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ECE/ENGRD 2100
Introduction to Circuits for ECE
Lecture 38
Mutual Inductance and Transformers
ECE/ENGRD 2100 2
Announcements
• Recommended Reading:– Textbook Chapter 6.4-6.5 and Chapter 9.10-9.11
• Upcoming due dates:– Homework 5 due by 11:59 pm on Monday April 29, 2019– Prelab 6 due by 11:59 pm on Monday April 29, 2019– Lab report 6 due by 11:59 pm on Friday May 3, 2019– Homework 6 due by 11:59 pm on Tuesday May 7, 2019
ECE/ENGRD 2100 3
Inductance (Self-Inductance)
𝑖"
𝑣"L 𝑣" = L𝑑𝑖"𝑑𝑡
L ≡𝑑𝜆"𝑑𝑖"
=𝜇𝐴𝑁-
𝑙 = 𝒫𝑁-
𝑖"
𝑣"
𝑣" =𝑑𝜆"𝑑𝑡Faraday’sLaw:
𝜆" = 𝑁Φ = 𝑁 𝐵𝐴 = 𝑁 𝜇𝐻 𝐴 = 𝜇𝑁𝑁𝑖"𝑙 𝐴
𝜆" =𝜇𝐴𝑁-𝑖"
𝑙
𝜆" = L𝑖"
𝑁
𝐴𝜇
𝑙
𝜇3 = 4𝜋×109:H/m
ECE/ENGRD 2100 4
Mutual Inductance
𝑖?
𝑣? 𝑣-
𝑖-
L? L-
M
• Use right-hand-rule to determine direction of flux
𝑁-
𝑖?
𝑣?
𝑖-
𝑣-
𝐴𝜇
𝑙
𝑁?
ECE/ENGRD 2100 5
Dot Convention
𝑖?
𝑣? 𝑣-
𝑖-
L? L-
M
• When the reference direction for a current enters the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is positive at its dotted terminal
• Currents into the dot generate flux inside the coupled coils in the same direction
𝑣? = L?𝑑𝑖?𝑑𝑡 +M
𝑑𝑖-𝑑𝑡
𝑖?
𝑣? 𝑣-
𝑖-
L? L-
M
𝑣- = M𝑑𝑖?𝑑𝑡 + L-
𝑑𝑖-𝑑𝑡
𝑁-
𝑖?
𝑣?
𝑖-
𝑣-
𝐴𝜇
𝑙
𝑁?
ECE/ENGRD 2100 6
Relationship Between Mutual and Self Inductances
L? =𝑑𝜆?𝑑𝑖?
= 𝒫?𝑁?- = 𝒫?? + 𝒫?- 𝑁?-
L- =𝑑𝜆-𝑑𝑖-
= 𝒫-𝑁-- = 𝒫-- +𝒫-? 𝑁--
M =𝑑𝜆-𝑑𝑖?
=𝑑𝜆?𝑑𝑖-
= 𝒫-?𝑁-𝑁? = 𝒫?-𝑁?𝑁-
L?L- = 𝒫?𝑁?-𝒫-𝑁-- = 𝒫?? + 𝒫?- 𝒫-- +𝒫-? 𝑁?-𝑁--
𝑁-
𝑖?
𝑣?
𝑖-
𝑣-
𝐴𝜇
𝑙
𝑁?
ECE/ENGRD 2100 7
Ideal Transformer
𝑁-
𝑖?
𝑣?
𝑖-
𝑣-
𝑁?
𝑣? =𝑑𝜆?𝑑𝑡
𝑣- =𝑑𝜆-𝑑𝑡
𝑣-𝑣?=
𝑁?: 𝑁-𝑖?
𝑣? 𝑣-
𝑖-
Ideal
ECE/ENGRD 2100 8
Ideal Transformer – Impedance Transformation
VD- VD?
=𝑁-𝑁?
IF- IF?
= −𝑁?𝑁-
𝑁?: 𝑁-IF?
VD? VD-
IF-
Ideal
𝑁?: 𝑁-
Ideal
Z"
ECE/ENGRD 2100 9
Ideal Transformer – Dependent Source Model
𝑣-𝑣?=𝑁-𝑁?
𝑖-𝑖?= −
𝑁?𝑁-
𝑁?: 𝑁-𝑖?
𝑣? 𝑣-
𝑖-
Ideal
ECE/ENGRD 2100 10
Non-Ideal Transformer Model
𝑣? = L?𝑑𝑖?𝑑𝑡 +M
𝑑𝑖-𝑑𝑡
𝑣- = M𝑑𝑖?𝑑𝑡 + L-
𝑑𝑖-𝑑𝑡
𝑁?: 𝑁-𝑖?
𝑣? 𝑣-
𝑖-
𝐿J?
𝑖J?
𝐿K-𝐿K?
Ideal