AC Circuits with AC Source

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AC Circuits with AC Source
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• AC Circuits with AC Source
• Instantaneous, peak, and root mean square (rms)
  values of current, voltage, and power
• The use of phasors
• LRC Series AC Circuit
• Ohms law for ac circuits
• Impedance
• Resonance
• Resonance in AC Circuits

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AC Source

• AC generator can produce
• the emf
• the current
• varying sinusoidally in time
• The instantaneous current
• The instantaneous voltage

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AC Source

• ?2pf, the angular frequency f, the frequency of
  the source.
• ?, the phase difference between current and
  voltage.
• Resistors, capacitors, and inductors have
  different phase relationships between current and
  voltage when placed in an ac circuit.

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A Resistor in an AC Circuit
The current through a resistor is in phase with
the voltage.
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A Resistor in an AC Circuit
Instantaneous power dissipated in the resistor
average value over a complete cycle
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A Resistor in an AC Circuit

• root mean square (rms) current
• rms voltage
• rms power

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An Inductor in an AC Circuit
Therefore, the current through an inductor lags
the voltage by 90.
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An Inductor in an AC Circuit
The voltage across the inductor is related to the
current through it in the form of Ohms law
The quantity XL is called the inductive
reactance, and has units of ohms
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An Inductor in an AC Circuit
The instantaneous power supplied to the inductor
is
The average power over a complete circle is zero.
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An Inductor in an AC Circuit
Reactance of a coil. A coil has a resistance R
1.00 O and an inductance of 0.300 H. Determine
the current in the coil if (a) 120-V dc is
applied to it, and (b) 120-V ac (rms) at 60.0 Hz
is applied.
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A Capacitor in an AC Circuit
The current in the circuit is charging the
capacitor, so idq/dt. Thus dqidt
The constant depends on the initial condition and
we choose it to be zero.
The quantity XC is called the capacitive
reactance, and (just like the inductive
reactance) has units of ohms
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A Capacitor in an AC Circuit
According to the loop rule, v-vc0
Therefore, the current leads the voltage by 90.
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A Capacitor in an AC Circuit
The voltage across the inductor is related to the
current through it in the form of Ohms law
The quantity XC is called the reactance of the
capacitor, and has units of ohms
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An Capacitor in an AC Circuit
The instantaneous power supplied to the capacitor
is
The average power over a complete circle is zero.
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A Capacitor in an AC Circuit
Capacitor reactance. What is the rms current in
the circuit shown if C 1.0 µF and Vrms 120 V?
Calculate (a) for f 60 Hz and then (b) for f
6.0 x 105 Hz.
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AC Circuits with AC Source
This figure shows a high-pass filter (allows an
ac signal to pass but blocks a dc voltage) and a
low-pass filter (allows a dc voltage to be
maintained but blocks higher-frequency
fluctuations).
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Phasors
Phasors are vectors. Vector A can be current I or
voltage V, varying sinusoidally in time, e.g.,
i0
i
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Phasors
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Phasors
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Phasors
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LRC Series AC Circuit
Analyzing the LRC series AC circuit is
complicated, as the voltages are not in phase
this means we cannot simply add them.
Furthermore, the reactances depend on the
frequency.
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LRC Series AC Circuit
The instantaneous current is the same at all
points in the circuit
The instantaneous voltages are, according to the
loop rule, given by
The vector sum of the voltage phasors is
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LRC Series AC Circuit
We calculate the voltage (and current) using
phasors these are vectors representing the
individual voltages.
Here, at t 0, the current and voltage are both
at a maximum. As time goes on, the phasors will
rotate counterclockwise.
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LRC Series AC Circuit
Some time t later, the phasors have rotated.
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LRC Series AC Circuit
The voltages across each device are given by the
x-component of each, and the current by its
x-component. The current is the same throughout
the circuit.
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LRC Series AC Circuit
We can write the relation between voltage and
current in the form of Ohms law with the
effective resistance, called the impedance
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LRC Series AC Circuit
The phase angle between the voltage and the
current is given by
or
The factor cos f is called the power factor of
the circuit.
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LRC Series AC Circuit
LRC circuit. Suppose R 25.0 O, L 30.0 mH, and
C 12.0 µF, and they are connected in series to
a 90.0-V ac (rms) 500-Hz source. Calculate (a)
the current in the circuit, (b) the voltmeter
readings (rms) across each element, (c) the phase
angle ?, and (d) the power dissipated in the
circuit.
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Resonance in AC Circuits
The rms current I in an ac circuit by a rms
voltage V is
Clearly, the impedance Z depends on the
frequency. Z reaches a minimum value R when
XLXC, i.e. ?L1/?C.
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Resonance in AC Circuits
This defines the resonance (angular) frequency
At this frequency, I reaches a maximum
The instantaneous current and voltage are in phase
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Power in AC Circuits
The instantaneous power delivered by the source
emf is
The average power over a complete circle is
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Power in AC Circuits
The first term in the integrand contributes
because the average of sin2(?t) is ½ whereas the
average of sin(?t)cos(?t) is zero. The average
power is, therefore,
since
The rms power becomes
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LRC Series AC Circuit
LRC circuit. Suppose R 5.00 O, L 60.0 mH, and
they are connected in series to a 30.0-V ac
(peak value) 60.0-Hz source. For what value of
the capacitance would the average rate at which
energy is dissipated be (a) a maximum and (b) a
minimum? What are (c) the maximum dissipation
rate and (d) the corresponding phase angle ? and
(e) power factor? What are (f) the minimum
dissipation rate and (g) the corresponding phase
angle ? and (h) power factor?
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Instantaneous and peak values of current and
voltage
Root mean square (rms) values of current and
voltage
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Ohms law in LRC circuits
The impedance and reactance
The phase angle
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The resonance angular frequency
The rms power delivered by the source emf
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Transformers
The transformer is a device that can raise or
lower the amplitude of ac potential differences.
It consists of two coils, either interwoven or
linked by an iron core. A changing emf in one
induces an emf in the other. The primary coil is
connected to the source. While the secondary coil
is connected to the load.
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Transformers
The flux through both coils are the same. The
emfs are
Take the ratio of the emfs, we find
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Transformers
Energy must be conserved in the absence of
losses, the power supplied by the primary coil
and loaded to the secondary coil are the same.
Therefore, the ratio of the currents must be the
inverse of the ratio of turns
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Transformers
Cell phone charger. The charger for a cell phone
contains a transformer that reduces 120-V ac to
5.0-V ac to charge the 3.7-V battery. (It also
contains diodes to change the 5.0-V ac to 5.0-V
dc.) Suppose the secondary coil contains 30 turns
and the charger supplies 700 mA. Calculate (a)
the number of turns in the primary coil, (b) the
current in the primary, and (c) the power
transformed.
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Transformers
Transformers work only if the current is
changing this is one reason why electricity is
transmitted as ac.
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Transformers
Transmission lines. An average of 120 kW of
electric power is sent to a small town from a
power plant 10 km away. The transmission lines
have a total resistance of 0.40 O. Calculate the
power loss if the power is transmitted at (a) 240
V and (b) 24,000 V.
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Impedance Matching
When one electrical circuit is connected to
another, maximum power is transmitted when the
output impedance of the first equals the input
impedance of the second.
The power delivered to the circuit will be a
minimum when dP/dt 0 this occurs when R1 R2.
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Three-Phase AC
Transmission lines usually transmit three-phase
ac power, with the phases being separated by
120. This makes the power flow much smoother
than if a single phase were used.
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Three-Phase AC
Three-phase circuit. In a three-phase circuit,
266 V rms exists between line 1 and ground. What
is the rms voltage between lines 2 and 3?