L15 - AC Representation _ Response.pptx

1. Basic Electrical Technology
[ELE 1051]
SINGLE PHASE AC CIRCUITS
L15 – AC Representation & Response
Monday, March 11, 2024 Dept. of Electrical & Electronics Engg., MIT - Manipal 1

2. Topics covered…
 Representation of AC
o Mathematical form
o Graphical form
o Phasors
 AC response of
o Pure Resistor
o Pure Inductor
o Pure Capacitor
Monday, March 11, 2024 Dept. of Electrical & Electronics Engg., MIT - Manipal 2

3. Representing AC
 Consider three sinusoidal signals 𝑥 𝑡 , 𝑦 𝑡 & 𝑧(𝑡) with same frequency
Monday, March 11, 2024 Dept. of Electrical & Electronics Engg., MIT - Manipal 3
𝒙(𝒕) = 𝑿𝒎𝒔𝒊𝒏(𝝎𝒕)
𝒚(𝒕) = 𝒀𝒎 𝐬𝐢𝐧 𝝎𝒕 − ∅
𝒛(𝒕) = 𝒁𝒎 𝐬𝐢𝐧 𝝎𝒕 + 𝜽
Graphical Representation
Mathematical Representation
Phasor Representation
Amplitude
→
Time →
𝒚(𝒕)
𝒙(𝒕)
𝜽
∅
𝒛(𝒕)
𝑿𝒓𝒎𝒔
𝒀𝒓𝒎𝒔
𝒁𝒓𝒎𝒔
𝜽
∅
 Representing the relationship between sinusoidal
signals with same frequency in graphical or
mathematical form is tedious
 Phasor representation is often used

4. R circuit response with AC supply
Monday, March 11, 2024 Dept. of Electrical & Electronics Engg., MIT - Manipal 4
𝑣(𝑡) = 𝑉
𝑚 sin 𝜔𝑡
𝑖(𝑡) = 𝐼𝑚sin(𝜔𝑡)
𝑽𝒓𝒎𝒔
𝑰𝒓𝒎𝒔
Amplitude
→
Time →
𝒗(𝒕)
𝐢(𝒕)
f = 50 Hz
𝒗(𝒕)
𝒊(𝒕)
𝑹
Graphical Representation Mathematical Representat
Phasor Representation
𝑖 𝑡 =
𝑣(𝑡)
𝑅
‘Current through the resistor
is in phase with the voltage across it’

5. 𝑃 𝑡
Power Associated - Pure Resistive Circuit
Monday, March 11, 2024 Dept. of Electrical & Electronics Engg., MIT - Manipal 5
0.02 0.04
𝑣 𝑡
0.06
𝑖 𝑡
Amplitude
Time(sec)
Instantaneous power,
𝑝 𝑡 = 𝑣 𝑡 . 𝑖(𝑡) = 𝑉
𝑚𝐼𝑚𝑠𝑖𝑛2𝜔𝑡 𝑨𝒗𝒆𝒓𝒂𝒈𝒆 𝑷𝒐𝒘𝒆𝒓, 𝑷 =
𝟏
𝑻
𝟎
𝑻
𝒑 𝒕 𝒅𝒕
𝑷𝒂𝒗𝒈 =
𝑽𝒎𝑰𝒎
𝟐
= 𝑽𝒓𝒎𝒔𝑰𝒓𝒎𝒔 =
𝑽𝒓𝒎𝒔
𝟐
𝑹
= 𝑰𝒓𝒎𝒔
𝟐 𝑹

6. L circuit response with AC supply
Monday, March 11, 2024 Dept. of Electrical & Electronics Engg., MIT - Manipal 6
𝑣(𝑡) = 𝑉
𝑚 sin 𝜔𝑡
𝑖(𝑡) = 𝐼𝑚sin(𝜔𝑡 − 90°)
𝑽𝒓𝒎𝒔
𝑰𝒓𝒎𝒔
Graphical Representation Mathematical Representat
Phasor Representation
f = 50 Hz
𝒗(𝒕)
𝒊(𝒕)
𝑳
𝒗(𝒕)
𝐢(𝒕)
Amplitude
→
Time →
𝑖 𝑡 =
1
𝐿
𝑣 𝑡 𝑑𝑡
‘Current through the inductor lags the
voltage across it by 90°’
𝑉 = 𝑉∠0° 𝐼 = 𝐼∠ − 90°
𝑽
𝑰
=
𝑽∠𝟎°
𝑰∠ − 𝟗𝟎°
= 𝒋𝑿𝑳 𝑤ℎ𝑒𝑟𝑒
𝑉
𝐼
= 𝑋𝐿
𝑿𝑳 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑰𝒏𝒅𝒖𝒄𝒕𝒊𝒗𝒆 𝑹𝒆𝒂𝒄𝒕𝒂𝒏𝒄𝒆

7. 𝑃 𝑡
Power Associated – Pure Inductive Circuit
Monday, March 11, 2024 Dept. of Electrical & Electronics Engg., MIT - Manipal 7
𝑖 𝑡
Amplitude
Time(sec)
0.04 0.06
𝑣 𝑡
0.02
Instantaneous power,
𝑝 𝑡 = 𝑣 𝑡 . 𝑖(𝑡) 𝑨𝒗𝒆𝒓𝒂𝒈𝒆 𝑷𝒐𝒘𝒆𝒓, 𝑷 =
𝟏
𝑻
𝟎
𝑻
𝒑 𝒕 𝒅𝒕
𝑷𝒂𝒗𝒈 = 𝟎
= 𝑉
𝑚𝐼𝑚 sin 𝜔𝑡 . sin(𝜔𝑡 − 90 °)
= −
𝑉
𝑚𝐼𝑚
2
sin 2𝜔𝑡

8. C circuit response with AC supply
Monday, March 11, 2024 Dept. of Electrical & Electronics Engg., MIT - Manipal 8
𝑣(𝑡) = 𝑉
𝑚 sin 𝜔𝑡
𝑖(𝑡) = 𝐼𝑚sin(𝜔𝑡 + 90°)
𝑽𝒓𝒎𝒔
𝑰𝒓𝒎𝒔
Graphical Representation Mathematical Representat
f = 50 Hz
𝒗(𝒕)
𝒊(𝒕)
𝑪
Phasor Representation
Amplitude
→
Time →
𝒗(𝒕)
𝐢(𝒕)
‘Current through the capacitor
leads the voltage across it by 90°’
𝑖 𝑡 = 𝐶
𝑑𝑣(𝑡)
𝑑𝑡
𝑉 = 𝑉∠0° 𝐼 = 𝐼∠90°
𝑽
𝑰
=
𝑽∠𝟎°
𝑰∠𝟗𝟎°
= −𝒋𝑿𝑪 𝑤ℎ𝑒𝑟𝑒
𝑉
𝐼
= 𝑋𝐶
𝑿𝑪 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑪𝒂𝒑𝒂𝒄𝒊𝒕𝒊𝒗𝒆 𝑹𝒆𝒂𝒄𝒕𝒂𝒏𝒄𝒆

9. 𝑃 𝑡
Monday, March 11, 2024 Dept. of Electrical & Electronics Engg., MIT - Manipal 9
𝑣 𝑡
0.02
Time(sec)
Amplitude
0.04 0.06
Power Associated – Pure capacitive Circuit
𝑖 𝑡
Instantaneous power,
𝑝 𝑡 = 𝑣 𝑡 . 𝑖(𝑡) 𝑨𝒗𝒆𝒓𝒂𝒈𝒆 𝑷𝒐𝒘𝒆𝒓, 𝑷 =
𝟏
𝑻
𝟎
𝑻
𝒑 𝒕 𝒅𝒕
𝑷𝒂𝒗𝒈 = 𝟎
= 𝑉
𝑚𝐼𝑚 sin 𝜔𝑡 . sin(𝜔𝑡 + 90 °)
=
𝑉
𝑚𝐼𝑚
2
sin 2𝜔𝑡

10. Summary
 Sinusoidal alternating signals of same frequency can be
represented graphically by Phasors
 Define: Inductive and capacitive Reactances
Monday, March 11, 2024 Dept. of Electrical & Electronics Engg., MIT - Manipal 10
R L C
Voltage, current
relationship
v(t) in phase with I(t) i(t) lags v(t) by
90°
i(t) leads v(t) by
90°
Power associated
𝐼2𝑅 =
𝑉2
𝑅
(Active Power)
𝐼2
𝑋𝐿
(Reactive Power)
𝐼2
𝑋𝐶
(Reactive Power)