DE4401
AC
PHASORS BASICS
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AC Resistors
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Resistor Current in phase with Voltage
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Resistor Power Phase
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Phasor Review
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Phase Difference
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Phasor Lag
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Phasor Sum
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Phasor Sum Resultant
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Phasor Subtraction
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AC Capacitor Breakdown voltage
• Above a particular electric field,
known as the dielectric strength Eds,
the dielectric in a capacitor becomes
conductive.
•
The voltage at which this occurs is called
the breakdown voltage Vbd of the device,
and is given by the product of the
dielectric strength and the separation
between the conductors,
•
d
Breakdown voltage limits the maximum
energy that can be stored safely in a
capacitor.
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Series and parallel circuits
• Capacitors in a parallel configuration
each have the same applied voltage.
Their capacitances add up.
– Charge is apportioned among them by size.
Using the schematic diagram to visualize
parallel plates, it is apparent that each capacitor
contributes to the total surface area.
• Connected in series, the schematic
diagram reveals that the separation
distance, not the plate area, adds up. The
total voltage difference from end to end is
apportioned to each capacitor according
to the inverse of its capacitance.
– The entire series acts as a capacitor smaller
than any of its components
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Steady-state analysis
• The response of a network has two parts:
– the natural response
– the forced response
• It was shown that the natural response (transition)
decays rapidly to zero. The forced response for
sinusoidal sources persists indefinitely, and therefore is
called the steady-state response. Because the natural
response quickly decays, the steady-state response is
often of more interest.
• We are going to work with steady-state responses for
sinusoidal sources, avoiding the problem of solving the
differential equations.
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Phasors
• The steady-state analysis is greatly simplified if the
currents and voltages are represented as vectors in the
complex-number plane. These vectors are called
phasors.
• Why doing this? Let’s assume we have to work with sine wave
voltage:
v(t) = Vmcos(wt+f)
• KVL and KCL may often give us an equations such as:
v(t) = 20cos(wt)+ 4cos(wt-45)+ 5sin(wt+90)
• To solve these, we need to do some tediously slow trigonometry.
• Using phasors and the techniques of phasor analysis, solving
circuits with sinusoidal sources gets much easier.
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Definition of a Phasor
• A phasor is a complex number, represented as a vector in
the complex-number plane.
• Phasor is a complex number whose magnitude is the
magnitude of a corresponding sinusoid, and whose phase
is the phase of that corresponding sinusoid.
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Impedance
• The impedance is the ratio of the phasor of the voltage
to the phasor of the current for that passive element.
• The unit for impedance is the same as the unit of
resistance, since it is a ratio of voltage to current.
– The impedance will behave like a resistance behaved in dc
circuits.
• We use the symbol Z for impedance.
Vxm
ZX 
I xm
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Admitance
The inverse of the impedance is called the admittance.
The admittance is the ratio of the phasor of the current to
the phasor of the voltage for that passive element. The
ratio of phasor current to phasor voltage will have units of
conductance, since it is a ratio of current to voltage. We
use the symbol Y for admittance. The admittance will
behave like a conductance behaved in dc circuits.
I xm
YX 
Vxm
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Terminology for Impedance and Admittance
• The impedance and the admittance for a combination of
elements will be complex. Thus, the impedance, or the
admittance, can have a real part and an imaginary part.
Alternatively, we can think of these values as having
magnitude and phase. We have names for the real and
imaginary parts. These names are shown below.
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Phasor Transforms of Resistors
• The phasor transform of a resistor is just a resistor.
Remember that a resistor is a device with a constant
ratio of voltage to current. If you take the ratio of the
phasor of the voltage to the phasor of the current for a
resistor, you get the resistance.
• For a resistor, the impedance and admittance are real.
ZR  R
YR  G  1
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R
Phasor Transforms of Capacitors:
Capacitive Reactance
• The phasor transform of a capacitor is a capacitor with an
admittance of jwC. In other words, the capacitor has an admittance
in the phasor domain which increases with frequency. This comes
from taking the ratio of phasor current to phasor voltage for a
capacitor, and is a direct result of the capacitive current being
proportional to the derivative of the voltage.
•
For a capacitor, the impedance and admittance are purely
imaginary. The impedance has a negative imaginary part, and the
admittance has a positive imaginary part.
• The impedance is:
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Capacitive reactance and frequency
Capacitive Reactance:
w is frequency in radians per second:
w=2pf
Impedance:
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Example
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Phasor Diagram for AC Capacitance
The current is always leading the voltage by 1/4 of a cycle or π/2 = 90o “out-ofphase” with the potential difference across the capacitor because of the
charging and discharging process. This effect can also be represented by a
phasor diagram where in a purely capacitive circuit the voltage “LAGS” the
current by 90o ( one quarter of a cycle).
For a pure capacitor, VC “lags” IC by 90o, or
we can say that IC “leads” VC by 90o
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Q factor
• The quality factor (or Q) of a capacitor at a given
frequency is a measure of its efficiency. The higher the
Q factor of the capacitor, the closer it approaches the
behaviour of an ideal, lossless, capacitor.
w is frequency in radians per second,
C is the capacitance
RC is the series resistance of the
capacitor
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