Alternating Current (AC)

1
Chapter 33

• Alternating Current (AC)
• R, L, C in AC circuits

2
AC, the description

• A DC power source, like the one from a battery,
  provides a potential difference (a voltage) that
  does not change its polarity with respect to a
  reference point (often the ground)
• An AC power source is sinusoidal voltage source
  which is described as
• Here

is the instantaneous voltage with respect to a
reference (often not the ground).
is the maximum voltage or amplitude.
is the angular frequency, related to frequency f
and period T as
The US AC system is 110V/60Hz. Many European and
Asian countries use 220V/50Hz.
3
Resistors in an AC Circuit, Ohms Law
The voltage over the resistor
Apply Ohms Law, the current through the resistor
The current is also a sinusoidal function of time
t. The current through and the voltage over the
resistor are in phase both reach their maximum
and minimum values at the same time.
The power consumed by the resistor is
We will come back to the power discussion later.
PLAY ACTIVE FIGURE
4
Phasor Diagram, a useful tool.
y

• The projection of a circular motion with a
  constant angular velocity on the y-axis is a
  sinusoidal function.
• To simplify the analysis of AC circuits, a
  graphical constructor called a phasor diagram is
  used. A phasor is a vector whose length is
  proportional to the maximum value of the variable
  it represents
• The phasor diagram of a resistor in AC is shown
  here. The vectors representing current and
  voltage overlap each other, because they are in
  phase.

R
x
O
The projection on the y-axis is
5
The power for a resistive AC circuit and the rms
current and voltage
When the AC voltage source is applied on the
resistor, the voltage over and current through
the resistor are
Both average to zero.
But the power over the resistor is
And it does not average to zero. The averaged
power is
6
The power for a resistive AC circuit and the rms
current and voltage
So the averaged power the resistor consumes is
If the power were averaged to zero, like the
current and voltage, could we use AC power source
here?
The averaged power can also be written as
Define a root mean square for the voltage and
current
or
One get back to the DC formula equivalent
7
Resistors in an AC Circuit, summary

• Ohms Law applies. Te current through and voltage
  over the resistor are in phase.
• The average power consumed by the resistor is
• From this we define the root mean square current
  and voltage. AC meters (V or I) read these
  values.
• The US AC system of 110V/60Hz, here the 110 V is
  the rms voltage, and the 60 Hz is the frequency
  f, so

8
Inductors in an AC circuit, voltage and current
The voltage over the inductor is
To find the current i through the inductor, we
start with Kirchhoffs loop rule
or
Solve the equation for i
or
9
Inductors in an AC circuit, voltage leads current
Examining the formulas for voltage over and
current through the inductor
Voltage leads the current by ¼ of a period (T/4
or 90 or p/2) . Or in a phasor diagram, the
rotating current vector is 90 behind the voltage
vector.
PLAY ACTIVE FIGURE
10
Inductive Reactance, the resistance the
inductor offers in the circuit.
Examining the formulas for voltage over and
current through the inductor again
This time pay attention to the relationship
between the maximum values of the current and the
voltage
This could be Ohms Law if we define a
resistance for the inductor to be
And this is called the inductive reactance.
Remember, it is the product of the inductance,
and the angular frequency of the AC source. I
guess that this is the reason for it to be called
a reactance instead of a passive resistance.
The following formulas may be useful
11
Capacitors in an AC circuit, voltage and current
The voltage over the capacitor is
To find the current i through the capacitor, we
start with Kirchhoffs loop rule
or
Solve the first equation for q, and take the
derivative for i
or
Here I still like to keep the Ohms Law type of
formula for voltage, current and a type of
resistance.
12
Capacitors in an AC circuit, current leads the
voltage
Examining the formulas for voltage over and
current through the capacitor
Current leads the voltage by ¼ of a period (T/4
or 90 or p/2) . Or in a phasor diagram, the
rotating voltage vector is 90 behind the current
vector.
PLAY ACTIVE FIGURE
13
Capacitive Reactance, the resistance the
capacitor offers in the circuit.
Examining the formulas for voltage over and
current through the capacitor again
This time pay attention to the relationship
between the maximum values of the current and the
voltage
This could be Ohms Law if we define a
resistance for the capacitor to be
And this is called the capacitive reactance. It
is the inverse of the product of the capacitance,
and the angular frequency of the AC source. The
following formulas may be useful
14
The RLC series circuit, current and voltage
The voltage over the RLC is
Now lets find the current. From the this
equation, write out each component
Simply solve for the current i
Overall resistance
Phase angle
Where
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The RLC series circuit, current and voltage,
solved with Phasor Diagrams
The RLC are in serial connection, the current i
is common and must be in phase
i
So use this as the base (the x-axis) for the
phasor diagrams
16
The RLC series circuit, current and voltage,
solved with Phasor Diagrams
Now overlap the three phasor diagrams, we have
17
The RLC series circuit, current and voltage,
solved with Phasor Diagrams
Now from final phasor diagram, we get the voltage
components in x- and y-axes
From
or
We have
Here Z is the overall resistance, called the
impedance.
From the diagram, the phase angle is
We have
PLAY ACTIVE FIGURE
18
Determining the Nature of the Circuit

• If f is positive
• XLgt XC (which occurs at high frequencies)
• The current lags the applied voltage
• The circuit is more inductive than capacitive
• If f is negative
• XLlt XC (which occurs at low frequencies)
• The current leads the applied voltage
• The circuit is more capacitive than inductive
• If f is zero
• XL XC (which occurs at
  )
• The circuit is purely resistive and the impedance
  is minimum, and current reaches maximum, the
  circuit resonates.
• Often this resonant frequency is called