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Your course, please

A. Science

B. Nanoscience

C. Theoretical Physics

Session ID: PY2P10EM Laptop: responseware.eu

AC Sources

• Most present-day household and industrial power distribution systems operate with alternating current (ac).

• Any appliance that you plug into a wall outlet uses ac.

• An ac source is a device that supplies a sinusoidally varying voltage.

Sinusoids

A sinusoidal is a signal that has the form of the sine or cosine function, which is a time-varying excitation

It is a period function of time 𝑡 with period 𝑇 =2𝜋

𝜔:

𝑣 𝑡 + 𝑛𝑇 = 𝑉𝑚 cos 𝜔 𝑡 + 𝑛𝑇 + 𝜙 = 𝑉𝑚 cos 𝜔𝑡 + 𝜙 + 𝑛𝜔𝑇

= 𝑉𝑚 cos 𝜔𝑡 + 𝜙 + 𝑛𝜔2𝜋

𝜔= 𝑉𝑚cos 𝜔𝑡 + 𝜙 + 𝑛2𝜋

= 𝑉𝑚cos(𝜔𝑡 + 𝜙) = 𝑣(𝑡)

𝑣 𝑡 = 𝑉𝑚 cos 𝜔𝑡 + 𝜙Sinusoidal

the amplitude

angular frequency measured in radians/s and the cyclic frequency 𝑓 in Hz, 𝜔 = 2𝜋𝑓

the phase

Two sinusoids cos(𝑥 + 𝜋/3) leads cos(𝑥) by 𝜋/3

cos(𝑥 − 𝜋/3) lags cos 𝑥 by 𝜋/3

𝑣1 = cos 𝑥 +𝜋

3, 𝜙1 =

𝜋

3

𝑣2 = cos 𝑥, 𝜙2 = 0

𝑣1 leads 𝑣2 by

Δ𝜙 = 𝜙1 − 𝜙2 =𝜋

3

Phase angleA sinusoid can be expressed in either sine or cosine form. When comparing two sinusoids, it is expedient to express both as either sine or cosine with positive amplitudes.

• Δ𝜙 > 0, 𝑣1 leads 𝑣2 by Δ𝜙• Δ𝜙 = 0, 𝑣1 and 𝑣2 are in phase

• Δ𝜙 < 0, 𝑣1 lags 𝑣2 by Δ𝜙

Δ𝜙 = 𝜙1 − 𝜙2

Phase angle

between two

signals:

𝑣1 = 𝑉𝑚1 cos(𝜔𝑡 + 𝜙1)

𝑣2 = 𝑉𝑚2 cos(𝜔𝑡 + 𝜙2)

Positive

Use cosine

−cos 𝑥 = cos(𝑥 ± 𝜋)

sin(𝑥) = cos(𝑥 −𝜋

2)

−𝝅 < 𝝓 < 𝝅

Q1: The phase angle between,

𝑣1 = −10 cos(𝜔𝑡 +𝜋

3) and

𝑣2 = 12 sin 𝜔𝑡 +𝜋

6

is: 𝜋

3−

𝜋

6=

𝜋

6. This means 𝑣1 leads 𝑣2 by

𝜋

6.

A. True

B. False

Answer to Q1:

Use the same form:

𝑣1 = −10 cos 𝜔𝑡 +𝜋

3= 10 cos 𝜔𝑡 +

𝜋

3− 𝜋 = 10 cos 𝜔𝑡 −

2𝜋

3

𝑣2 = 12 sin 𝜔𝑡 +𝜋

6= 12 cos(𝜔𝑡 +

𝜋

6−𝜋

2) = 12 cos(𝜔𝑡 −

𝜋

3)

Compare:

Δ𝜙 = −2𝜋

3−𝜋

3= −

𝜋

3< 0

Therefore, 𝑣1 lags 𝑣2 by 𝜋

3.

Phasor

• A phasor is a complex number that represents the amplitude (e.g. 𝑉𝑚) and phase (𝜙) of a sinusoid, 𝑣 𝑡 = 𝑉𝑚 cos(𝜔𝑡 + 𝜙).

• The real part of a phasor represents the sinusoid signal 𝑣 𝑡 .

• Since we consider a single frequency, the phasor can be written as 𝑽 = 𝑉𝑚𝑒

𝑖𝜙 , i.e. 𝑒𝑖𝜔𝑡 is implicitly present.

Phasor : 𝑉𝑚𝑒𝑖 𝜔𝑡+𝜙 = 𝑉𝑚 cos(𝜔𝑡 + 𝜙) + 𝑖𝑉𝑚 sin(𝜔𝑡 + 𝜙)

Exponential representation Rectangular representation

The sinusoid signal 𝑣(𝑡)

Q2: The phasor of 𝑣1 = −10 cos(𝜔𝑡 +𝜋

3) is

A. −10𝑒𝑖𝜋

3

B. 10𝑒𝑖𝜋

3

C. 10𝑒−𝑖𝜋

3

D. 10𝑒−𝑖2𝜋

3

Q3: The ac sinusoid voltage 𝑣(𝑡) (𝜔 = 4 rads/s) that corresponds to a phasor 𝑽 = 3V is

A. 3 V

B. cos(4𝑡 + 3)V

C. 3 cos(4𝑡)V

D. 3 sin(4𝑡) V

E. 3 cos(4𝑡 + 𝜋)V

Answer to Q2-Q3

Q2: To wirte the phasor for 𝑣1 = −10 cos(𝜔𝑡 +𝜋

3) , we first need to

convert it into the conventional form, i.e. a cosine with a positive amplitude,

𝑣1 = −10 cos 𝜔𝑡 +𝜋

3= 10 cos 𝜔𝑡 +

𝜋

3− 𝜋 = 10 cos(𝜔𝑡 −

2𝜋

3)

Therefore, 𝑽 = 10𝑒−2𝜋

3𝑖 = 10∠ − 120O

Q3: Note 𝑽 = 3 = 3e𝑖⋅0, i.e. the amplitude is 3 V, the phase is 0, and 𝜔 = 4

Therefore𝑣 𝑡 = 3 cos 𝜔𝑡 + 0 = 3 cos(4𝑡) V

Phasor diagram

• To represent sinusoidallyvarying voltages and currents, we define rotating vectors in the Argand plane called phasors.

• Shown is a phasor diagramfor sinusoidal voltage and current with their initial phases 𝜙 and −𝜃.

Time domain and phasor (frequency) domain

Time-domain representation is time dependent and always real, and its phasor (or frequency) domain counterpart is time-independent, generally complex. The phasor domain is for a constant 𝜔, i.e. we consider signals which have the same frequency. Circuit response depends on 𝜔. If we switch from one frequency to another, the circuit responses changes.

𝑉𝑚𝑒𝑖𝜙

Time-independent and complex

Phasor Time

Time-dependent and real

𝑉𝑚 cos(𝜔𝑡 + 𝜙)

Derivative and integral in phasor domain

In phasor representation, the time derivative of a sinusoid becomes just multiplication by the constant 𝑖𝜔; integrating a

phasor corresponds to multiplication by 1

𝑖𝜔.

PhasorTime

𝑣 𝑡 = 𝑉𝑚 cos(𝜔𝑡 + 𝜙)𝑑𝑣 𝑡

𝑑𝑡= −𝑉𝑚𝜔 sin 𝜔𝑡 + 𝜙 = 𝑉𝑚𝜔 cos(𝜔𝑡 + 𝜙 +

𝜋

2)

න𝑣 𝑡 𝑑𝑡 =𝑉𝑚𝜔sin(𝜔𝑡 + 𝜙) =

𝑉𝑚𝜔cos(𝜔𝑡 + 𝜙 −

𝜋

2)

𝑽 = 𝑉𝑚𝑒𝑖𝜙

𝑉𝑚𝜔𝑒𝑖 𝜙+

𝜋2 = (𝑉𝑚𝑒

𝑖𝜙)𝜔𝑒𝑖𝜋2 = 𝑖𝜔𝑽

𝑉𝑚𝜔𝑒𝑖 𝜙−

𝜋2 = 𝑉𝑚𝑒

𝑖𝜙𝑒−𝑖

𝜋2

𝜔=

𝑽

𝑖𝜔

Try this: 𝑑 𝑉𝑚𝑒𝑖 𝜔𝑡+𝜙

𝑑𝑡

Q4: In an ac circuit, the voltage across a 4Ωresistor is 𝑣 𝑡 = 4 cos(10𝑡 + 𝜋/3), the phase of the current through the resistor is

A. 0

B. −𝜋

3

C.𝜋

3

D. None of the above

Q5: For a resistor, its voltage and current are always in phase.

A. True

B. False

Resistor in an ac circuit

• The resistance does not depend on the frequency of the ac source.

• The voltage and current are related by Ohm’s law: 𝑣𝑅(𝑡) = 𝑖𝑅 𝑡 𝑅 and Ohm’s law holds in phasor domain.

• Current and voltage are in phase.

PhasorTime

𝑖𝑅 𝑡 = 𝐼𝑚 cos(𝜔𝑡 + 𝜙)𝑣𝑅 𝑡 = 𝑖𝑅 𝑡 𝑅 = 𝐼𝑚𝑅 cos(𝜔𝑡 + 𝜙)

𝑰𝑹 = 𝐼𝑚𝑒𝑖𝜙

𝑽𝑹 = 𝐼𝑚𝑅𝑒𝑖𝜙 = 𝑅𝑰𝑹

Q6: For an inductor, its voltage and current are always in phase.

A. True

B. False

Inductor in an ac circuit

• The inductance does not depend on the frequency of the ac source.

• The voltage and current are related by :

𝑣𝐿(𝑡) = 𝐿𝑑𝑖𝐿 𝑡

𝑑𝑡.

• Voltage leads current by 𝜋/2

PhasorTime

𝑖𝐿 𝑡 = 𝐼𝑚 cos(𝜔𝑡 + 𝜙)

𝑣𝐿 𝑡 = 𝐿𝑑𝑖𝐿 𝑡

𝑑𝑡= −𝐿𝜔𝐼𝑚 sin(𝜔𝑡 + 𝜙)

𝑰𝑳 = 𝐼𝑚𝑒𝑖𝜙

𝑽𝑳 = (𝑖𝜔𝐿)𝑰𝑳

Q7: For a capacitor, its voltage and current are always in phase.

A. True

B. False

Capacitor in an ac circuit

• The capacitance does not depend on the frequency of the ac source.

• The voltage and current are related by :

𝑖𝐶(𝑡) = 𝐶𝑑𝑣𝐶 𝑡

𝑑𝑡.

• Current leads voltage by 𝜋/2.

PhasorTime

𝑣𝑐 𝑡 = 𝑉𝑚 cos(𝜔𝑡 + 𝜙)

𝑖𝐶 𝑡 = 𝐶𝑑𝑣𝑐 𝑡

𝑑𝑡= −𝐶𝜔𝑉𝑚 sin(𝜔𝑡 + 𝜙)

𝑰𝑪 = 𝐼𝑚𝑒𝑖𝜙

𝑽𝑪 =𝑰𝑪𝑖𝜔𝐶

Impedance and admittance

• Impedance represents the opposition to the flow of sinusoidal current.

• 𝒁 is generally a complex number (Ω).

• Admittance, 𝒀 = 𝟏/𝒁 is the inverse of impedance (S).

𝑽 = 𝒁𝑰

𝑽 = 𝑅𝑰𝑽 = 𝑖𝜔𝐿 𝑰

𝑽 =1

𝑖𝜔𝐶𝑰

Phasor

Impedance

Continued on next page

• 𝑋 < 0, capacitive/leading reactance, e.g. 𝒁 = −1

𝜔𝐶𝑖

• 𝑋 > 0, inductive/lagging reactance, e.g. 𝒁 = 𝜔𝐿 𝑖

Impedance:

Admittance:

𝒁 = 𝑅 + 𝑖𝑋

𝒀 =𝟏

𝒁= 𝐺 + 𝑖𝐵

Resistance Reactance

Conductance Susceptance

Capacitor Inductor

𝜔 → 0 𝒁 = −1

𝜔𝐶𝑖 → ∞

open circuit

𝒁 = 𝜔𝐿 𝑖 → 0short circuit

𝜔 → ∞ 𝒁 = −1

𝜔𝐶𝑖 → 0

short circuit

𝒁 = 𝜔𝐿 𝑖 → ∞open circuit

Circuit response depends on the frequency