10: Sine waves and phasors • Sine Waves • Rotating Rod • Phasors • PhasorExamples + • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary 10: Sine waves and phasors
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 1 / 11 Sine Waves
10: Sine waves and phasors dv di • Sine Waves For inductors and capacitors i = C dt and v = L dt so we need to • Rotating Rod differentiate i(t) and v(t) when analysing circuits containing them. • Phasors • PhasorExamples + • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 2 / 11 Sine Waves
10: Sine waves and phasors dv di • Sine Waves For inductors and capacitors i = C dt and v = L dt so we need to • Rotating Rod differentiate i(t) and v(t) when analysing circuits containing them. • Phasors • PhasorExamples + • Phasor arithmetic Usually differentiation changes the 1 • Complex Impedances shape of a waveform. 0 • PhasorAnalysis + -1 0 1 2 3 4 • CIVIL t • Impedance and 5 Admittance 0 • Summary -5 0 1 2 3 4 t
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 2 / 11 Sine Waves
10: Sine waves and phasors dv di • Sine Waves For inductors and capacitors i = C dt and v = L dt so we need to • Rotating Rod differentiate i(t) and v(t) when analysing circuits containing them. • Phasors • PhasorExamples + • Phasor arithmetic Usually differentiation changes the 1 • Complex Impedances shape of a waveform. 0 • PhasorAnalysis + -1 0 1 2 3 4 • CIVIL t • Impedance and For bounded waveforms there is 5 Admittance 0 • Summary only one exception: -5 0 1 2 3 4 t
v(t) = sin t dv = cos t ⇒ dt
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 2 / 11 Sine Waves
10: Sine waves and phasors dv di • Sine Waves For inductors and capacitors i = C dt and v = L dt so we need to • Rotating Rod differentiate i(t) and v(t) when analysing circuits containing them. • Phasors • PhasorExamples + • Phasor arithmetic Usually differentiation changes the 1 • Complex Impedances shape of a waveform. 0 • PhasorAnalysis + -1 0 1 2 3 4 • CIVIL t • Impedance and For bounded waveforms there is 5 Admittance 0 • Summary only one exception: -5 0 1 2 3 4 t
dv 1 v(t) = sin t dt = cos t 0 ⇒ v(t) -1 0 5 10 15 t 1
0 dv/dt -1 0 5 10 15 t
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 2 / 11 Sine Waves
10: Sine waves and phasors dv di • Sine Waves For inductors and capacitors i = C dt and v = L dt so we need to • Rotating Rod differentiate i(t) and v(t) when analysing circuits containing them. • Phasors • PhasorExamples + • Phasor arithmetic Usually differentiation changes the 1 • Complex Impedances shape of a waveform. 0 • PhasorAnalysis + -1 0 1 2 3 4 • CIVIL t • Impedance and For bounded waveforms there is 5 Admittance 0 • Summary only one exception: -5 0 1 2 3 4 t
dv 1 v(t) = sin t dt = cos t 0 ⇒ v(t) same shape but with a time shift. -1 0 5 10 15 t 1
0 dv/dt -1 0 5 10 15 t
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 2 / 11 Sine Waves
10: Sine waves and phasors dv di • Sine Waves For inductors and capacitors i = C dt and v = L dt so we need to • Rotating Rod differentiate i(t) and v(t) when analysing circuits containing them. • Phasors • PhasorExamples + • Phasor arithmetic Usually differentiation changes the 1 • Complex Impedances shape of a waveform. 0 • PhasorAnalysis + -1 0 1 2 3 4 • CIVIL t • Impedance and For bounded waveforms there is 5 Admittance 0 • Summary only one exception: -5 0 1 2 3 4 t
dv 1 v(t) = sin t dt = cos t 0 ⇒ v(t) same shape but with a time shift. -1 0 5 10 15 t sin t completes one full period 1 0 dv/dt every time t increases by 2π. -1 0 5 10 15 t
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 2 / 11 Sine Waves
10: Sine waves and phasors dv di • Sine Waves For inductors and capacitors i = C dt and v = L dt so we need to • Rotating Rod differentiate i(t) and v(t) when analysing circuits containing them. • Phasors • PhasorExamples + • Phasor arithmetic Usually differentiation changes the 1 • Complex Impedances shape of a waveform. 0 • PhasorAnalysis + -1 0 1 2 3 4 • CIVIL t • Impedance and For bounded waveforms there is 5 Admittance 0 • Summary only one exception: -5 0 1 2 3 4 t
dv 1 v(t) = sin t dt = cos t 0 ⇒ v(t) same shape but with a time shift. -1 0 5 10 15 t sin t completes one full period 1 0 dv/dt every time t increases by 2π. -1 0 5 10 15 t sin 2πft 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 (2017-10213) Phasors: 10 – 2 / 11 Sine Waves
10: Sine waves and phasors dv di • Sine Waves For inductors and capacitors i = C dt and v = L dt so we need to • Rotating Rod differentiate i(t) and v(t) when analysing circuits containing them. • Phasors • PhasorExamples + • Phasor arithmetic Usually differentiation changes the 1 • Complex Impedances shape of a waveform. 0 • PhasorAnalysis + -1 0 1 2 3 4 • CIVIL t • Impedance and For bounded waveforms there is 5 Admittance 0 • Summary only one exception: -5 0 1 2 3 4 t
dv 1 v(t) = sin t dt = cos t 0 ⇒ v(t) same shape but with a time shift. -1 0 5 10 15 t sin t completes one full period 1 0 dv/dt every time t increases by 2π. -1 0 5 10 15 t sin 2πft 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. 50Hz 314 rad.s−1. ≃ E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 2 / 11 Rotating Rod
10: Sine waves and phasors A useful way to think of a cosine wave is as the • Sine Waves • Rotating Rod projection of a rotating rod onto the horizontal axis. • Phasors • PhasorExamples + For a unit-length rod, the projection has length θ. • Phasor arithmetic cos • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 3 / 11 Rotating Rod
10: Sine waves and phasors A useful way to think of a cosine wave is as the • Sine Waves • Rotating Rod projection of a rotating rod onto the horizontal axis. • Phasors • PhasorExamples + For a unit-length rod, the projection has length θ. • Phasor arithmetic cos • Complex Impedances • PhasorAnalysis + If the rod is rotating at a speed of f revolutions per • CIVIL • Impedance and second, then θ increases uniformly with time: Admittance • Summary θ = 2πft.
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 3 / 11 Rotating Rod
10: Sine waves and phasors A useful way to think of a cosine wave is as the • Sine Waves • Rotating Rod projection of a rotating rod onto the horizontal axis. • Phasors • PhasorExamples + For a unit-length rod, the projection has length θ. • Phasor arithmetic cos • Complex Impedances • PhasorAnalysis + If the rod is rotating at a speed of f revolutions per • CIVIL • Impedance and second, then θ increases uniformly with time: Admittance • Summary θ = 2πft. The only difference between cos and sin is the starting position of the rod:
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 3 / 11 Rotating Rod
10: Sine waves and phasors A useful way to think of a cosine wave is as the • Sine Waves • Rotating Rod projection of a rotating rod onto the horizontal axis. • Phasors • PhasorExamples + For a unit-length rod, the projection has length θ. • Phasor arithmetic cos • Complex Impedances • PhasorAnalysis + If the rod is rotating at a speed of f revolutions per • CIVIL • Impedance and second, then θ increases uniformly with time: Admittance • Summary θ = 2πft. The only difference between cos and sin is the starting position of the rod:
1
0
-1 0 5 10 15 t
v = cos 2πft
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 3 / 11 Rotating Rod
10: Sine waves and phasors A useful way to think of a cosine wave is as the • Sine Waves • Rotating Rod projection of a rotating rod onto the horizontal axis. • Phasors • PhasorExamples + For a unit-length rod, the projection has length θ. • Phasor arithmetic cos • Complex Impedances • PhasorAnalysis + If the rod is rotating at a speed of f revolutions per • CIVIL • Impedance and second, then θ increases uniformly with time: Admittance • Summary θ = 2πft. The only difference between cos and sin is the starting position of the rod:
1 1
0 0
-1 -1 0 5 10 15 0 5 10 15 t t
v = cos 2πft v = sin 2πft
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 3 / 11 Rotating Rod
10: Sine waves and phasors A useful way to think of a cosine wave is as the • Sine Waves • Rotating Rod projection of a rotating rod onto the horizontal axis. • Phasors • PhasorExamples + For a unit-length rod, the projection has length θ. • Phasor arithmetic cos • Complex Impedances • PhasorAnalysis + If the rod is rotating at a speed of f revolutions per • CIVIL • Impedance and second, then θ increases uniformly with time: Admittance • Summary θ = 2πft. The only difference between cos and sin is the starting position of the rod:
1 1
0 0
-1 -1 0 5 10 15 0 5 10 15 t t
v = cos 2πft v = sin 2πft = cos 2πft π − 2
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 3 / 11 Rotating Rod
10: Sine waves and phasors A useful way to think of a cosine wave is as the • Sine Waves • Rotating Rod projection of a rotating rod onto the horizontal axis. • Phasors • PhasorExamples + For a unit-length rod, the projection has length θ. • Phasor arithmetic cos • Complex Impedances • PhasorAnalysis + If the rod is rotating at a speed of f revolutions per • CIVIL • Impedance and second, then θ increases uniformly with time: Admittance • Summary θ = 2πft. The only difference between cos and sin is the starting position of the rod:
1 1
0 0
-1 -1 0 5 10 15 0 5 10 15 t t
v = cos 2πft v = sin 2πft = cos 2πft π − 2
◦ π 1 sin 2πft lags cos 2πft by 90 (or 2 radians) because its peaks occurs 4 of a cycle later (equivalently cos leads sin).
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 3 / 11 Phasors
10: Sine waves and phasors If the rod has length A and starts at an angle φ then the projection onto the • Sine Waves • Rotating Rod horizontal axis is • Phasors • PhasorExamples + A cos (2πft + φ) • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 4 / 11 Phasors
10: Sine waves and phasors If the rod has length A and starts at an angle φ then the projection onto the • Sine Waves • Rotating Rod horizontal axis is • Phasors • PhasorExamples + A cos (2πft + φ) • Phasor arithmetic • Complex Impedances = A cos φ cos 2πft A sin φ sin 2πft • PhasorAnalysis + − • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 4 / 11 Phasors
10: Sine waves and phasors If the rod has length A and starts at an angle φ then the projection onto the • Sine Waves • Rotating Rod horizontal axis is • Phasors • PhasorExamples + A cos (2πft + φ) • Phasor arithmetic • Complex Impedances = A cos φ cos 2πft A sin φ sin 2πft • PhasorAnalysis + − • CIVIL = X cos 2πft Y sin 2πft • Impedance and − Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 4 / 11 Phasors
10: Sine waves and phasors If the rod has length A and starts at an angle φ then the projection onto the • Sine Waves • Rotating Rod horizontal axis is • Phasors • PhasorExamples + A cos (2πft + φ) • Phasor arithmetic • Complex Impedances = A cos φ cos 2πft A sin φ sin 2πft • PhasorAnalysis + − • CIVIL = X cos 2πft Y sin 2πft • Impedance and − Admittance • Summary At time t = 0, the tip of the rod has coordinates (X, Y )=(A cos φ, A sin φ).
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 4 / 11 Phasors
10: Sine waves and phasors If the rod has length A and starts at an angle φ then the projection onto the • Sine Waves • Rotating Rod horizontal axis is • Phasors • PhasorExamples + A cos (2πft + φ) • Phasor arithmetic • Complex Impedances = A cos φ cos 2πft A sin φ sin 2πft • PhasorAnalysis + − • CIVIL = X cos 2πft Y sin 2πft • Impedance and − Admittance • Summary 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 (2017-10213) Phasors: 10 – 4 / 11 Phasors
10: Sine waves and phasors If the rod has length A and starts at an angle φ then the projection onto the • Sine Waves • Rotating Rod horizontal axis is • Phasors • PhasorExamples + A cos (2πft + φ) • Phasor arithmetic • Complex Impedances = A cos φ cos 2πft A sin φ sin 2πft • PhasorAnalysis + − • CIVIL = X cos 2πft Y sin 2πft • Impedance and − Admittance • Summary 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 = √X2 + Y 2, gives the amplitude (peak value) of the sine wave.
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 4 / 11 Phasors
10: Sine waves and phasors If the rod has length A and starts at an angle φ then the projection onto the • Sine Waves • Rotating Rod horizontal axis is • Phasors • PhasorExamples + A cos (2πft + φ) • Phasor arithmetic • Complex Impedances = A cos φ cos 2πft A sin φ sin 2πft • PhasorAnalysis + − • CIVIL = X cos 2πft Y sin 2πft • Impedance and − Admittance • Summary 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 = √X2 + Y 2, gives the amplitude (peak value) of the sine wave. Y The argument of the phasor, φ = arctan X , gives the phase shift relative to cos 2πft. If φ > 0, it is leading and if φ < 0, it is lagging relative to cos 2πft.
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 4 / 11 Phasor Examples +
10: Sine waves and phasors V , f • Sine Waves = 1 = 50Hz • Rotating Rod • Phasors • PhasorExamples + • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j • Complex Impedances − • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V = 1 0.5j • Summary − −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j • Summary = 1 0 5 v(t) = − cos− 2πft + 0.5 sin 2πft −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j . ∠ ◦ • Summary = 1 0 5 = 1 12 153 v(t) = − cos− 2πft + 0.5 sin− 2πft −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j . ∠ ◦ • Summary = 1 0 5 = 1 12 153 v(t) = − cos− 2πft + 0.5 sin− 2πft = 1−.12 cos (2πft 2.68) −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j . ∠ ◦ • Summary = 1 0 5 = 1 12 153 v(t) = − cos− 2πft + 0.5 sin− 2πft = 1−.12 cos (2πft 2.68) − V = X + jY
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j . ∠ ◦ • Summary = 1 0 5 = 1 12 153 v(t) = − cos− 2πft + 0.5 sin− 2πft = 1−.12 cos (2πft 2.68) − V = X + jY v(t) = X cos 2πft Y sin 2πft −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j . ∠ ◦ • Summary = 1 0 5 = 1 12 153 v(t) = − cos− 2πft + 0.5 sin− 2πft = 1−.12 cos (2πft 2.68) − V = X + jY v(t) = X cos 2πft Y sin 2πft Beware minus sign. −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j . ∠ ◦ • Summary = 1 0 5 = 1 12 153 v(t) = − cos− 2πft + 0.5 sin− 2πft = 1−.12 cos (2πft 2.68) − V = X + jY V = A∠φ v(t) = X cos 2πft Y sin 2πft Beware minus sign. −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j . ∠ ◦ • Summary = 1 0 5 = 1 12 153 v(t) = − cos− 2πft + 0.5 sin− 2πft = 1−.12 cos (2πft 2.68) − V = X + jY V = A∠φ v(t) = X cos 2πft Y sin 2πft v(t) = A cos (2πft + φ) Beware minus sign. −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j . ∠ ◦ • Summary = 1 0 5 = 1 12 153 v(t) = − cos− 2πft + 0.5 sin− 2πft = 1−.12 cos (2πft 2.68) − V = X + jY V = A∠φ = Aejφ v(t) = X cos 2πft Y sin 2πft v(t) = A cos (2πft + φ) Beware minus sign. −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j . ∠ ◦ • Summary = 1 0 5 = 1 12 153 v(t) = − cos− 2πft + 0.5 sin− 2πft = 1−.12 cos (2πft 2.68) − V = X + jY V = A∠φ = Aejφ v(t) = X cos 2πft Y sin 2πft v(t) = A cos (2πft + 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 (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j . ∠ ◦ • Summary = 1 0 5 = 1 12 153 v(t) = − cos− 2πft + 0.5 sin− 2πft = 1−.12 cos (2πft 2.68) − V = X + jY V = A∠φ = Aejφ v(t) = X cos 2πft Y sin 2πft v(t) = A cos (2πft + 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 (2017-10213) Phasors: 10 – 5 / 11 Phasor Examples +
10: Sine waves and phasors V , f 1 • Sine Waves = 1 = 50Hz 0 • Rotating Rod v(t) = cos 2πft • Phasors -1 0 0.02 0.04 0.06 • PhasorExamples + t • Phasor arithmetic V = j 1 • Complex Impedances − 0 • PhasorAnalysis + v(t) = sin 2πft • -1 CIVIL 0 0.02 0.04 0.06 • Impedance and t Admittance V . j . ∠ ◦ • Summary = 1 0 5 = 1 12 153 v(t) = − cos− 2πft + 0.5 sin− 2πft = 1−.12 cos (2πft 2.68) − V = X + jY V = A∠φ = Aejφ v(t) = X cos 2πft Y sin 2πft v(t) = A cos (2πft + 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 (2017-10213) → → → Phasors: 10 – 5 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + a v(t) • CIVIL × • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance v1(t) + v2(t) • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance V1 + V2 v1(t) + v2(t) • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance V1 + V2 v1(t) + v2(t) • Summary Adding or scaling is the same for waveforms and phasors.
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance V1 + V2 v1(t) + v2(t) • Summary Adding or scaling is the same for waveforms and phasors.
dv = ωP sin ωt ωQ cos ωt dt − −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance V1 + V2 v1(t) + v2(t) • Summary Adding or scaling is the same for waveforms and phasors.
dv dt = ωP sin ωt ωQ cos ωt =(− ωQ) cos ωt− (ωP ) sin ωt − −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance V1 + V2 v1(t) + v2(t) • Summary Adding or scaling is the same for waveforms and phasors.
dv = ωP sin ωt ωQ cos ωt ˙ dt V =( ωQ) + j (ωP ) =(− ωQ) cos ωt− (ωP ) sin ωt − − −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance V1 + V2 v1(t) + v2(t) • Summary Adding or scaling is the same for waveforms and phasors.
dv = ωP sin ωt ωQ cos ωt ˙ dt V =( ωQ) + j (ωP ) =(− ωQ) cos ωt− (ωP ) sin ωt = jω−(P + jQ) − −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance V1 + V2 v1(t) + v2(t) • Summary Adding or scaling is the same for waveforms and phasors.
dv = ωP sin ωt ωQ cos ωt ˙ dt V =( ωQ) + j (ωP ) =(− ωQ) cos ωt− (ωP ) sin ωt = jω−(P + jQ) − − = jωV
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance V1 + V2 v1(t) + v2(t) • Summary Adding or scaling is the same for waveforms and phasors.
dv = ωP sin ωt ωQ cos ωt ˙ dt V =( ωQ) + j (ωP ) =(− ωQ) cos ωt− (ωP ) sin ωt = jω−(P + jQ) − − = jωV Differentiating waveforms corresponds to multiplying phasors by jω.
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance V1 + V2 v1(t) + v2(t) • Summary Adding or scaling is the same for waveforms and phasors.
dv = ωP sin ωt ωQ cos ωt ˙ dt V =( ωQ) + j (ωP ) =(− ωQ) cos ωt− (ωP ) sin ωt = jω−(P + jQ) − − = jωV Differentiating waveforms corresponds to multiplying phasors by jω.
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Phasor arithmetic
10: Sine waves and phasors Phasors Waveforms • Sine Waves • Rotating Rod • Phasors V = P + jQ v(t) = P cos ωt Q sin ωt • PhasorExamples + where ω πf. − • Phasor arithmetic = 2 • Complex Impedances • PhasorAnalysis + aV a v(t) = aP cos ωt aQ sin ωt • CIVIL × − • Impedance and Admittance V1 + V2 v1(t) + v2(t) • Summary Adding or scaling is the same for waveforms and phasors.
dv = ωP sin ωt ωQ cos ωt ˙ dt V =( ωQ) + j (ωP ) =(− ωQ) cos ωt− (ωP ) sin ωt = jω−(P + jQ) − − = jωV Differentiating waveforms corresponds to multiplying phasors by jω. Rotate anti-clockwise 90◦ and scale by ω = 2πf.
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 6 / 11 Complex Impedances
10: Sine waves and phasors Resistor: • Sine Waves • Rotating Rod • Phasors v(t) = Ri(t) • PhasorExamples + • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 7 / 11 Complex Impedances
10: Sine waves and phasors Resistor: • Sine Waves • Rotating Rod • Phasors v(t) = Ri(t) V = RI • PhasorExamples + ⇒ • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 7 / 11 Complex Impedances
10: Sine waves and phasors Resistor: • Sine Waves • Rotating Rod V • Phasors v(t) = Ri(t) V = RI I = R • PhasorExamples + ⇒ ⇒ • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 7 / 11 Complex Impedances
10: Sine waves and phasors Resistor: • Sine Waves • Rotating Rod V • Phasors v(t) = Ri(t) V = RI I = R • PhasorExamples + ⇒ ⇒ • Phasor arithmetic • Complex Impedances • PhasorAnalysis + Inductor: • CIVIL • Impedance and Admittance v(t) = L di • Summary dt
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 7 / 11 Complex Impedances
10: Sine waves and phasors Resistor: • Sine Waves • Rotating Rod V • Phasors v(t) = Ri(t) V = RI I = R • PhasorExamples + ⇒ ⇒ • Phasor arithmetic • Complex Impedances • PhasorAnalysis + Inductor: • CIVIL • Impedance and Admittance v(t) = L di V = jωLI • Summary dt ⇒
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 7 / 11 Complex Impedances
10: Sine waves and phasors Resistor: • Sine Waves • Rotating Rod V • Phasors v(t) = Ri(t) V = RI I = R • PhasorExamples + ⇒ ⇒ • Phasor arithmetic • Complex Impedances • PhasorAnalysis + Inductor: • CIVIL • Impedance and Admittance v(t) = L di V = jωLI V = jωL • Summary dt ⇒ ⇒ I
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 7 / 11 Complex Impedances
10: Sine waves and phasors Resistor: • Sine Waves • Rotating Rod V • Phasors v(t) = Ri(t) V = RI I = R • PhasorExamples + ⇒ ⇒ • Phasor arithmetic • Complex Impedances • PhasorAnalysis + Inductor: • CIVIL • Impedance and Admittance v(t) = L di V = jωLI V = jωL • Summary dt ⇒ ⇒ I
Capacitor: dv i(t) = C dt
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 7 / 11 Complex Impedances
10: Sine waves and phasors Resistor: • Sine Waves • Rotating Rod V • Phasors v(t) = Ri(t) V = RI I = R • PhasorExamples + ⇒ ⇒ • Phasor arithmetic • Complex Impedances • PhasorAnalysis + Inductor: • CIVIL • Impedance and Admittance v(t) = L di V = jωLI V = jωL • Summary dt ⇒ ⇒ I
Capacitor: i(t) = C dv I = jωCV dt ⇒
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 7 / 11 Complex Impedances
10: Sine waves and phasors Resistor: • Sine Waves • Rotating Rod V • Phasors v(t) = Ri(t) V = RI I = R • PhasorExamples + ⇒ ⇒ • Phasor arithmetic • Complex Impedances • PhasorAnalysis + Inductor: • CIVIL • Impedance and Admittance v(t) = L di V = jωLI V = jωL • Summary dt ⇒ ⇒ I
Capacitor: i(t) = C dv I = jωCV V = 1 dt ⇒ ⇒ I jωC
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 7 / 11 Complex Impedances
10: Sine waves and phasors Resistor: • Sine Waves • Rotating Rod V • Phasors v(t) = Ri(t) V = RI I = R • PhasorExamples + ⇒ ⇒ • Phasor arithmetic • Complex Impedances • PhasorAnalysis + Inductor: • CIVIL • Impedance and Admittance v(t) = L di V = jωLI V = jωL • Summary dt ⇒ ⇒ I
Capacitor: i(t) = C dv I = jωCV V = 1 dt ⇒ ⇒ I 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 (2017-10213) Phasors: 10 – 7 / 11 Complex Impedances
10: Sine waves and phasors Resistor: • Sine Waves • Rotating Rod V • Phasors v(t) = Ri(t) V = RI I = R • PhasorExamples + ⇒ ⇒ • Phasor arithmetic • Complex Impedances • PhasorAnalysis + Inductor: • CIVIL • Impedance and Admittance v(t) = L di V = jωLI V = jωL • Summary dt ⇒ ⇒ I
Capacitor: i(t) = C dv I = jωCV V = 1 dt ⇒ ⇒ I 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 (2017-10213) Phasors: 10 – 7 / 11 Phasor Analysis +
10: Sine waves and phasors Given v = 10 sin ωt where ω = 2π 1000, • Sine Waves × • Rotating Rod find vC (t). • Phasors • PhasorExamples + • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • 10 Summary v
0
-10 0 0.5 1 1.5 2 t (ms)
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 8 / 11 Phasor Analysis +
10: Sine waves and phasors Given v = 10 sin ωt where ω = 2π 1000, • Sine Waves × • Rotating Rod find vC (t). • Phasors • PhasorExamples + • (1) Find capacitor complex impedance Phasor arithmetic 1 1 • Complex Impedances Z = jωC = 6.28j×10−4 = 1592j • PhasorAnalysis + − • CIVIL • Impedance and Admittance • 10 Summary v
0
-10 0 0.5 1 1.5 2 t (ms)
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 8 / 11 Phasor Analysis +
10: Sine waves and phasors Given v = 10 sin ωt where ω = 2π 1000, • Sine Waves × • Rotating Rod find vC (t). • Phasors • PhasorExamples + • (1) Find capacitor complex impedance Phasor arithmetic 1 1 • Complex Impedances Z = jωC = 6.28j×10−4 = 1592j • PhasorAnalysis + − • CIVIL • Impedance and Admittance • Summary (2) Solve circuit with phasors 10 Z v VC = V R+Z × 0
-10 0 0.5 1 1.5 2 t (ms)
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 8 / 11 Phasor Analysis +
10: Sine waves and phasors Given v = 10 sin ωt where ω = 2π 1000, • Sine Waves × • Rotating Rod find vC (t). • Phasors • PhasorExamples + • (1) Find capacitor complex impedance Phasor arithmetic 1 1 • Complex Impedances Z = jωC = 6.28j×10−4 = 1592j • PhasorAnalysis + − • CIVIL • Impedance and Admittance • Summary (2) Solve circuit with phasors 10 Z v VC = V R+Z × 0 = 10j −1592j − × 1000−1592j -10 0 0.5 1 1.5 2 t (ms)
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 8 / 11 Phasor Analysis +
10: Sine waves and phasors Given v = 10 sin ωt where ω = 2π 1000, • Sine Waves × • Rotating Rod find vC (t). • Phasors • PhasorExamples + • (1) Find capacitor complex impedance Phasor arithmetic 1 1 • Complex Impedances Z = jωC = 6.28j×10−4 = 1592j • PhasorAnalysis + − • CIVIL • Impedance and Admittance • Summary (2) Solve circuit with phasors 10 Z v VC = V × R+Z −1592j 0 = 10j 1000−1592j − × ◦ . . j . ∠ -10 = 4 5 7 2 = 8 47 122 0 0.5 1 1.5 2 − − − t (ms)
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 8 / 11 Phasor Analysis +
10: Sine waves and phasors Given v = 10 sin ωt where ω = 2π 1000, • Sine Waves × • Rotating Rod find vC (t). • Phasors • PhasorExamples + • (1) Find capacitor complex impedance Phasor arithmetic 1 1 • Complex Impedances Z = jωC = 6.28j×10−4 = 1592j • PhasorAnalysis + − • CIVIL • Impedance and Admittance • (2) Solve circuit with phasors 10 Summary v Z v C
VC = V C × R+Z −1592j 0 = 10j 1000−1592j − × ◦ . . j . ∠ -10 = 4 5 7 2 = 8 47 122 0 0.5 1 1.5 2 − − ◦ − vC = 8.47 cos (ωt 122 ) t (ms) −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 8 / 11 Phasor Analysis +
10: Sine waves and phasors Given v = 10 sin ωt where ω = 2π 1000, • Sine Waves × • Rotating Rod find vC (t). • Phasors • PhasorExamples + • (1) Find capacitor complex impedance Phasor arithmetic 1 1 • Complex Impedances Z = jωC = 6.28j×10−4 = 1592j • PhasorAnalysis + − • CIVIL • Impedance and Admittance • (2) Solve circuit with phasors 10 Summary v Z v C
VC = V C × R+Z −1592j 0 = 10j 1000−1592j − × ◦ . . j . ∠ -10 = 4 5 7 2 = 8 47 122 0 0.5 1 1.5 2 − − ◦ − vC = 8.47 cos (ωt 122 ) t (ms) − (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 (2017-10213) Phasors: 10 – 8 / 11 Phasor Analysis +
10: Sine waves and phasors Given v = 10 sin ωt where ω = 2π 1000, • Sine Waves × • Rotating Rod find vC (t). • Phasors • PhasorExamples + • (1) Find capacitor complex impedance Phasor arithmetic 1 1 • Complex Impedances Z = jωC = 6.28j×10−4 = 1592j • PhasorAnalysis + − • CIVIL • Impedance and Admittance • (2) Solve circuit with phasors 10 Summary v Z R v VC = V v C × R+Z R −1592j 0 = 10j 1000−1592j C − × ◦ . . j . ∠ -10 = 4 5 7 2 = 8 47 122 0 0.5 1 1.5 2 − − ◦ − vC = 8.47 cos (ωt 122 ) t (ms) − (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 (2017-10213) Phasors: 10 – 8 / 11 Phasor Analysis +
10: Sine waves and phasors Given v = 10 sin ωt where ω = 2π 1000, • Sine Waves × • Rotating Rod find vC (t). • Phasors • PhasorExamples + • (1) Find capacitor complex impedance Phasor arithmetic 1 1 • Complex Impedances Z = jωC = 6.28j×10−4 = 1592j • PhasorAnalysis + − • CIVIL • Impedance and Admittance • (2) Solve circuit with phasors 10 Summary v Z R v VC = V v C × R+Z R −1592j 0 = 10j 1000−1592j C − × ◦ . . j . ∠ -10 = 4 5 7 2 = 8 47 122 0 0.5 1 1.5 2 − − ◦ − vC = 8.47 cos (ωt 122 ) t (ms) − (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 (2017-10213) Phasors: 10 – 8 / 11 CIVIL
10: Sine waves and phasors dv • Sine Waves Capacitors: i = C dt I leads V • ⇒ Rotating Rod di • Phasors Inductors: v = L dt V leads I • PhasorExamples + ⇒ • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 9 / 11 CIVIL
10: Sine waves and phasors dv • Sine Waves Capacitors: i = C dt I leads V • ⇒ Rotating Rod di • Phasors Inductors: v = L dt V leads I • PhasorExamples + ⇒ • Phasor arithmetic Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”. • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 9 / 11 CIVIL
10: Sine waves and phasors dv • Sine Waves Capacitors: i = C dt I leads V • ⇒ Rotating Rod di • Phasors Inductors: v = L dt V leads I • PhasorExamples + ⇒ • Phasor arithmetic Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”. • Complex Impedances • PhasorAnalysis + • CIVIL COMPLEX ARITHMETIC TRICKS: • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 9 / 11 CIVIL
10: Sine waves and phasors dv • Sine Waves Capacitors: i = C dt I leads V • ⇒ Rotating Rod di • Phasors Inductors: v = L dt V leads I • PhasorExamples + ⇒ • Phasor arithmetic Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”. • Complex Impedances • PhasorAnalysis + • CIVIL COMPLEX ARITHMETIC TRICKS: • Impedance and Admittance • Summary (1) j j = j j = 1 × − × − −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 9 / 11 CIVIL
10: Sine waves and phasors dv • Sine Waves Capacitors: i = C dt I leads V • ⇒ Rotating Rod di • Phasors Inductors: v = L dt V leads I • PhasorExamples + ⇒ • Phasor arithmetic Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”. • Complex Impedances • PhasorAnalysis + • CIVIL COMPLEX ARITHMETIC TRICKS: • Impedance and Admittance • Summary (1) j j = j j = 1 1 × − × − − (2) = j j −
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 9 / 11 CIVIL
10: Sine waves and phasors dv • Sine Waves Capacitors: i = C dt I leads V • ⇒ Rotating Rod di • Phasors Inductors: v = L dt V leads I • PhasorExamples + ⇒ • Phasor arithmetic Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”. • Complex Impedances • PhasorAnalysis + • CIVIL COMPLEX ARITHMETIC TRICKS: • Impedance and Admittance • Summary (1) j j = j j = 1 1 × − × − − (2) = j j − (3) a + jb = r∠θ = rejθ where r = √a2 + b2 and θ = arctan b ( 180◦ if a < 0) a ±
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 9 / 11 CIVIL
10: Sine waves and phasors dv • Sine Waves Capacitors: i = C dt I leads V • ⇒ Rotating Rod di • Phasors Inductors: v = L dt V leads I • PhasorExamples + ⇒ • Phasor arithmetic Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”. • Complex Impedances • PhasorAnalysis + • CIVIL COMPLEX ARITHMETIC TRICKS: • Impedance and Admittance • Summary (1) j j = j j = 1 1 × − × − − (2) = j j − (3) a + jb = r∠θ = rejθ where r = √a2 + b2 and θ = arctan b ( 180◦ if a < 0) a ± (4) r∠θ = rejθ =(r cos θ) + j (r sin θ)
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 9 / 11 CIVIL
10: Sine waves and phasors dv • Sine Waves Capacitors: i = C dt I leads V • ⇒ Rotating Rod di • Phasors Inductors: v = L dt V leads I • PhasorExamples + ⇒ • Phasor arithmetic Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”. • Complex Impedances • PhasorAnalysis + • CIVIL COMPLEX ARITHMETIC TRICKS: • Impedance and Admittance • Summary (1) j j = j j = 1 1 × − × − − (2) = j j − (3) a + jb = r∠θ = rejθ where r = √a2 + b2 and θ = arctan b ( 180◦ if a < 0) a ± (4) r∠θ = rejθ =(r cos θ) + j (r sin θ) ∠ ∠ ∠ a∠θ a ∠ (5) a θ b φ = ab (θ + φ) and b∠φ = b (θ φ). ×Multiplication and division are much easier in− polar form.
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 9 / 11 CIVIL
10: Sine waves and phasors dv • Sine Waves Capacitors: i = C dt I leads V • ⇒ Rotating Rod di • Phasors Inductors: v = L dt V leads I • PhasorExamples + ⇒ • Phasor arithmetic Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”. • Complex Impedances • PhasorAnalysis + • CIVIL COMPLEX ARITHMETIC TRICKS: • Impedance and Admittance • Summary (1) j j = j j = 1 1 × − × − − (2) = j j − (3) a + jb = r∠θ = rejθ where r = √a2 + b2 and θ = arctan b ( 180◦ if a < 0) a ± (4) r∠θ = rejθ =(r cos θ) + j (r sin θ) ∠ ∠ ∠ a∠θ a ∠ (5) a θ b φ = ab (θ + φ) and b∠φ = b (θ 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 (2017-10213) Phasors: 10 – 9 / 11 CIVIL
10: Sine waves and phasors dv • Sine Waves Capacitors: i = C dt I leads V • ⇒ Rotating Rod di • Phasors Inductors: v = L dt V leads I • PhasorExamples + ⇒ • Phasor arithmetic Mnemonic: CIVIL = “In a capacitor I lead V but V leads I in an inductor”. • Complex Impedances • PhasorAnalysis + • CIVIL COMPLEX ARITHMETIC TRICKS: • Impedance and Admittance • Summary (1) j j = j j = 1 1 × − × − − (2) = j j − (3) a + jb = r∠θ = rejθ where r = √a2 + b2 and θ = arctan b ( 180◦ if a < 0) a ± (4) r∠θ = rejθ =(r cos θ) + j (r sin θ) ∠ ∠ ∠ a∠θ a ∠ (5) a θ b φ = ab (θ + φ) and b∠φ = b (θ 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, 1 , x ,x∗) in CMPLX mode. − × ÷ x | | Learn how to use this: it will save lots of time and errors.
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 9 / 11 Impedance and Admittance
10: Sine waves and phasors For any network (resistors+capacitors+inductors): • Sine Waves • Rotating Rod • Phasors • PhasorExamples + • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 10 / 11 Impedance and Admittance
10: Sine waves and phasors For any network (resistors+capacitors+inductors): • Sine Waves • Rotating Rod • Phasors (1) Impedance = Resistance + j Reactance • PhasorExamples + Z R jX ( ) × • Phasor arithmetic = + Ω • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 10 / 11 Impedance and Admittance
10: Sine waves and phasors For any network (resistors+capacitors+inductors): • Sine Waves • Rotating Rod • Phasors (1) Impedance = Resistance + j Reactance • PhasorExamples + Z R jX ( ) × • Phasor arithmetic = + Ω 2 2 2 X • Complex Impedances ∠ Z = R + X Z = arctan R • PhasorAnalysis + | | • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 10 / 11 Impedance and Admittance
10: Sine waves and phasors For any network (resistors+capacitors+inductors): • Sine Waves • Rotating Rod • Phasors (1) Impedance = Resistance + j Reactance • PhasorExamples + Z R jX ( ) × • Phasor arithmetic = + Ω 2 2 2 X • Complex Impedances ∠ Z = R + X Z = arctan R • PhasorAnalysis + | | • CIVIL 1 • Impedance and (2) Admittance = Impedance = Conductance + j Susceptance Admittance 1 × • Summary Y = Z = G + jB Siemens (S)
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 10 / 11 Impedance and Admittance
10: Sine waves and phasors For any network (resistors+capacitors+inductors): • Sine Waves • Rotating Rod • Phasors (1) Impedance = Resistance + j Reactance • PhasorExamples + Z R jX ( ) × • Phasor arithmetic = + Ω 2 2 2 X • Complex Impedances ∠ Z = R + X Z = arctan R • PhasorAnalysis + | | • CIVIL 1 • Impedance and (2) Admittance = Impedance = Conductance + j Susceptance Admittance 1 × • Summary Y = Z = G + jB Siemens (S) 2 1 2 2 B Y = 2 = G + B ∠Y = ∠Z = arctan | | |Z| − G
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 10 / 11 Impedance and Admittance
10: Sine waves and phasors For any network (resistors+capacitors+inductors): • Sine Waves • Rotating Rod • Phasors (1) Impedance = Resistance + j Reactance • PhasorExamples + Z R jX ( ) × • Phasor arithmetic = + Ω 2 2 2 X • Complex Impedances ∠ Z = R + X Z = arctan R • PhasorAnalysis + | | • CIVIL 1 • Impedance and (2) Admittance = Impedance = Conductance + j Susceptance Admittance 1 × • Summary Y = Z = G + jB Siemens (S) 2 1 2 2 B Y = 2 = G + B ∠Y = ∠Z = arctan | | |Z| − G Note: 1 Y = G + jB = Z
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 10 / 11 Impedance and Admittance
10: Sine waves and phasors For any network (resistors+capacitors+inductors): • Sine Waves • Rotating Rod • Phasors (1) Impedance = Resistance + j Reactance • PhasorExamples + Z R jX ( ) × • Phasor arithmetic = + Ω 2 2 2 X • Complex Impedances ∠ Z = R + X Z = arctan R • PhasorAnalysis + | | • CIVIL 1 • Impedance and (2) Admittance = Impedance = Conductance + j Susceptance Admittance 1 × • Summary Y = Z = G + jB Siemens (S) 2 1 2 2 B Y = 2 = G + B ∠Y = ∠Z = arctan | | |Z| − G Note: 1 1 Y = G + jB = Z = R+jX
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 10 / 11 Impedance and Admittance
10: Sine waves and phasors For any network (resistors+capacitors+inductors): • Sine Waves • Rotating Rod • Phasors (1) Impedance = Resistance + j Reactance • PhasorExamples + Z R jX ( ) × • Phasor arithmetic = + Ω 2 2 2 X • Complex Impedances ∠ Z = R + X Z = arctan R • PhasorAnalysis + | | • CIVIL 1 • Impedance and (2) Admittance = Impedance = Conductance + j Susceptance Admittance 1 × • Summary Y = Z = G + jB Siemens (S) 2 1 2 2 B Y = 2 = G + B ∠Y = ∠Z = arctan | | |Z| − G Note: 1 1 R −X Y = G + jB = Z = R+jX = R2+X2 + j R2+X2
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 10 / 11 Impedance and Admittance
10: Sine waves and phasors For any network (resistors+capacitors+inductors): • Sine Waves • Rotating Rod • Phasors (1) Impedance = Resistance + j Reactance • PhasorExamples + Z R jX ( ) × • Phasor arithmetic = + Ω 2 2 2 X • Complex Impedances ∠ Z = R + X Z = arctan R • PhasorAnalysis + | | • CIVIL 1 • Impedance and (2) Admittance = Impedance = Conductance + j Susceptance Admittance 1 × • Summary Y = Z = G + jB Siemens (S) 2 1 2 2 B Y = 2 = G + B ∠Y = ∠Z = arctan | | |Z| − G Note: 1 1 R −X Y = G + jB = Z = R+jX = R2+X2 + j R2+X2 R R So G 2 2 2 = R +X = |Z| −X −X B 2 2 2 = R +X = |Z|
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 10 / 11 Impedance and Admittance
10: Sine waves and phasors For any network (resistors+capacitors+inductors): • Sine Waves • Rotating Rod • Phasors (1) Impedance = Resistance + j Reactance • PhasorExamples + Z R jX ( ) × • Phasor arithmetic = + Ω 2 2 2 X • Complex Impedances ∠ Z = R + X Z = arctan R • PhasorAnalysis + | | • CIVIL 1 • Impedance and (2) Admittance = Impedance = Conductance + j Susceptance Admittance 1 × • Summary Y = Z = G + jB Siemens (S) 2 1 2 2 B Y = 2 = G + B ∠Y = ∠Z = arctan | | |Z| − G Note: 1 1 R −X Y = G + jB = Z = R+jX = R2+X2 + j R2+X2 R R So G 2 2 2 = R +X = |Z| −X −X B 2 2 2 = R +X = |Z|
Beware: G = 1 unless X = 0. 6 R
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 10 / 11 Summary
10: Sine waves and phasors Sine waves are the only bounded signals whose shape is unchanged by • Sine Waves • • Rotating Rod differentiation. • Phasors • PhasorExamples + • Phasor arithmetic • Complex Impedances • PhasorAnalysis + • CIVIL • Impedance and Admittance • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 11 / 11 Summary
10: Sine waves and phasors Sine waves are the only bounded signals whose shape is unchanged by • Sine Waves • • Rotating Rod differentiation. • Phasors • PhasorExamples + Think of a sine wave as the projection of a rotating rod onto the • Phasor arithmetic • • Complex Impedances horizontal (or real) axis. • PhasorAnalysis + • CIVIL A phasor is a complex number representing the length and position • Impedance and ◦ Admittance of the rod at time t = 0. • Summary
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 11 / 11 Summary
10: Sine waves and phasors Sine waves are the only bounded signals whose shape is unchanged by • Sine Waves • • Rotating Rod differentiation. • Phasors • PhasorExamples + Think of a sine wave as the projection of a rotating rod onto the • Phasor arithmetic • • Complex Impedances horizontal (or real) axis. • PhasorAnalysis + • CIVIL A phasor is a complex number representing the length and position • Impedance and ◦ Admittance of the rod at time t = 0. • Summary If V = a + jb = r∠θ = rejθ, then ◦ v(t) = a cos ωt b sin ωt = r cos(ωt + θ) = V ejωt − ℜ
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 11 / 11 Summary
10: Sine waves and phasors Sine waves are the only bounded signals whose shape is unchanged by • Sine Waves • • Rotating Rod differentiation. • Phasors • PhasorExamples + Think of a sine wave as the projection of a rotating rod onto the • Phasor arithmetic • • Complex Impedances horizontal (or real) axis. • PhasorAnalysis + • CIVIL A phasor is a complex number representing the length and position • Impedance and ◦ Admittance of the rod at time t = 0. • Summary If V = a + jb = r∠θ = rejθ, then ◦ v(t) = a cos ωt b sin ωt = r cos(ωt + θ) = V ejωt − ℜ The angular frequency ω = 2πf is assumed known. ◦
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 11 / 11 Summary
10: Sine waves and phasors Sine waves are the only bounded signals whose shape is unchanged by • Sine Waves • • Rotating Rod differentiation. • Phasors • PhasorExamples + Think of a sine wave as the projection of a rotating rod onto the • Phasor arithmetic • • Complex Impedances horizontal (or real) axis. • PhasorAnalysis + • CIVIL A phasor is a complex number representing the length and position • Impedance and ◦ Admittance of the rod at time t = 0. • Summary If V = a + jb = r∠θ = rejθ, then ◦ v(t) = a cos ωt b sin ωt = r cos(ωt + θ) = V ejωt − ℜ 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 (2017-10213) Phasors: 10 – 11 / 11 Summary
10: Sine waves and phasors Sine waves are the only bounded signals whose shape is unchanged by • Sine Waves • • Rotating Rod differentiation. • Phasors • PhasorExamples + Think of a sine wave as the projection of a rotating rod onto the • Phasor arithmetic • • Complex Impedances horizontal (or real) axis. • PhasorAnalysis + • CIVIL A phasor is a complex number representing the length and position • Impedance and ◦ Admittance of the rod at time t = 0. • Summary If V = a + jb = r∠θ = rejθ, then ◦ v(t) = a cos ωt b sin ωt = r cos(ωt + θ) = V ejωt − ℜ 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: Use complex impedances: jωL and 1 ◦ jωC
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 11 / 11 Summary
10: Sine waves and phasors Sine waves are the only bounded signals whose shape is unchanged by • Sine Waves • • Rotating Rod differentiation. • Phasors • PhasorExamples + Think of a sine wave as the projection of a rotating rod onto the • Phasor arithmetic • • Complex Impedances horizontal (or real) axis. • PhasorAnalysis + • CIVIL A phasor is a complex number representing the length and position • Impedance and ◦ Admittance of the rod at time t = 0. • Summary If V = a + jb = r∠θ = rejθ, then ◦ v(t) = a cos ωt b sin ωt = r cos(ωt + θ) = V ejωt − ℜ 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: Use complex impedances: jωL and 1 ◦ jωC Mnemonic: CIVIL tells you whether I leads V or vice versa ◦ (“leads” means “reaches its peak before”).
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 11 / 11 Summary
10: Sine waves and phasors Sine waves are the only bounded signals whose shape is unchanged by • Sine Waves • • Rotating Rod differentiation. • Phasors • PhasorExamples + Think of a sine wave as the projection of a rotating rod onto the • Phasor arithmetic • • Complex Impedances horizontal (or real) axis. • PhasorAnalysis + • CIVIL A phasor is a complex number representing the length and position • Impedance and ◦ Admittance of the rod at time t = 0. • Summary If V = a + jb = r∠θ = rejθ, then ◦ v(t) = a cos ωt b sin ωt = r cos(ωt + θ) = V ejωt − ℜ 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: Use complex impedances: jωL and 1 ◦ 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 (2017-10213) Phasors: 10 – 11 / 11 Summary
10: Sine waves and phasors Sine waves are the only bounded signals whose shape is unchanged by • Sine Waves • • Rotating Rod differentiation. • Phasors • PhasorExamples + Think of a sine wave as the projection of a rotating rod onto the • Phasor arithmetic • • Complex Impedances horizontal (or real) axis. • PhasorAnalysis + • CIVIL A phasor is a complex number representing the length and position • Impedance and ◦ Admittance of the rod at time t = 0. • Summary If V = a + jb = r∠θ = rejθ, then ◦ v(t) = a cos ωt b sin ωt = r cos(ωt + θ) = V ejωt − ℜ 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: Use complex impedances: jωL and 1 ◦ 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 (2017-10213) Phasors: 10 – 11 / 11 Summary
10: Sine waves and phasors Sine waves are the only bounded signals whose shape is unchanged by • Sine Waves • • Rotating Rod differentiation. • Phasors • PhasorExamples + Think of a sine wave as the projection of a rotating rod onto the • Phasor arithmetic • • Complex Impedances horizontal (or real) axis. • PhasorAnalysis + • CIVIL A phasor is a complex number representing the length and position • Impedance and ◦ Admittance of the rod at time t = 0. • Summary If V = a + jb = r∠θ = rejθ, then ◦ v(t) = a cos ωt b sin ωt = r cos(ωt + θ) = V ejωt − ℜ 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: Use complex impedances: jωL and 1 ◦ 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 (2017-10213) Phasors: 10 – 11 / 11 Summary
10: Sine waves and phasors Sine waves are the only bounded signals whose shape is unchanged by • Sine Waves • • Rotating Rod differentiation. • Phasors • PhasorExamples + Think of a sine wave as the projection of a rotating rod onto the • Phasor arithmetic • • Complex Impedances horizontal (or real) axis. • PhasorAnalysis + • CIVIL A phasor is a complex number representing the length and position • Impedance and ◦ Admittance of the rod at time t = 0. • Summary If V = a + jb = r∠θ = rejθ, then ◦ v(t) = a cos ωt b sin ωt = r cos(ωt + θ) = V ejωt − ℜ 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: Use complex impedances: jωL and 1 ◦ 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 Ch 10 or Irwin Ch 8
E1.1 Analysis of Circuits (2017-10213) Phasors: 10 – 11 / 11