10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
10: Sine waves and phasors
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 1 / 11
Sine Waves
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
di
For inductors and capacitors i = C dv
dt and v = L dt so we need to
differentiate i(t) and v(t) when analysing circuits containing them.
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 2 / 11
Sine Waves
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
di
For inductors and capacitors i = C dv
dt and v = L dt so we need to
differentiate i(t) and v(t) when analysing circuits containing them.
Usually differentiation changes the
shape of a waveform.
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
1
0
-1
0
1
2
t
3
4
1
2
t
3
4
5
0
-5
0
Phasors: 10 – 2 / 11
Sine Waves
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
di
For inductors and capacitors i = C dv
dt and v = L dt so we need to
differentiate i(t) and v(t) when analysing circuits containing them.
Usually differentiation changes the
shape of a waveform.
For bounded waveforms there is
only one exception:
v(t) = sin t ⇒
E1.1 Analysis of Circuits (2015-7199)
dv
dt
1
0
-1
0
1
2
t
3
4
1
2
t
3
4
5
0
-5
0
= cos t
Phasors: 10 – 2 / 11
Sine Waves
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
di
For inductors and capacitors i = C dv
dt and v = L dt so we need to
differentiate i(t) and v(t) when analysing circuits containing them.
Usually differentiation changes the
shape of a waveform.
1
0
-1
0
For bounded waveforms there is
only one exception:
v(t) = sin t ⇒
dv
dt
1
2
t
3
4
1
2
t
3
4
5
0
-5
0
= cos t
1
v(t)
10: Sine waves and phasors
0
-1
0
5
10
15
10
15
t
dv/dt
1
0
-1
0
5
t
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 2 / 11
Sine Waves
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
di
For inductors and capacitors i = C dv
dt and v = L dt so we need to
differentiate i(t) and v(t) when analysing circuits containing them.
Usually differentiation changes the
shape of a waveform.
1
0
-1
0
For bounded waveforms there is
only one exception:
v(t) = sin t ⇒
dv
dt
2
t
3
4
1
2
t
3
4
0
-5
0
= cos t
same shape but with a time shift.
1
5
1
v(t)
10: Sine waves and phasors
0
-1
0
5
10
15
10
15
t
dv/dt
1
0
-1
0
5
t
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 2 / 11
Sine Waves
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
di
For inductors and capacitors i = C dv
dt and v = L dt so we need to
differentiate i(t) and v(t) when analysing circuits containing them.
Usually differentiation changes the
shape of a waveform.
1
0
-1
0
For bounded waveforms there is
only one exception:
v(t) = sin t ⇒
dv
dt
2
t
3
4
1
2
t
3
4
0
-5
0
= cos t
same shape but with a time shift.
1
5
1
v(t)
10: Sine waves and phasors
0
-1
0
5
10
15
10
15
t
1
dv/dt
sin t completes one full period
every time t increases by 2π .
0
-1
0
5
t
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 2 / 11
Sine Waves
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
di
For inductors and capacitors i = C dv
dt and v = L dt so we need to
differentiate i(t) and v(t) when analysing circuits containing them.
Usually differentiation changes the
shape of a waveform.
1
0
-1
0
For bounded waveforms there is
only one exception:
v(t) = sin t ⇒
dv
dt
2
t
3
4
1
2
t
3
4
0
-5
0
= cos t
same shape but with a time shift.
1
5
1
v(t)
10: Sine waves and phasors
0
-1
0
5
10
15
10
15
t
1
dv/dt
sin t completes one full period
every time t increases by 2π .
0
-1
0
5
t
sin 2πf t makes f complete repetitions every time t increases by 1; this
gives a frequency of f cycles per second, or f Hz.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 2 / 11
Sine Waves
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
di
For inductors and capacitors i = C dv
dt and v = L dt so we need to
differentiate i(t) and v(t) when analysing circuits containing them.
Usually differentiation changes the
shape of a waveform.
1
0
-1
0
For bounded waveforms there is
only one exception:
v(t) = sin t ⇒
dv
dt
2
t
3
4
1
2
t
3
4
0
-5
0
= cos t
same shape but with a time shift.
1
5
1
v(t)
10: Sine waves and phasors
0
-1
0
5
10
15
10
15
t
1
dv/dt
sin t completes one full period
every time t increases by 2π .
0
-1
0
5
t
sin 2πf t makes f complete repetitions every time t increases by 1; this
gives a frequency of f cycles per second, or f Hz.
We often use the angular frequency, ω = 2πf instead.
ω is measured in radians per second. E.g. 50 Hz ≃ 314 rad.s−1 .
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 2 / 11
Rotating Rod
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
A useful way to think of a cosine wave is as the
projection of a rotating rod onto the horizontal axis.
For a unit-length rod, the projection has length cos θ .
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 3 / 11
Rotating Rod
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
A useful way to think of a cosine wave is as the
projection of a rotating rod onto the horizontal axis.
For a unit-length rod, the projection has length cos θ .
If the rod is rotating at a speed of f revolutions per
second, then θ increases uniformly with time:
θ = 2πf t.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 3 / 11
Rotating Rod
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
A useful way to think of a cosine wave is as the
projection of a rotating rod onto the horizontal axis.
For a unit-length rod, the projection has length cos θ .
If the rod is rotating at a speed of f revolutions per
second, then θ increases uniformly with time:
θ = 2πf t.
The only difference between cos and sin is the starting position of the rod:
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 3 / 11
Rotating Rod
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
A useful way to think of a cosine wave is as the
projection of a rotating rod onto the horizontal axis.
For a unit-length rod, the projection has length cos θ .
If the rod is rotating at a speed of f revolutions per
second, then θ increases uniformly with time:
θ = 2πf t.
The only difference between cos and sin is the starting position of the rod:
1
0
-1
0
5
10
15
t
v = cos 2πf t
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 3 / 11
Rotating Rod
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
A useful way to think of a cosine wave is as the
projection of a rotating rod onto the horizontal axis.
For a unit-length rod, the projection has length cos θ .
If the rod is rotating at a speed of f revolutions per
second, then θ increases uniformly with time:
θ = 2πf t.
The only difference between cos and sin is the starting position of the rod:
1
1
0
0
-1
0
5
10
-1
0
15
t
v = cos 2πf t
E1.1 Analysis of Circuits (2015-7199)
5
10
15
t
v = sin 2πf t
Phasors: 10 – 3 / 11
Rotating Rod
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
A useful way to think of a cosine wave is as the
projection of a rotating rod onto the horizontal axis.
For a unit-length rod, the projection has length cos θ .
If the rod is rotating at a speed of f revolutions per
second, then θ increases uniformly with time:
θ = 2πf t.
The only difference between cos and sin is the starting position of the rod:
1
1
0
0
-1
0
5
10
t
v = cos 2πf t
E1.1 Analysis of Circuits (2015-7199)
15
-1
0
5
10
15
t
v = sin 2πf t = cos 2πf t −
π
2
Phasors: 10 – 3 / 11
Rotating Rod
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
A useful way to think of a cosine wave is as the
projection of a rotating rod onto the horizontal axis.
For a unit-length rod, the projection has length cos θ .
If the rod is rotating at a speed of f revolutions per
second, then θ increases uniformly with time:
θ = 2πf t.
The only difference between cos and sin is the starting position of the rod:
1
1
0
0
-1
0
5
10
t
v = cos 2πf t
15
-1
0
5
10
v = sin 2πf t = cos 2πf t −
sin 2πf t lags cos 2πf t by 90◦ (or π2 radians) because its peaks occurs
of a cycle later (equivalently cos leads sin) .
E1.1 Analysis of Circuits (2015-7199)
15
t
π
2
1
4
Phasors: 10 – 3 / 11
Phasors
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
If the rod has length A and starts at an angle φ then the projection onto the
horizontal axis is
A cos (2πf t + φ)
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 4 / 11
Phasors
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
If the rod has length A and starts at an angle φ then the projection onto the
horizontal axis is
A cos (2πf t + φ)
= A cos φ cos 2πf t − A sin φ sin 2πf t
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 4 / 11
Phasors
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
If the rod has length A and starts at an angle φ then the projection onto the
horizontal axis is
A cos (2πf t + φ)
= A cos φ cos 2πf t − A sin φ sin 2πf t
= X cos 2πf t − Y sin 2πf t
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 4 / 11
Phasors
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
If the rod has length A and starts at an angle φ then the projection onto the
horizontal axis is
A cos (2πf t + φ)
= A cos φ cos 2πf t − A sin φ sin 2πf t
= X cos 2πf t − Y sin 2πf t
At time t = 0, the tip of the rod has coordinates
(X, Y ) = (A cos φ, A sin φ).
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 4 / 11
Phasors
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
If the rod has length A and starts at an angle φ then the projection onto the
horizontal axis is
A cos (2πf t + φ)
= A cos φ cos 2πf t − A sin φ sin 2πf t
= X cos 2πf t − Y sin 2πf t
At time t = 0, the tip of the rod has coordinates
(X, Y ) = (A cos φ, A sin φ).
If we think of the plane as an Argand Diagram (or complex plane), then the
complex number X + jY corresponding to the tip of the rod at t = 0 is
called a phasor .
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 4 / 11
Phasors
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
If the rod has length A and starts at an angle φ then the projection onto the
horizontal axis is
A cos (2πf t + φ)
= A cos φ cos 2πf t − A sin φ sin 2πf t
= X cos 2πf t − Y sin 2πf t
At time t = 0, the tip of the rod has coordinates
(X, Y ) = (A cos φ, A sin φ).
If we think of the plane as an Argand Diagram (or complex plane), then the
complex number X + jY corresponding to the tip of the rod at t = 0 is
called a phasor .
√
The magnitude of the phasor, A = X 2 + Y 2 , gives the amplitude (peak
value) of the sine wave.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 4 / 11
Phasors
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
If the rod has length A and starts at an angle φ then the projection onto the
horizontal axis is
A cos (2πf t + φ)
= A cos φ cos 2πf t − A sin φ sin 2πf t
= X cos 2πf t − Y sin 2πf t
At time t = 0, the tip of the rod has coordinates
(X, Y ) = (A cos φ, A sin φ).
If we think of the plane as an Argand Diagram (or complex plane), then the
complex number X + jY corresponding to the tip of the rod at t = 0 is
called a phasor .
√
The magnitude of the phasor, A = X 2 + Y 2 , gives the amplitude (peak
value) of the sine wave.
Y
, gives the phase shift relative
The argument of the phasor, φ = arctan X
to cos 2πf t.
If φ > 0, it is leading and if φ < 0, it is lagging relative to cos 2πf t.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 4 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
V = 1, f = 50 Hz
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
t
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
t
V = −j
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
0.02
t
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
0.02
t
V = −1−0.5j
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
0.02
t
V = −1−0.5j
v(t) = − cos 2πf t + 0.5 sin 2πf t
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
V = −1−0.5j = 1.12∠−153◦
v(t) = − cos 2πf t + 0.5 sin 2πf t
E1.1 Analysis of Circuits (2015-7199)
0.02
t
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
V = −1−0.5j = 1.12∠−153◦
v(t) = − cos 2πf t + 0.5 sin 2πf t
= 1.12 cos (2πf t − 2.68)
E1.1 Analysis of Circuits (2015-7199)
0.02
t
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
V = −1−0.5j = 1.12∠−153◦
v(t) = − cos 2πf t + 0.5 sin 2πf t
= 1.12 cos (2πf t − 2.68)
0.02
t
V = X + jY
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
V = −1−0.5j = 1.12∠−153◦
v(t) = − cos 2πf t + 0.5 sin 2πf t
= 1.12 cos (2πf t − 2.68)
0.02
t
V = X + jY
v(t) = X cos 2πf t − Y sin 2πf t
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
V = −1−0.5j = 1.12∠−153◦
v(t) = − cos 2πf t + 0.5 sin 2πf t
= 1.12 cos (2πf t − 2.68)
0.02
t
V = X + jY
v(t) = X cos 2πf t − Y sin 2πf t
Beware minus sign.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
t
V = −1−0.5j = 1.12∠−153◦
v(t) = − cos 2πf t + 0.5 sin 2πf t
= 1.12 cos (2πf t − 2.68)
V = X + jY
v(t) = X cos 2πf t − Y sin 2πf t
0.02
V = A∠φ
Beware minus sign.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
V = −1−0.5j = 1.12∠−153◦
v(t) = − cos 2πf t + 0.5 sin 2πf t
= 1.12 cos (2πf t − 2.68)
V = X + jY
v(t) = X cos 2πf t − Y sin 2πf t
0.02
t
V = A∠φ
v(t) = A cos (2πf t + φ)
Beware minus sign.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
V = −1−0.5j = 1.12∠−153◦
v(t) = − cos 2πf t + 0.5 sin 2πf t
= 1.12 cos (2πf t − 2.68)
V = X + jY
v(t) = X cos 2πf t − Y sin 2πf t
0.02
t
V = A∠φ = Aejφ
v(t) = A cos (2πf t + φ)
Beware minus sign.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
V = −1−0.5j = 1.12∠−153◦
v(t) = − cos 2πf t + 0.5 sin 2πf t
= 1.12 cos (2πf t − 2.68)
V = X + jY
v(t) = X cos 2πf t − Y sin 2πf t
0.02
t
V = A∠φ = Aejφ
v(t) = A cos (2πf t + φ)
Beware minus sign.
A phasor represents an entire waveform (encompassing all time) as a
single complex number. We assume the frequency, f , is known.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
0.02
t
V = −1−0.5j = 1.12∠−153◦
v(t) = − cos 2πf t + 0.5 sin 2πf t
= 1.12 cos (2πf t − 2.68)
V = X + jY
v(t) = X cos 2πf t − Y sin 2πf t
V = A∠φ = Aejφ
v(t) = A cos (2πf t + φ)
Beware minus sign.
A phasor represents an entire waveform (encompassing all time) as a
single complex number. We assume the frequency, f , is known.
A phasor is not time-varying, so we use a capital letter: V .
A waveform is time-varying, so we use a small letter: v(t).
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor Examples
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
V = 1, f = 50 Hz
v(t) = cos 2πf t
1
0
-1
0
0.02
0.04
0.06
0.04
0.06
t
V = −j
v(t) = sin 2πf t
1
0
-1
0
0.02
t
V = −1−0.5j = 1.12∠−153◦
v(t) = − cos 2πf t + 0.5 sin 2πf t
= 1.12 cos (2πf t − 2.68)
V = X + jY
v(t) = X cos 2πf t − Y sin 2πf t
V = A∠φ = Aejφ
v(t) = A cos (2πf t + φ)
Beware minus sign.
A phasor represents an entire waveform (encompassing all time) as a
single complex number. We assume the frequency, f , is known.
A phasor is not time-varying, so we use a capital letter: V .
A waveform is time-varying, so we use a small letter: v(t).
Casio: Pol(X, Y ) → A, φ, Rec(A, φ) → X, Y . Saved → X & Y mems.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 5 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
a × v(t)
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
a × v(t) = aX cos ωt − aY sin ωt
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
Admittance
v1 (t) + v2 (t)
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
V1 + V2
v1 (t) + v2 (t)
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
V1 + V2
v1 (t) + v2 (t)
• Summary
Adding or scaling is the same for waveforms and phasors.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
V1 + V2
v1 (t) + v2 (t)
• Summary
Adding or scaling is the same for waveforms and phasors.
dv
dt
E1.1 Analysis of Circuits (2015-7199)
= −ωX sin ωt − ωY cos ωt
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
V1 + V2
v1 (t) + v2 (t)
• Summary
Adding or scaling is the same for waveforms and phasors.
dv
dt
E1.1 Analysis of Circuits (2015-7199)
= −ωX sin ωt − ωY cos ωt
= (−ωY ) cos ωt − (ωX) sin ωt
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
V1 + V2
v1 (t) + v2 (t)
• Summary
Adding or scaling is the same for waveforms and phasors.
V̇ = (−ωY ) + j (ωX)
E1.1 Analysis of Circuits (2015-7199)
dv
dt
= −ωX sin ωt − ωY cos ωt
= (−ωY ) cos ωt − (ωX) sin ωt
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
V1 + V2
v1 (t) + v2 (t)
• Summary
Adding or scaling is the same for waveforms and phasors.
V̇ = (−ωY ) + j (ωX)
= jω (X + jY )
E1.1 Analysis of Circuits (2015-7199)
dv
dt
= −ωX sin ωt − ωY cos ωt
= (−ωY ) cos ωt − (ωX) sin ωt
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
V1 + V2
v1 (t) + v2 (t)
• Summary
Adding or scaling is the same for waveforms and phasors.
V̇ = (−ωY ) + j (ωX)
= jω (X + jY )
= jωV
E1.1 Analysis of Circuits (2015-7199)
dv
dt
= −ωX sin ωt − ωY cos ωt
= (−ωY ) cos ωt − (ωX) sin ωt
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
V1 + V2
v1 (t) + v2 (t)
• Summary
Adding or scaling is the same for waveforms and phasors.
V̇ = (−ωY ) + j (ωX)
= jω (X + jY )
= jωV
dv
dt
= −ωX sin ωt − ωY cos ωt
= (−ωY ) cos ωt − (ωX) sin ωt
Differentiating waveforms corresponds to multiplying
phasors by jω .
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
V1 + V2
v1 (t) + v2 (t)
• Summary
Adding or scaling is the same for waveforms and phasors.
V̇ = (−ωY ) + j (ωX)
= jω (X + jY )
= jωV
dv
dt
= −ωX sin ωt − ωY cos ωt
= (−ωY ) cos ωt − (ωX) sin ωt
Differentiating waveforms corresponds to multiplying
phasors by jω .
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 6 / 11
Phasor arithmetic
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
Phasors
Waveforms
V = X + jY
v(t) = X cos ωt − Y sin ωt
where ω = 2πf .
aV
a × v(t) = aX cos ωt − aY sin ωt
V1 + V2
v1 (t) + v2 (t)
• Summary
Adding or scaling is the same for waveforms and phasors.
V̇ = (−ωY ) + j (ωX)
= jω (X + jY )
= jωV
dv
dt
= −ωX sin ωt − ωY cos ωt
= (−ωY ) cos ωt − (ωX) sin ωt
Differentiating waveforms corresponds to multiplying
phasors by jω .
Rotate anti-clockwise 90◦ and scale by ω = 2πf .
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 6 / 11
Complex Impedances
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Resistor:
v(t) = Ri(t)
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 7 / 11
Complex Impedances
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Resistor:
v(t) = Ri(t) ⇒ V = RI
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 7 / 11
Complex Impedances
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Resistor:
v(t) = Ri(t) ⇒ V = RI
⇒
V
I
=R
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 7 / 11
Complex Impedances
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Resistor:
v(t) = Ri(t) ⇒ V = RI
⇒
V
I
=R
Inductor:
di
v(t) = L dt
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 7 / 11
Complex Impedances
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Resistor:
v(t) = Ri(t) ⇒ V = RI
⇒
V
I
=R
Inductor:
di
v(t) = L dt
⇒ V = jωLI
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 7 / 11
Complex Impedances
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Resistor:
v(t) = Ri(t) ⇒ V = RI
⇒
V
I
=R
⇒
V
I
= jωL
Inductor:
di
v(t) = L dt
⇒ V = jωLI
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 7 / 11
Complex Impedances
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Resistor:
v(t) = Ri(t) ⇒ V = RI
⇒
V
I
=R
⇒
V
I
= jωL
Inductor:
di
v(t) = L dt
⇒ V = jωLI
Capacitor:
i(t) = C dv
dt
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 7 / 11
Complex Impedances
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Resistor:
v(t) = Ri(t) ⇒ V = RI
⇒
V
I
=R
⇒
V
I
= jωL
Inductor:
di
v(t) = L dt
⇒ V = jωLI
Capacitor:
i(t) = C dv
dt ⇒ I = jωCV
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 7 / 11
Complex Impedances
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Resistor:
v(t) = Ri(t) ⇒ V = RI
⇒
V
I
=R
⇒
V
I
= jωL
⇒
V
I
=
Inductor:
di
v(t) = L dt
⇒ V = jωLI
Capacitor:
i(t) = C dv
dt ⇒ I = jωCV
E1.1 Analysis of Circuits (2015-7199)
1
jωC
Phasors: 10 – 7 / 11
Complex Impedances
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Resistor:
v(t) = Ri(t) ⇒ V = RI
⇒
V
I
=R
⇒
V
I
= jωL
⇒
V
I
=
Inductor:
di
v(t) = L dt
⇒ V = jωLI
Capacitor:
i(t) = C dv
dt ⇒ I = jωCV
1
jωC
For all three components, phasors obey Ohm’s law if we use the complex
1
impedances jωL and jωC
as the “resistance” of an inductor or capacitor.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 7 / 11
Complex Impedances
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Resistor:
v(t) = Ri(t) ⇒ V = RI
⇒
V
I
=R
⇒
V
I
= jωL
⇒
V
I
=
Inductor:
di
v(t) = L dt
⇒ V = jωLI
Capacitor:
i(t) = C dv
dt ⇒ I = jωCV
1
jωC
For all three components, phasors obey Ohm’s law if we use the complex
1
impedances jωL and jωC
as the “resistance” of an inductor or capacitor.
If all sources in a circuit are sine waves having the same frequency, we can
do circuit analysis exactly as before by using complex impedances.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 7 / 11
Phasor Analysis
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Given v = 10 sin ωt where ω = 2π × 1000,
find vC (t).
Admittance
• Summary
10
v
0
-10
0
E1.1 Analysis of Circuits (2015-7199)
0.5
1
t (ms)
1.5
2
Phasors: 10 – 8 / 11
Phasor Analysis
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Given v = 10 sin ωt where ω = 2π × 1000,
find vC (t).
(1) Find capacitor complex impedance
Z=
1
jωC
=
1
6.28j×10−4
= −1592j
Admittance
• Summary
10
v
0
-10
0
E1.1 Analysis of Circuits (2015-7199)
0.5
1
t (ms)
1.5
2
Phasors: 10 – 8 / 11
Phasor Analysis
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Given v = 10 sin ωt where ω = 2π × 1000,
find vC (t).
(1) Find capacitor complex impedance
Z=
1
jωC
=
1
6.28j×10−4
= −1592j
(2) Solve circuit with phasors
VC = V ×
10
0
-10
0
E1.1 Analysis of Circuits (2015-7199)
v
Z
R+Z
0.5
1
t (ms)
1.5
2
Phasors: 10 – 8 / 11
Phasor Analysis
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Given v = 10 sin ωt where ω = 2π × 1000,
find vC (t).
(1) Find capacitor complex impedance
Z=
1
jωC
=
1
6.28j×10−4
= −1592j
(2) Solve circuit with phasors
VC = V ×
10
= −10j ×
−1592j
1000−1592j
0
-10
0
E1.1 Analysis of Circuits (2015-7199)
v
Z
R+Z
0.5
1
t (ms)
1.5
2
Phasors: 10 – 8 / 11
Phasor Analysis
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Given v = 10 sin ωt where ω = 2π × 1000,
find vC (t).
(1) Find capacitor complex impedance
Z=
1
jωC
=
1
6.28j×10−4
= −1592j
(2) Solve circuit with phasors
VC = V ×
−1592j
= −10j × 1000−1592j
= −4.5 − 7.2j = 8.47∠ − 122◦
E1.1 Analysis of Circuits (2015-7199)
10
v
Z
R+Z
0
-10
0
0.5
1
t (ms)
1.5
2
Phasors: 10 – 8 / 11
Phasor Analysis
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Given v = 10 sin ωt where ω = 2π × 1000,
find vC (t).
(1) Find capacitor complex impedance
Z=
1
jωC
=
1
6.28j×10−4
= −1592j
(2) Solve circuit with phasors
VC = V ×
Z
R+Z
−1592j
= −10j × 1000−1592j
= −4.5 − 7.2j = 8.47∠ − 122◦
vC = 8.47 cos (ωt − 122◦ )
E1.1 Analysis of Circuits (2015-7199)
10
v
C
10: Sine waves and phasors
vC
0
-10
0
0.5
1
t (ms)
1.5
2
Phasors: 10 – 8 / 11
Phasor Analysis
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
Given v = 10 sin ωt where ω = 2π × 1000,
find vC (t).
(1) Find capacitor complex impedance
Z=
1
jωC
=
1
6.28j×10−4
= −1592j
(2) Solve circuit with phasors
VC = V ×
Z
R+Z
−1592j
= −10j × 1000−1592j
= −4.5 − 7.2j = 8.47∠ − 122◦
vC = 8.47 cos (ωt − 122◦ )
10
v
C
10: Sine waves and phasors
vC
0
-10
0
0.5
1
t (ms)
1.5
2
(3) Draw a phasor diagram showing KVL:
V = −10j
VC = −4.5 − 7.2j
VR = V − VC = 4.5 − 2.8j = 5.3∠ − 32◦
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 8 / 11
Phasor Analysis
Admittance
• Summary
Given v = 10 sin ωt where ω = 2π × 1000,
find vC (t).
(1) Find capacitor complex impedance
Z=
1
jωC
=
1
6.28j×10−4
= −1592j
(2) Solve circuit with phasors
VC = V ×
Z
R+Z
−1592j
= −10j × 1000−1592j
= −4.5 − 7.2j = 8.47∠ − 122◦
vC = 8.47 cos (ωt − 122◦ )
10
v
R
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
vR
vC
0
C
10: Sine waves and phasors
-10
0
0.5
1
t (ms)
1.5
2
(3) Draw a phasor diagram showing KVL:
V = −10j
VC = −4.5 − 7.2j
VR = V − VC = 4.5 − 2.8j = 5.3∠ − 32◦
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 8 / 11
Phasor Analysis
Admittance
• Summary
Given v = 10 sin ωt where ω = 2π × 1000,
find vC (t).
(1) Find capacitor complex impedance
Z=
1
jωC
=
1
6.28j×10−4
= −1592j
(2) Solve circuit with phasors
VC = V ×
Z
R+Z
−1592j
= −10j × 1000−1592j
= −4.5 − 7.2j = 8.47∠ − 122◦
vC = 8.47 cos (ωt − 122◦ )
10
v
R
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
vR
vC
0
C
10: Sine waves and phasors
-10
0
0.5
1
t (ms)
1.5
2
(3) Draw a phasor diagram showing KVL:
V = −10j
VC = −4.5 − 7.2j
VR = V − VC = 4.5 − 2.8j = 5.3∠ − 32◦
Phasors add like vectors
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 8 / 11
CIVIL
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
⇒ I leads V
Capacitors: i = C dv
dt
di
⇒ V leads I
Inductors: v = L dt
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 9 / 11
CIVIL
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
⇒ I leads V
Capacitors: i = C dv
dt
di
⇒ V leads I
Inductors: v = L dt
Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”.
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 9 / 11
CIVIL
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
⇒ I leads V
Capacitors: i = C dv
dt
di
⇒ V leads I
Inductors: v = L dt
Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”.
COMPLEX ARITHMETIC TRICKS:
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 9 / 11
CIVIL
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
⇒ I leads V
Capacitors: i = C dv
dt
di
⇒ V leads I
Inductors: v = L dt
Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”.
COMPLEX ARITHMETIC TRICKS:
Admittance
• Summary
(1) j × j = −j × −j = −1
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 9 / 11
CIVIL
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
⇒ I leads V
Capacitors: i = C dv
dt
di
⇒ V leads I
Inductors: v = L dt
Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”.
COMPLEX ARITHMETIC TRICKS:
Admittance
• Summary
(1) j × j = −j × −j = −1
(2) 1j = −j
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 9 / 11
CIVIL
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
⇒ I leads V
Capacitors: i = C dv
dt
di
⇒ V leads I
Inductors: v = L dt
Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”.
COMPLEX ARITHMETIC TRICKS:
Admittance
• Summary
(1) j × j = −j × −j = −1
(2) 1j = −j
(3) a + jb = r∠θ =√rejθ
where r = a2 + b2 and θ = arctan ab (±180◦ if a < 0)
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 9 / 11
CIVIL
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
⇒ I leads V
Capacitors: i = C dv
dt
di
⇒ V leads I
Inductors: v = L dt
Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”.
COMPLEX ARITHMETIC TRICKS:
Admittance
• Summary
(1) j × j = −j × −j = −1
(2) 1j = −j
(3) a + jb = r∠θ =√rejθ
where r = a2 + b2 and θ = arctan ab (±180◦ if a < 0)
(4) r∠θ = rejθ = (r cos θ) + j (r sin θ)
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 9 / 11
CIVIL
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
⇒ I leads V
Capacitors: i = C dv
dt
di
⇒ V leads I
Inductors: v = L dt
Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”.
COMPLEX ARITHMETIC TRICKS:
Admittance
• Summary
(1) j × j = −j × −j = −1
(2) 1j = −j
(3) a + jb = r∠θ =√rejθ
where r = a2 + b2 and θ = arctan ab (±180◦ if a < 0)
(4) r∠θ = rejθ = (r cos θ) + j (r sin θ)
a∠θ
= ab ∠ (θ − φ).
(5) a∠θ × b∠φ = ab∠ (θ + φ) and b∠φ
Multiplication and division are much easier in polar form.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 9 / 11
CIVIL
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
⇒ I leads V
Capacitors: i = C dv
dt
di
⇒ V leads I
Inductors: v = L dt
Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”.
COMPLEX ARITHMETIC TRICKS:
Admittance
• Summary
(1) j × j = −j × −j = −1
(2) 1j = −j
(3) a + jb = r∠θ =√rejθ
where r = a2 + b2 and θ = arctan ab (±180◦ if a < 0)
(4) r∠θ = rejθ = (r cos θ) + j (r sin θ)
a∠θ
= ab ∠ (θ − φ).
(5) a∠θ × b∠φ = ab∠ (θ + φ) and b∠φ
Multiplication and division are much easier in polar form.
(6) All scientific calculators will convert rectangular to/from polar form.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 9 / 11
CIVIL
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
⇒ I leads V
Capacitors: i = C dv
dt
di
⇒ V leads I
Inductors: v = L dt
Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”.
COMPLEX ARITHMETIC TRICKS:
Admittance
• Summary
(1) j × j = −j × −j = −1
(2) 1j = −j
(3) a + jb = r∠θ =√rejθ
where r = a2 + b2 and θ = arctan ab (±180◦ if a < 0)
(4) r∠θ = rejθ = (r cos θ) + j (r sin θ)
a∠θ
= ab ∠ (θ − φ).
(5) a∠θ × b∠φ = ab∠ (θ + φ) and b∠φ
Multiplication and division are much easier in polar form.
(6) All scientific calculators will convert rectangular to/from polar form.
Casio fx-991 (available in all exams except Maths) will do complex
arithmetic (+, −, ×, ÷, x2 , x1 , |x|, x∗ ) in CMPLX mode.
Learn how to use this: it will save lots of time and errors.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 9 / 11
Impedance and Admittance
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
For any network (resistors+capacitors+inductors):
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 10 / 11
Impedance and Admittance
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
For any network (resistors+capacitors+inductors):
(1) Impedance = Resistance + j× Reactance
Z = R + jX (Ω)
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 10 / 11
Impedance and Admittance
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
For any network (resistors+capacitors+inductors):
(1) Impedance = Resistance + j× Reactance
Z = R + jX (Ω)
|Z|2 = R2 + X 2
∠Z = arctan X
R
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 10 / 11
Impedance and Admittance
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
For any network (resistors+capacitors+inductors):
(1) Impedance = Resistance + j× Reactance
Z = R + jX (Ω)
|Z|2 = R2 + X 2
∠Z = arctan X
R
1
= Conductance + j× Susceptance
(2) Admittance = Impedance
Y =
E1.1 Analysis of Circuits (2015-7199)
1
Z
= G + jB Seimens (S)
Phasors: 10 – 10 / 11
Impedance and Admittance
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
For any network (resistors+capacitors+inductors):
(1) Impedance = Resistance + j× Reactance
Z = R + jX (Ω)
|Z|2 = R2 + X 2
∠Z = arctan X
R
1
= Conductance + j× Susceptance
(2) Admittance = Impedance
Y = Z1 = G + jB Seimens (S)
1
2
2
∠Y = −∠Z = arctan B
|Y |2 = |Z|
2 = G + B
G
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 10 / 11
Impedance and Admittance
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
For any network (resistors+capacitors+inductors):
(1) Impedance = Resistance + j× Reactance
Z = R + jX (Ω)
|Z|2 = R2 + X 2
∠Z = arctan X
R
1
= Conductance + j× Susceptance
(2) Admittance = Impedance
Y = Z1 = G + jB Seimens (S)
1
2
2
∠Y = −∠Z = arctan B
|Y |2 = |Z|
2 = G + B
G
Note:
Y = G + jB =
E1.1 Analysis of Circuits (2015-7199)
1
Z
Phasors: 10 – 10 / 11
Impedance and Admittance
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
For any network (resistors+capacitors+inductors):
(1) Impedance = Resistance + j× Reactance
Z = R + jX (Ω)
|Z|2 = R2 + X 2
∠Z = arctan X
R
1
= Conductance + j× Susceptance
(2) Admittance = Impedance
Y = Z1 = G + jB Seimens (S)
1
2
2
∠Y = −∠Z = arctan B
|Y |2 = |Z|
2 = G + B
G
Note:
Y = G + jB =
E1.1 Analysis of Circuits (2015-7199)
1
1
=
Z
R+jX
Phasors: 10 – 10 / 11
Impedance and Admittance
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
For any network (resistors+capacitors+inductors):
(1) Impedance = Resistance + j× Reactance
Z = R + jX (Ω)
|Z|2 = R2 + X 2
∠Z = arctan X
R
1
= Conductance + j× Susceptance
(2) Admittance = Impedance
Y = Z1 = G + jB Seimens (S)
1
2
2
∠Y = −∠Z = arctan B
|Y |2 = |Z|
2 = G + B
G
Note:
Y = G + jB =
E1.1 Analysis of Circuits (2015-7199)
1
1
R
=
=
2
Z
R+jX
R +X 2
+ j R2−X
+X 2
Phasors: 10 – 10 / 11
Impedance and Admittance
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
For any network (resistors+capacitors+inductors):
(1) Impedance = Resistance + j× Reactance
Z = R + jX (Ω)
|Z|2 = R2 + X 2
∠Z = arctan X
R
1
= Conductance + j× Susceptance
(2) Admittance = Impedance
Y = Z1 = G + jB Seimens (S)
1
2
2
∠Y = −∠Z = arctan B
|Y |2 = |Z|
2 = G + B
G
Note:
Y = G + jB =
So G =
B=
E1.1 Analysis of Circuits (2015-7199)
1
1
R
=
=
2
Z
R+jX
R +X 2
R
R
=
2
2
R +X
|Z|2
−X
R2 +X 2
=
+ j R2−X
+X 2
−X
|Z|2
Phasors: 10 – 10 / 11
Impedance and Admittance
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
For any network (resistors+capacitors+inductors):
(1) Impedance = Resistance + j× Reactance
Z = R + jX (Ω)
|Z|2 = R2 + X 2
∠Z = arctan X
R
1
= Conductance + j× Susceptance
(2) Admittance = Impedance
Y = Z1 = G + jB Seimens (S)
1
2
2
∠Y = −∠Z = arctan B
|Y |2 = |Z|
2 = G + B
G
Note:
Y = G + jB =
So G =
B=
1
1
R
=
=
2
Z
R+jX
R +X 2
R
R
=
2
2
R +X
|Z|2
−X
R2 +X 2
=
+ j R2−X
+X 2
−X
|Z|2
1
unless X = 0.
Beware: G 6= R
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 10 / 11
Summary
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
• Sine waves are the only bounded signals whose shape is unchanged by
differentiation.
Admittance
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 11 / 11
Summary
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Sine waves are the only bounded signals whose shape is unchanged by
differentiation.
• Think of a sine wave as the projection of a rotating rod onto the
horizontal (or real) axis.
◦ A phasor is a complex number representing the length and position
of the rod at time t = 0.
• Summary
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 11 / 11
Summary
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
• Sine waves are the only bounded signals whose shape is unchanged by
differentiation.
• Think of a sine wave as the projection of a rotating rod onto the
horizontal (or real) axis.
◦ A phasor is a complex number representing the length and position
of the rod at time t = 0.
◦ If V = a + jb = r∠θ = rejθ , then
v(t) = a cos ωt − b sin ωt = r cos (ωt + θ) = ℜ V ejωt
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 11 / 11
Summary
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
• Sine waves are the only bounded signals whose shape is unchanged by
differentiation.
• Think of a sine wave as the projection of a rotating rod onto the
horizontal (or real) axis.
◦ A phasor is a complex number representing the length and position
of the rod at time t = 0.
◦ If V = a + jb = r∠θ = rejθ , then
v(t) = a cos ωt − b sin ωt = r cos (ωt + θ) = ℜ V ejωt
◦ The angular frequency ω = 2πf is assumed known.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 11 / 11
Summary
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
• Sine waves are the only bounded signals whose shape is unchanged by
differentiation.
• Think of a sine wave as the projection of a rotating rod onto the
horizontal (or real) axis.
◦ A phasor is a complex number representing the length and position
of the rod at time t = 0.
◦ If V = a + jb = r∠θ = rejθ , then
v(t) = a cos ωt − b sin ωt = r cos (ωt + θ) = ℜ V ejωt
◦ The angular frequency ω = 2πf is assumed known.
• If all sources in a linear circuit are sine waves having the same
frequency, we can use phasors for circuit analysis:
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 11 / 11
Summary
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
• Sine waves are the only bounded signals whose shape is unchanged by
differentiation.
• Think of a sine wave as the projection of a rotating rod onto the
horizontal (or real) axis.
◦ A phasor is a complex number representing the length and position
of the rod at time t = 0.
◦ If V = a + jb = r∠θ = rejθ , then
v(t) = a cos ωt − b sin ωt = r cos (ωt + θ) = ℜ V ejωt
◦ The angular frequency ω = 2πf is assumed known.
• If all sources in a linear circuit are sine waves having the same
frequency, we can use phasors for circuit analysis:
1
◦ Use complex impedances: jωL and jωC
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 11 / 11
Summary
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
• Sine waves are the only bounded signals whose shape is unchanged by
differentiation.
• Think of a sine wave as the projection of a rotating rod onto the
horizontal (or real) axis.
◦ A phasor is a complex number representing the length and position
of the rod at time t = 0.
◦ If V = a + jb = r∠θ = rejθ , then
v(t) = a cos ωt − b sin ωt = r cos (ωt + θ) = ℜ V ejωt
◦ The angular frequency ω = 2πf is assumed known.
• If all sources in a linear circuit are sine waves having the same
frequency, we can use phasors for circuit analysis:
1
◦ Use complex impedances: jωL and jωC
◦ Mnemonic: CIVIL tells you whether I leads V or vice versa
(“leads” means “reaches its peak before”).
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 11 / 11
Summary
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
• Sine waves are the only bounded signals whose shape is unchanged by
differentiation.
• Think of a sine wave as the projection of a rotating rod onto the
horizontal (or real) axis.
◦ A phasor is a complex number representing the length and position
of the rod at time t = 0.
◦ If V = a + jb = r∠θ = rejθ , then
v(t) = a cos ωt − b sin ωt = r cos (ωt + θ) = ℜ V ejωt
◦ The angular frequency ω = 2πf is assumed known.
• If all sources in a linear circuit are sine waves having the same
frequency, we can use phasors for circuit analysis:
1
◦ Use complex impedances: jωL and jωC
◦ Mnemonic: CIVIL tells you whether I leads V or vice versa
(“leads” means “reaches its peak before”).
◦ Phasors eliminate time from equations ,
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 11 / 11
Summary
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
• Sine waves are the only bounded signals whose shape is unchanged by
differentiation.
• Think of a sine wave as the projection of a rotating rod onto the
horizontal (or real) axis.
◦ A phasor is a complex number representing the length and position
of the rod at time t = 0.
◦ If V = a + jb = r∠θ = rejθ , then
v(t) = a cos ωt − b sin ωt = r cos (ωt + θ) = ℜ V ejωt
◦ The angular frequency ω = 2πf is assumed known.
• If all sources in a linear circuit are sine waves having the same
frequency, we can use phasors for circuit analysis:
1
◦ Use complex impedances: jωL and jωC
◦ Mnemonic: CIVIL tells you whether I leads V or vice versa
(“leads” means “reaches its peak before”).
◦ Phasors eliminate time from equations ,, converts simultaneous
differential equations into simultaneous linear equations ,,,.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 11 / 11
Summary
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
• Sine waves are the only bounded signals whose shape is unchanged by
differentiation.
• Think of a sine wave as the projection of a rotating rod onto the
horizontal (or real) axis.
◦ A phasor is a complex number representing the length and position
of the rod at time t = 0.
◦ If V = a + jb = r∠θ = rejθ , then
v(t) = a cos ωt − b sin ωt = r cos (ωt + θ) = ℜ V ejωt
◦ The angular frequency ω = 2πf is assumed known.
• If all sources in a linear circuit are sine waves having the same
frequency, we can use phasors for circuit analysis:
1
◦ Use complex impedances: jωL and jωC
◦ Mnemonic: CIVIL tells you whether I leads V or vice versa
(“leads” means “reaches its peak before”).
◦ Phasors eliminate time from equations ,, converts simultaneous
differential equations into simultaneous linear equations ,,,.
◦ Needs complex numbers / but worth it.
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 11 / 11
Summary
10: Sine waves and phasors
• Sine Waves
• Rotating Rod
• Phasors
• Phasor Examples
• Phasor arithmetic
• Complex Impedances
• Phasor Analysis
• CIVIL
• Impedance and
Admittance
• Summary
• Sine waves are the only bounded signals whose shape is unchanged by
differentiation.
• Think of a sine wave as the projection of a rotating rod onto the
horizontal (or real) axis.
◦ A phasor is a complex number representing the length and position
of the rod at time t = 0.
◦ If V = a + jb = r∠θ = rejθ , then
v(t) = a cos ωt − b sin ωt = r cos (ωt + θ) = ℜ V ejωt
◦ The angular frequency ω = 2πf is assumed known.
• If all sources in a linear circuit are sine waves having the same
frequency, we can use phasors for circuit analysis:
1
◦ Use complex impedances: jωL and jωC
◦ Mnemonic: CIVIL tells you whether I leads V or vice versa
(“leads” means “reaches its peak before”).
◦ Phasors eliminate time from equations ,, converts simultaneous
differential equations into simultaneous linear equations ,,,.
◦ Needs complex numbers / but worth it.
see Hayt Chapter 10
E1.1 Analysis of Circuits (2015-7199)
Phasors: 10 – 11 / 11