430.213A Introduction to Circuit Theory and Lab.
, Spring 2024, Seoul National University
Lecture 11.
AC Steady-State Power
Wooyeol Choi, PhD
Assistant Professor
Department of Electrical and Computer Engineering
Seoul National University, Seoul, Korea
Copyright Statement: The materials provided by the instructor in this course are for the use of the students enrolled in the
course. Copyrighted course materials may not be further disseminated.
Outline
▪ Reading – Textbook Chapter 11
▪ Goals
• Part 1 – Learn definition and physical meaning of various AC power
• Part 2 – Learn the operation of transformers
▪ Contents
• Power – Instantaneous, average, complex
• Effective voltage and current – root mean square
• Power factor and (correction)
• Superposition (of power)
• Maximum power transfer (of AC)
• Transformers
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 2
From lecture 7
Inductor
▪ Ampere’s circuital law relates
current flow and magnetic flux [Wb,
Weber]
Φ = 𝐿𝑖
▪ Faraday’s induction law relates
magnetic flux and electromotive force
(emf) [V]
𝑑Φ
𝜀=−
𝑑𝑡
Negative sign indicates the direction
for generated emf
▪ Combining them
𝑑Φ 𝑡 𝑑𝑖 𝑡
𝑣 𝑡 = =𝐿
𝑑𝑡 𝑑𝑡
▪ Integration form
1 𝑡
𝑖 𝑡 = න 𝑣(𝜏) 𝑑𝜏
𝐿 −∞
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 3
Mutual Inductance
Magnetic Field
AC current AC voltage
▪ Faraday’s law does not care if the flux is from itself or somewhere else
▪ If it is from itself – self inductance
▪ If it is from another inductor – mutual inductance
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 4
Right Hand Rule and Dot Convention
Current flows from a to b Current flows from b to a
𝑣𝑎 > 𝑣𝑏 𝑣𝑎 < 𝑣𝑏
▪ Mutual inductance sign is dependent on the direction of current flow and
magnetic flux
▪ Right hand rule
▪ Dot indicates the node where the higher voltage is induced
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 5
When 𝑖1 applied,
𝜙1 = 𝑐1 𝑁1 𝑖1
𝑑𝑖1 𝑑𝜙1 𝑑𝑖1
𝑣1 = 𝐿1 = 𝑁1 = 𝑐1 𝑁12
𝑑𝑡 𝑑𝑡 𝑑𝑡
2
𝐿1 = 𝑐1 𝑁1
𝜙2 = 𝜙1
𝑑𝜙2 𝑑𝑖1
𝑣2 = 𝑁2 = 𝑐𝑀 𝑁1 𝑁2
𝑑𝑡 𝑑𝑡
In the same manner, Superposition!
when 𝑖2 applied 𝑑𝑖1 𝑑𝑖2
𝑑𝜙2 𝑑𝑖2 𝑣1 = 𝐿1 +𝑀
𝑣2 = 𝑁2 2
= 𝑐2 𝑁2 𝑑𝑡 𝑑𝑡
𝑑𝑡 𝑑𝑡 𝑑𝑖1 𝑑𝑖2
𝑑𝜙1 𝑑𝑖2 𝑣2 = 𝑀 + 𝐿2
𝑣1 = 𝑁1 = 𝑐𝑀 𝑁1 𝑁2 𝑑𝑡 𝑑𝑡
𝑑𝑡 𝑑𝑡
𝐿1 = 𝑐1 𝑁12
𝐿2 = 𝑐2 𝑁22
𝑀 = 𝑐𝑀 𝑁1 𝑁2
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 6
Dot Convention Again
𝑑𝑖1 𝑑𝑖2 𝐕1 = 𝑗𝜔𝐿1 𝐈1 + 𝑗𝜔𝑀𝐈2
𝑣1 = 𝐿1 +𝑀
𝑑𝑡 𝑑𝑡
𝑑𝑖1 𝑑𝑖2
𝑣2 = 𝑀 + 𝐿2 𝐕2 = 𝑗𝜔𝑀𝐈1 + 𝑗𝜔𝐿2 𝐈2
𝑑𝑡 𝑑𝑡
𝑑𝑖1 𝑑𝑖2 𝐕1 = 𝑗𝜔𝐿1 𝐈1 − 𝑗𝜔𝑀𝐈2
𝑣1 = 𝐿1 −𝑀
𝑑𝑡 𝑑𝑡
𝑑𝑖1 𝑑𝑖2
𝑣2 = −𝑀 + 𝐿2 𝐕2 = −𝑗𝜔𝑀𝐈1 + 𝑗𝜔𝐿2 𝐈2
𝑑𝑡 𝑑𝑡
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 7
How to Model Coupled Inductors
𝐕1 = 𝑗𝜔𝐿1 𝐈1 + 𝑗𝜔𝑀𝐈2
Current Sontrolled
𝐕2 = 𝑗𝜔𝑀𝐈1 + 𝑗𝜔𝐿2 𝐈2 Voltage Source
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 8
Self Inductance and Mutual Inductance
▪ There is no source
= no negative energy
𝑡
▪ 𝑤 = −∞ 𝑣1 𝑖1 + 𝑣2 𝑖2 𝑑𝜏
𝑑𝑖1 𝑑𝑖2
𝑣1 = 𝐿1 ±𝑀
𝑑𝑡 𝑑𝑡
𝑑𝑖1 𝑑𝑖2
𝑣2 = ±𝑀 + 𝐿2
𝑑𝑡 𝑑𝑡
𝑡 𝑡
𝑑𝑖1 𝑑𝑖2 𝑑𝑖1 𝑑𝑖2
𝑤 = න 𝐿1 𝑖1 ± 𝑀 𝑖1 𝑑𝜏 + න ±𝑀𝑖2 + 𝐿 2 𝑖2 𝑑𝜏
𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡
−∞ −∞
1 1
= 𝐿1 𝑖1 + 𝐿2 𝑖22 ± 𝑀𝑖1 𝑖2
2
2 2
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 9
1 1
𝑤= 𝐿1 𝑖12 + 𝐿2 𝑖22 ± 𝑀𝑖1 𝑖2
2 2
Suppose 𝑖1 𝑖2 ≥ 0
1 1
𝐿1 𝑖1 + 𝐿2 𝑖22 + 𝑀𝑖1 𝑖2 ≥ 0
2
2 2
1 1
𝐿1 𝑖12 + 𝐿2 𝑖22 − 𝑀𝑖1 𝑖2 = 0
2 2
1 1
𝐿1 𝑖12 + 𝐿2 𝑖22 − 𝑀𝑖1 𝑖2 + 𝐿1 𝐿2 𝑖1 𝑖2 − 𝐿1 𝐿2 𝑖1 𝑖2
2 2
2
𝐿1 𝐿2
= 𝑖 − 𝑖 + 𝑖1 𝑖2 −𝑀 + 𝐿1 𝐿2
2 1 2 2
For this to be equal to or greater than zero,
𝑀
𝐿1 𝐿2 − 𝑀 ≥ 0 → 𝑘 = ≤1
𝐿1 𝐿2
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 10
Coupling Coefficient
▪ Coupling coefficient indicates the relative amount of coupled magnetic flux
𝐿1 = 𝑐1 𝑁12
𝐿2 = 𝑐2 𝑁22
𝑀 = 𝑐𝑀 𝑁1 𝑁2
𝑀 𝑐𝑀
𝑘= =
𝐿1 𝐿2 𝑐1 𝑐2
▪ Coupling coefficient ranges
0≤𝑘≤1
▪ In an ideal transformer
𝑘=1
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 11
Transformer
Using phasor and impedance
𝑉1 = 𝑗𝜔𝐿1 𝐼1 + 𝑗𝜔𝑀 −𝐼2 (1)
𝑉2 = 𝑗𝜔𝑀𝐼1 + 𝑗𝜔𝐿2 (−𝐼2 ) (2)
𝑉2 = 𝐼2 𝑍2 (3)
Using (2) and (3), 𝐼2 = 𝑍
𝑗𝜔𝑀
𝐼1 If 𝑐1 = 𝑐2
2 +𝑗𝜔𝐿2 𝑁2
𝜔2 𝑀2 𝑗𝜔𝐿1 𝑍2 −𝜔2 𝐿1 𝐿2 +𝜔2 𝑀2 𝑉2 = 𝑉
𝑉1 = 𝑗𝜔𝐿1 + 𝑍 +𝑗𝜔𝐿
2 2
𝐼1 =
2 𝑍 +𝑗𝜔𝐿
2
𝐼1 𝑁1 1
2
𝜔 𝐿2 𝑀 𝑗𝜔𝑀𝑍2 −𝜔 𝐿2 𝑀+𝜔2 𝐿2 𝑀
2
𝑉2 = 𝑗𝜔𝑀 + 𝑍 +𝑗𝜔𝐿 𝐼1 = 𝐼1
2 2 𝑍2 +𝑗𝜔𝐿2
𝑉 𝑉
When 𝑘 = 0, 𝐼1 = 𝑗𝜔𝐿1 , 𝐼2 = 0
1 2
𝑉2 𝑀 𝐿1 𝐿2 𝐿2 𝑐2 𝑁22
When 𝑘 = 1 → 𝑀 = 𝐿1 𝐿2 → = = = =
𝑉1 𝐿1 𝐿1 𝐿1 𝑐1 𝑁12
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 12
KCL/KVL with Coupled Inductors
𝑣𝑆 = 5.94 cos(3𝑡 + 140°) → 𝑉𝑆 = 5.94𝑒 𝑗140° , 𝜔 = 3
▪ KCL and KVL should be valid
▪ Mind the position of dots
−𝑉𝑆 + 𝑉5Ω + 𝑉4𝐻 + 𝑉𝑜 = 0
𝑉5Ω = 5𝐼
𝑉4𝐻 = 𝑗𝜔 4 𝐼4𝐻 + 𝑗𝜔 2 𝐼5𝐻
Vo = 𝑉5𝐻 = 𝑗𝜔 2 𝐼4𝐻 + 𝑗𝜔 5 𝐼5𝐻
𝐼4𝐻 = 𝐼5𝐻 = 𝐼
−5.94𝑒 𝑗140° + 12𝑗 + 6𝑗 + 6𝑗 + 15𝑗 𝐼 = 0
𝐼 = 0.151𝑒 𝑗57°
𝑉𝑜 = 3.17𝑒 𝑗147°
𝑣𝑜 = 3.17 cos(3𝑡 + 147°)
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 13
Ideal Transformer
▪ Transformer with k=1
𝑁2
𝑉2 = 𝑉 = 𝑛𝑉1
𝑁1 1
▪ No energy lost in the transformer
(no loss)
→ power into port 1
= power from port2
𝑉1 𝐼1∗ 𝑉2 −𝐼2 ∗
=
2 2
𝑉1 𝑁1 1
𝐼2 = − 𝐼1 = − 𝐼1 = − 𝐼1
𝑉2 𝑁2 𝑛
▪ Impedance looking from V1
𝑉2
𝑉1 𝑛 1
= = 𝑍
𝐼1 −𝑛𝐼2 𝑛2 2
▪ Impedance transformation!
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 14
Maximum Power Transfer (Impedance Matching)
▪ We learned that both real and
imaginary part of the load should be
transformed to achieve maximum
power transfer
▪ Imaginary part (reactance) can be
easily manipulated by adding pure L
or C, but how about the real part?
→ Transformer
▪ If the transformer is lossless, all the
𝑅𝑆 = 𝑅𝑖𝑛 power delivered to Rin is same as
1 that delivered to RL
𝑅𝑖𝑛 = 2 𝑅𝐿
𝑛
𝑅𝐿
𝑛=
𝑅𝑆
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 15
Summary
▪ Inductors can share magnetic flux = Mutual inductance
▪ Dot convention is used to incorporate the direction of magnetic flux coupling
▪ Coupling coefficient k is used to quantify the amount of flux coupling
(ranges from 0 to 1)
▪ Ideal transformer scales voltage and current according to the turns ration
▪ Voltage and current scaling results in (lossless) impedance transformation
(can be used for impedance matching to achieve maximum power transfer)
Spring 2024 Intro. Circ. Theory and Lab., Lect 11b - transformers 16
