltwltw0, UL(w ) become large w
w0, UL(w ) QU w gtw 0,current begin to become
small,but the speech is slow. XL continue to
become large ,UL still have the trend of
becoming large ,but when UL(w ) reach the maximal
value ,then it begin to reduce ? w ??,XL??,
UL(?)U?
Similarly we can discuss UC(w )?
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Base on math analysis ,when ? ? Cm,UC(?) reach
maximal valuewhen ? ? Lm,UL(?) reach maximal
value ? UC(? Cm)UL(? Lm)?
w Lmw Cm w 0?
The higher Q,the more wLm and wCm is close to w0?
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9. 9 Parallel- connection Resonance Circuit
??similar G?C?L parallel connection circuit
R L C series connection
G C L parallel connection
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R L C series connection
G C L parallel connection
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R L C series connection
G C L parallel connection
Voltage resonance
Current resonance
IL(w 0) IC(w 0) QIS
UL(w 0)UC (w 0)QU
deduce
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? ?Inductance Loop Paralleled Connected Capacity
In fact the preceding current resonance do not
realize,because inductance loop exist resistance
,circuit become series and parallel connection
circuit ,resonance phenomenon is complexity.
Arise resonance B0,
Solve
Is decided by circuit parameter?
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Only under a certain condition ,circuit can arise
resonance ,if the parameter is not appropriate it
can not arise resonance ?
When circuit parameter is certain ,whether
circuit can arise resonance by changing source
frequency ? It is decided by the following
conditions
When circuit arise resonance ,the circuit is
equal to a resistance
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Supplement Resonance of Series-Parallel
Connection Circuit
Example
( a) circuit can arise series connection
resonance (X0),also can arise parallel
connection resonance (X?)?Solve the
series-parallel connection resonance frequency by
solving input impedance?
(a)
To (a) circuit,L1?C2 parallel circuit,in low
frequency it is inductive ?With frequency
becoming large ,at a w1arise parallel connection
resonance?w gtw1,the parallel part is capacitive
,at a w2 can arise series connection resonance
with L3?
Quantitative analysis
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When Z(w )0,namely numerator 0
solution
When Y(w )0,namely denominator0,
So, w 1ltw 2?
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To (b) circuit we can do similar qualitative
analysis ? L1?C2 parallel connection,at low
frequency it is inductive ? At w1it can arise
series connection resonance with ?When w gtw1,with
the frequency becoming large ,parallel part
become capacitive from inductive, At w2 arise
parallel connection resonance?
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(b)
Respectively numerator 0? denominator0
Series resonance
Parallel resonance
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Frequency Characteristic of Resistance
Z (? )jX(? )
(a)
(b)
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Supplement Application of LC Series-Parallel
Connection Circuit
Compose kinds of passive filter circuit?
example
source u1(t),including two frequency w1?w2
(w1ltw2)
u1(t) u11(w1)u12(w2)
Require response u2(t) only include the voltage
with w1?
How to come true?
by the following passive filter circuit to come
true
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Parallel resonance,open circuit
Series resonance,short circuit
w1 single is added to loads directly ?
in the circuit w2 gtw1 ,filter high frequency,get
low frequency?
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Other Forms of Filter Circuit
band-pass filter
band elimination filter
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Conclusion Exercises of Sine Steady Circuit

• Requirements
• 1.the basis concepts of sinusoid three
  key factor of sinusoid?phase difference?waving
  and so.
• 2. complex impedance?complex
  admittance
• 3. quantitative computerphasor
  method
• 4. qualitative analysis phasor
  diagram
• 5. power computerreal power?reactive
  power?apparent power?power factor ?complex power
  and so

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??judge the result right or wrong ,if it is wrong
,please correct ?
1.
2.
If
so
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??
Circuit shown as follows ,write loop current
equation and nodes voltage equation in phasor?
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??
IfA is 1.5A(rms value)? solve(1)US? (2)solve
circuit absorb P and Q
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Electric source release
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??
If
Solve the P and Q is released by two source
respectively?
solve
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??
IfU220V,f50HZ,A1 is 4A, A2is 2A,A3is 3A,Z3is
inductive load?solveR2 ?Z3?
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Solve Z3 solution?.
4cos? j4sin? 23cos? 3j3sin? 3
Thus
16 (23cos? 3)2(3sin? 3)2 412cos?
39(cos? 3)29(sin? 3)2 412cos? 39
from (1)2(2)2
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??
phase method,assume
so
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U2 can use phase graph
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(abandon negative value)
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• Assignments9- 8. 10. 15. 24. 26(A).28.
  32 ?38

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Problems in homeworknotice writing
Judge Right Or Wrong
in R-L-C series circuit,assume

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Judge Right Or Wrong
in R-L-C series circuit,assume
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Judge right or wrong
in R-L-C series circuit,assume