Superimposed DC and AC voltages
Non-Sinusoidal Waveforms
Pulse Waveforms
Vavg = baseline + (duty cycle)(amplitude)
Square Waves
Vrms  V p 
V pp
2
Triangle Waves
Vrms 
Harmonics
Odd harmonics = odd multiples of fundamental
Even harmonics = even multiples of fundamental
Vp
3

V pp
2 3
The Oscilloscope
Phasors
Phasors provide a convenient method for representing sine waves. The angle θ represents the angular
position, and the length of the phasor indicates the magnitude:
The position of the phasor is directly related to the sinewave it represents:
The instantaneous value of the associated sine wave at any point is given by the vertical distance from the tip
of the phasor to the horizontal axis.
Positive and Negative Phase Angles
Phasor Diagrams
Angular Velocity
The rate at which the phasors are rotated around the circle is referred to as angular velocity
(ω), and is measured in radians/second (rads/s). The angular velocity of a phasor is directly
related to the frequency of the associated sinewave:
  2f in rads/s
Using the relationship which exists between the angle of a phasor and its angular velocity
(   t ), it is possible to write the following useful equivalent form of the sine wave
expression:
v  Vp sin(2ft )
Example: Construct a phasor diagram to represent the sine waves in the following figure. If
the frequency of the sinewaves is 18kHz, find the corresponding angular velocity of the
phasors.
ω = 113 krad/s
Complex Numbers
Complex numbers are often used to represent phasors in order to simplify their analysis –
complex numbers provide a useful means to determine the addition, subtraction,
multiplication or division of two or more phasors.
Rectangular Form:
A + jB
Polar form: C
b
where C  a 2  b2 and   Arc tan 
a
Capacitors in AC Circuits
dq
Since current is defined as the rate of flow of charge   and charge may be related to
 dt 
capacitance (q = Cv) it follows that the current in a capacitor is proportional to the rate
of change of the voltage across the capacitor:
 dv 
i  C 
 dt 
In a purely capacitive circuit, the current leads the voltage by 90°
Another way to look at this is to realize that since a capacitor will NOT allow you to
change the voltage across it instantaneously, the current ends up leading the voltage
across a capacitor
Capacitive Reactance
Just like resistors provide a resistance to the flow of current in a dc (or ac) circuit,
Capacitors provide a resistance to the flow of current in an ac circuit. In this case, the
resistance is referred to as Capacitive Reactance (Xc). Just like resistance, the unit of
reactance is ohms (Ω).
Unlike resistance, reactance is frequency dependent, and is given by the following
relationship:
XC 
1
2fC
This equation shows that as the frequency of the applied voltage increases, the reactance
decreases. Similarly, as the frequency decreases, the reactance increases. This makes sense
since at DC the reactance is infinite, as expected.
Analysis of Capacitive AC Circuits
Since there is always a 90° phase shift between voltage and current in a purely capacitive
circuit, capacitive reactance Xc will always have an angle of -90° associated with it:
XC 
VS 0  VS 
    90
I90  I 
Example: determine the rms current in the following circuit:
Xc = 2.84kΩ|_-90;
Irms = 1.76 |_90
Power in a Capacitor
Instantaneous Power = the product of instantaneous voltage and current
Pinst  Vinst  I inst
The shaded curve in the figure represents the instantaneous power. Positive power indicates
that energy is supplied to the capacitor, whereas negative power indicates that power is
returned by the capacitor to the source. The figure shows that the power cycle occurs at
twice the frequency of either the voltage or current waveforms.
True Power = power which is dissipated in the form of heat. In an ideal capacitor the true
power would be zero, but in reality some power is lost due to internal resistance and leakage.
Reactive Power = a measure of the energy stored and released by the capacitor. In a purely
capacitive circuit, the reactive power may be calculated using the rms values for voltage and
current:
Pr  Vrms I rms
V 2 rms
Pr 
Xc
Pr  I 2 rms X c
Example: calculate the reactive power in the following circuit: