AC Circuits

1
AC Circuits

• Chapter 23

2
AC Circuits

• Capacitive Reactance
• Phasor Diagrams
• Inductive Reactance
• RCL Circuits
• Resonance

3
Resistive Loads in AC Circuits

• Ohms Law
• R is constant does not depend on frequency
• No phase difference between V and I

4
Capacitive Reactance

• At the moment a capacitor is connected to a
  voltage source
• Current is at its maximum
• Voltage across capacitor is zero

5
Capacitive Reactance

• After a long time, the capacitor is charged
• Current is zero
• Voltage is at its maximum ( supply voltage)

6
Capacitive Reactance

• Now, we reverse the polarity of the applied
  voltage
• Current is at its maximum (but reversed)
• Voltage hasnt changed yet

7
Capacitive Reactance

• Time passes the capacitor becomes fully charged
• Current is zero
• Voltage has reversed to match the applied polarity

8
Capacitive Reactance

• Apply an AC voltage source
• an AC current is present in the circuit
• a 90 phase difference is found between the
  voltage and the current

9
Capacitive Reactance

• We want to find a relationship between the
  voltage and the current that we can use like
  Ohms Law for an AC circuit with a capacitive
  load
• We call XC the capacitive reactance, and
  calculate it as
• units of capacitive reactance ohms (W)

10
Capacitive Reactance

• A particular example

11
Capacitive Reactance
12
Capacitive Reactance
13
Capacitive Reactance
14
Capacitive Reactance

• Power is zero each time either the voltage or
  current is zero
• Power is positive whenever V and I have the same
  sign
• Power is negative whenever V and I have opposite
  signs
• Power spends equal amounts of time being negative
  and positive
• Average power over time zero

15
Capacitive Reactance

• The larger the capacitance, the smaller the
  capacitive reactance
• As frequency increases, reactance decreases
• DC capacitor is an open circuit and
• high frequency capacitor is a short circuit
• and

16
Phasor Diagrams

• Consider a vector which rotates counterclockwise
  with an angular speed
• This vector is called
• a phasor. It is a
• visualization tool.

17
Phasor Diagrams

• For a resistive load the current is always
  proportional to the voltage

18
Phasor Diagrams

• For a capacitive load the current leads the
  voltage by p/2 (or 90)

19
Inductive Reactance

• A coil or inductor also acts as a reactive load
  in an AC circuit.

20
Inductive Reactance

• For a coil with a self-inductance L

21
Inductive Reactance

• As the current increases through zero, its time
  rate of change is a maximum and so is the
  induced EMF

22
Inductive Reactance

• As the current reaches its maximum value, its
  rate of change decreases to zero and so does
  the induced EMF

23
Inductive Reactance

• The voltage leads the current in the inductor
  by p/2 (or 90)

24
Inductive Reactance

• The inductive reactance is the Ohms Law constant
  of proportionality
• 
  units of inductive reactance
• 
  ohms (W)

25
Inductive Reactance

• The voltage-current relationship in an inductive
  load in an AC circuit can be represented by a
  phasor diagram
• 
  

26
Inductive Reactance

• Mnemonic for remembering what leads what
• ELI the ICEman
• 
  

EMF (voltage)
EMF (voltage)
current (I)
inductor (L)
capacitor (C)
current (I)
27
Inductive Reactance

• Larger inductance larger reactance (more induced
  EMF to oppose the applied AC voltage)
• Higher frequency larger impedance (higher
  frequency means higher time rate of change of
  current, which means more induced EMF to oppose
  the applied AC voltage)
• 
  

28
RCL Circuit

• Here is an AC circuit containing series-connected
  resistive, capacitive, and inductive loads
• The voltages across the loads at any instant are
  different, but a common current is present.
• 
  

29
RCL Circuit

• The current is in phase with voltage in the
  resistor.
• The capacitor voltage trails the current the
  inductor voltage leads it.
• We want to calculate the entire applied voltage
  from the generator.
• 
  

30
RCL Circuit

• We will add the voltage phasors as vectors (which
  is what they are.)
• We start out by adding the reactive voltages
  (across the capacitor and the inductor).
• This is easy because those phasors are opposite
  in direction. The resultants magnitude is the
  difference of the two, and its direction is that
  of the larger one.
  

31
RCL Circuit

• Now we use Pythagoras Theorem to add the VL VC
  phasor to the VR phasor.
• 
  

32
RCL Circuit

• The current phasor is unaffected by our addition
  of the voltage phasors.
• It now makes an angle f with the overall applied
  voltage phasor.
• 
  

33
RCL Circuit

• We can make Ohms Law substitutions for the
  voltages
• 
  

34
RCL Circuit

• Our result
• suggests an Ohms Law relationship for the
  combined loads in the series RCL circuit
• Z is called the impedance of the RCL circuit.
• SI units ohms (W)
• 
  

35
RCL Circuit -- Power

• If the load is purely resistive, the average
  power dissipated is
• We can use the phasor diagram to relate R to Z
  trigonometrically
• 
  

power factor
36
RCL Circuit -- Resonance

• Series-connected inductor and capacitor
• 
  

37
RCL Circuit -- Resonance

• 
  

38
RCL Circuit -- Resonance

• 
  

39
RCL Circuit -- Resonance

• 
  

40
RCL Circuit -- Resonance

• Energy is alternately stored in the capacitor (in
  the form of the electrical potential energy of
  separated charges) and in the inductor (in its
  magnetic field). When the magnetic field
  collapses, it charges the capacitor when the
  capacitor discharges, it builds the magnetic
  field in the inductor.
• 
  

41
RCL Circuit -- Resonance

• This LC oscillator or tuned tank circuit
  oscillates at a natural or resonant frequency of
• 
  

42
RCL Circuit -- Resonance

• At the resonant frequency, how are the inductive
  and capacitive reactances related?
• The reactances are equal to each other.
• 
  

43
RCL Circuit -- Resonance

• At the resonant frequency, when the inductive and
  capacitive reactances are equal, what is the
  situation in the circuit?
• 
  

44
RCL Circuit -- Resonance

• At the resonant frequency, when the inductive and
  capacitive reactances are equal, what is the
  impedance of the circuit?
• At resonance, the circuits impedance is simply
  equal to its resistance, and its voltage and
  current are in phase.
• If the resistance is small, the current may be
  quite large.