AC Review

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AC Review

• Discussion D12.2

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Passive Circuit Elements
i
i
i

-
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Energy stored in the capacitor
The instantaneous power delivered to the
capacitor is
The energy stored in the capacitor is thus
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Energy stored in the capacitor
Assuming the capacitor was uncharged at t -?,
and knowing that
represents the energy stored in the electric
field established between the two plates of the
capacitor. This energy can be retrieved. And,
in fact, the word capacitor is derived from this
elements ability (or capacity) to store energy.
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Parallel Capacitors
Thus, the equivalent capacitance of N capacitors
in parallel is the sum of the individual
capacitances. Capacitors in parallel act like
resistors in series.
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Series Capacitors
The equivalent capacitance of N series connected
capacitors is the reciprocal of the sum of the
reciprocals of the individual capacitors.
Capacitors in series act like resistors in
parallel.
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Energy stored in an inductor
The instantaneous power delivered to an inductor
is
The energy stored in the magnetic field is thus
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Series Inductors
The equivalent inductance of series connected
inductors is the sum of the individual
inductances. Thus, inductances in series combine
in the same way as resistors in series.
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Parallel Inductors
The equivalent inductance of parallel connected
inductors is the reciprocal of the sum of the
reciprocals of the individual inductances.
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Complex Numbers
Complex Plane
imag
A
real
j
Euler's equation
j measured positive counter-clockwise
Note
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Relationship between sin and cos
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Comparing Sinusoids
Note positive angles are counter-clockwise
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Phasors
A phasor is a complex number that represents the
amplitude and phase of a sinusoid.
XM
j
Recall that when we substituted
in the differential equations, the
cancelled out.
We are therefore left with just the phasors
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Impedance
Impedance
Units ohms
Note that impedance is a complex number
containing a real, or resistive component, and an
imaginary, or reactive, component.
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Admittance
Admittance
Units siemens
susceptance
conductance
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Im
V
V
I
I
Re
I in phase with V
Im
V
I
V
Re
I lags V
I
Im
V
I
V
I
Re
I leads V
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We see that if we replace Z by R the impedances
add like resistances.
Impedances in series add like resistors in series
Impedances in parallel add like resistors in
parallel
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Voltage Division
But
Therefore
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Instantaneous Power
Note twice the frequency
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Average Power
Purely reactive circuit
Purely resistive circuit
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Effective or RMS Values
We define the effective or rms value of a
periodic current (voltage) source to be the dc
current (voltage) that delivers the same average
power to a resistor.
root-mean-square
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Effective or RMS Values
and
Using
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Ideal Transformer - Voltage
The input AC voltage, v1, produces a flux
f
This changing flux through coil 2 induces a
voltage, v2 across coil 2
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Ideal Transformer - Current
Magnetomotive force, mmf
f
The total mmf applied to core is
For ideal transformer, the reluctance R is zero.
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Ideal Transformer - Impedance
Load impedance
Input impedance
Turns ratio
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Ideal Transformer - Power
Power delivered to primary
Power delivered to load
Power delivered to an ideal transformer by the
source is transferred to the load.
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Force on current in a magnetic field
Force on moving charge q -- Lorentz force
Current density, j, is the amount of charge
passing per unit area per unit time. N number
of charges, q, per unit volume moving with mean
velocity, v.
j
Force per unit length on a wire is
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Rotating Machine
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Back emf
B
B
Force out
r
i
a
q
l

-
b
i
Force in
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Back emf
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Armature with four coil loops
S
N
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Motor Circuit
Power and Torque