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