DC and AC
• So far we have been primarily talking about constant voltages and currents.
• We call these Direct Current (DC) voltages and currents.
• Voltages and currents may be time varying i.e. v(t), i(t).
• If the voltages and currents are sinusoidal functions of time, we call them
Alternating Current (AC) voltages and currents.
• Sinusoids arise naturally in many physical systems.
• Rotating electric machines naturally generate sinusoidal voltages.
1
Sinusoidal functions
1
θ (rad)
0
π/2
π
3π/2 2π
−1
This is a plot of y = sin(θ).
• θ is the angle in radians
• Peak values is 1
• The function repeats itself every 2π radians
2
Sine and cosine
1
sin(θ)
cos(θ)
θ (rad)
0
π/2
π
3π/2 2π
−1
• cos(θ) = sin(θ + π/2)
• Cos function is the sin function shifted left π/2 radians.
3
Derivatives of sine and cosine
• Sinusoidal signals are important because of the properties of their derivative:
d
π
sin θ = cos θ = sin θ +
dθ
2
d
π
cos θ = − sin θ = cos θ +
dθ
2
2
d
sin θ = − sin θ = sin (θ + π)
dθ2
• A sinusoid is its own derivative (with a phase shift).
• This means it is a natural solution to a differential equation.
4
Derivatives of sinusoidal signals
1
θ (rad)
0
π/2
π
sin(θ)
cos(θ)
− sin(θ)
3π/2 2π
−1
d
sin θ = cos θ
dθ
d
cos θ = − sin θ
dθ
d2
sin θ = − sin θ
dθ2
5
Sinusoidal functions of time
1
ωt (rad)
0
π/2
π
3π/2 2π
−1
• We are particularly interested in sinusoidal functions of time:
v(t) = cos(ωt)
• The function repeats itself every ωt = 2π radians
• i.e. every t = 2π/ω seconds.
• 2π/ω is called the period .
6
Frequency
1
cos(20t)
cos(50t)
t (s)
0
0.2
0.4
−1
The function repeats itself every t = 2π/ω seconds.
• 2π/ω is called the period , T .
• ω is called the angular frequency (radians per second).
ω
• f = T1 = 2π
is called the frequency (cycles per second on Hertz, Hz).
7
Phase
1
cos(θ)
cos(θ + π/4)
π/4
0
π/2
θ (rad)
π
3π/2
2π
−1
• cos(ωt + ϕ) is cos(ωt) shifted left by ϕ radians
• i.e. shifted forwards in time by ϕ/ω seconds.
• ϕ is called the phase (or phase shift).
8
Amplitude
• cos(ωt + ϕ) has a peak value of 1.
• A. cos(ωt + ϕ) has a peak value of A.
• A is called the peak amplitude of the signal.
• The peak-to-peak amplitude of the signal is 2A.
A
ωt (rad)
0
π/2
π
3π/2 2π
−A
9
AC voltage and current
For AC voltages and currents the following still apply:
P
• KCL ( ij (t) = 0)
P
• KVL ( vj (t) = 0)
• Ohm’s law (vR (t) = iR (t) × R)
For an AC current in a conductor, the electrons rush backward and forward
without any net current in either direction over time.
Can still transmit energy, just like water rushing back and forward past a paddle
wheel could still be used to do useful work.
10
Root mean square
• If we apply a voltage of v(t) = V cos(ωt + ϕ) across a resistor value R, the
current will be
i(t) =
v(t)
V
= cos(ωt + ϕ) = I cos(ωt + ϕ)
R
R
• The instantaneous power dissipated in the resistor will be:
p(t) = v(t).i(t) = v(t).v(t)/R =
V2
cos2 (ωt + ϕ)
R
• We can calculate the average full power dissipation by averaging over one full
period:
Z
Z
1 T
1 T V2
pAV =
p(t).dt =
cos2 (ωt + ϕ)
T 0
T 0 R
1V2
=
2 R
11
Root mean square
1
pAV =
T
Z T
0
1
p(t).dt =
T
Z T
0
1
v (t)/R.dt =
R.T
2
Z T
v 2 (t).dt
0
• We define the Root Mean Square voltage as
s
Z
1 T 2
VRM S =
v (t).dt
T 0
so that pAV =
• Then, for sinusoidal voltages
s
VRM S =
1
T
Z T
0
2
VRM
S
R
V
V 2 cos2 (ωt + ϕ).dt = √
2
12
Root mean square
For sinusoidal voltages
pAV =
√
where VRM S = V / 2 ≈ 0.707V .
V2
1V2
= RM S
2 R
R
13