lecture 31: basics of magnetically coupled circuits—part 1 ...ee202/lecture/s18/lec 31...

LECTURE 31: Basics of Magnetically Coupled Circuits—Part 1

1 Question 1: What is a magnetically coupled circuit? Answer: voltages and currents of two nearby inductors affect each other when excited by a sinusoidal current.

Question 2: Right! Can you be more rayspecific Professor. Answer: OK then. A changing current (non-constant derivative) entering inductor L1 induces a voltage across its neighboring inductor L2 according to

a formula v2 = M

di1dt

where M is a measure of their coupling—a measure of

how neighborly they really are. Question 3: Am taking marriage and the family 101 so would these two inductors become co-dependent? Answer: exactly, but they still maintain their own independent identities unlike the co-dependency illness you learn about in M&F 101. Question 4: Sew what? ☺ Answer: changing currents in neighboring and neighborly inductors induce voltages in each other in proportion to their degree of coupling, denoted by

Lecture 31 Sp 15 2 © R. A. DeCarlo

M , with polarities determined by how the coils of wire are wound relative to each other. Question 5: And how are we supposed to know how the coils are wound relative to each other? Answer: Using the dot convention which unfortunately does not take place in LasVegas, but on your HW and test problems?? Question 6: What are some applications of magnetically coupled circuits? (a) Transformers hanging on poles with high voltage lines attached; they step voltages down for household use. (b) Isolation between circuits; they were once used in audio to isolate speakers from tube amplifiers, back in the time of Edison, maybe Columbus. (c) Electric drives or motors that haul coal across the rockies. (d) Tuning in AM radios—broad and narrow band bandpass filter designs. Question 7: Some former 202 students told me I better learn the “dot” convention. Can you tell me how to connect the dots? Ha ha. Well, what is the dot convention? Answer 1: A CURRENT entering the dotted terminal of one coil (say P) induces an open circuit VOLTAGE at the terminals of the neighboring coil (say S) whose positive voltage reference direction is with respect to the dotted terminal (on S).


Now let’s flip the polarity on the secondary voltage measurement which we

take independent of Mr. Coupled Inductors. Ahhhh, the value of the

measurement becomes negative.

Answer 2 (Property Definition): A CURRENT entering the UNdotted terminal of one coil (say P) induces an open circuit VOLTAGE at the terminals of the neighboring coil (say S) whose positive voltage reference direction is with respect to the UNdotted terminal (on S). Exhibit 1 for the Defense:


Exhibit 2 for the Offense:

Question 8: OK, enough of the dot your eyes stuff—what are the equations—you are always telling us we have to know the equations? Answer: Consider the two neighborly inductors, L1 and L2 , in the figure having a coupling inductance M which is hiding behind a white appearance, but it is there. One salient characteristic of the circuit is that the dots on the


secondary (i.e., L2 ) are labeled as A or B meaning that the equations are going to be of the “what if” form: what if dot is at A and what if dot is at B.

SEW:

(i) The unmarried relationships: vk = Lk

dikdt

(assuming passive sign

convention) hold (each inductor maintains its own identity).

(ii) The induced voltage by the neighborly neighbor inductor is:

v1 = ±M

di2dt

and v2 = M

di1dt

.

(III) CONCLUSION: By the Principle of Superposition the two effects

combine (Oh Deere) L1 and L2 in which case:

v1 = L1

di1dt

± Mdi2dt

v2 = ±M

di1dt

+ L2di2dt


Question 9: And in the s-domain Professor Ray????? ASSUMING

ZERO IC’s and the passive sign convention on each inductor, Laplace-

transform the time domain equations to obtain:

V1(s) = L1sI1(s) ± MsI2(s)V2(s) = ±MsI1(s)+ L2sI2(s)

having the matrix form

V1(s)

V2(s)

⎡

⎣⎢⎢

⎤

⎦⎥⎥=

L1s ±Ms

±Ms L2s

⎡

⎣⎢⎢

⎤

⎦⎥⎥

I1(s)

I2(s)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

5. Question 10: Well professor Ray, maybe it’s time for an example??? It’s

not that we don’t like theory, but examples are helpful. You do understand

that don’t you Professor Ray?

Raysponse: Hmmmm. Smart donkeys you are.

Example 1: Consider the circuit below in which there is no internal stored

energy. Let, L1 = 1 H, L2 = 2 H, M = 1 H, and i1(t) = cos(2t)u(t) A in which

case I1(s) = s

s2 + 4. Find v2(t) .


Step 1. Coupled inductor equations. Clearly, ☺

V1(s)

V2(s)

⎡

⎣⎢⎢

⎤

⎦⎥⎥= s s

s 2s⎡

⎣⎢

⎤

⎦⎥

I1(s)

I2(s)

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Step 2. Terminal constraint. The 4 Ω resistor constrains the voltage and

current of the secondary. Specifically, V2(s) = −4I2(s) or equivalently

I2(s) = −0.25V2(s) .

Step 3. Hence,

V2(s) = sI1(s)+ 2sI2(s) = s2

s2 + 4− 0.5sV2(s)

After some algebra,

V2(s) = 2s2

(s+ 2)(s2 + 4)= 1

s+ 2+ s

s2 + 4− 2

s2 + 4

in which case


v2(t) = e−2tu(t)+ cos(2t)u(t)− sin(2t)u(t) V

Exercise. Identify the steady state and transient parts of this zero-state

response.

Example 2: Consider the circuit of example 1 above in which there is no

internal stored energy. Again let L1 = 1 H, L2 = 2 H, and M = 1 H. Find

Zin(s) =

V1(s)I1(s)

.

Step 1. Equation for V1(s) is: V1(s) = sI1(s)+ sI2(s) .

Step 2. Find I2(s) in terms of I1(s) . This time we use the terminal condition

differently. This time:

V2(s) = −4I2(s) = sI1(s)+ 2sI2(s)

implies

I2(s) = − 0.5s

s+ 2I1(s)

Step 3. Combine ingredients and blend.

V1(s) = sI1(s)+ sI2(s) = sI1(s)− 0.5s2

s+ 2I1(s)

imples


Zin(s) =

V1I1

= 0.5s2 + 2ss+ 2

= 0.5 s(s+ 4)s+ 2


Worksheet 2. Find

Zin(s) = for the configuration below.

Step 1. Write the coupled inductor equations in the s-world. Insert

appropriate signs in the equations below:

V1(s) = L1sI1 MsI2

V2(s) = MsI1 L2sI2

Step 2. (i) V2(s) =

(ii) Determine I2(s) in terms of I1(s) using second equation of step 1.

V2(s) = = MsI1 L2sI2 implies I2(s) = I1(s)

Step 3. Find V1(s) in terms of I1(s) by substituting. Then determine Zin(s) .


Worksheet 3. Find Zin(s) =

VinIin

.

1. Vin =V1 ±?? V2

2. I1 = ±?? Iin

3. I2 = ±?? Iin

4. V1 = ±?? L1sI1 ±?? MsI2 = ⎡

⎣⎤⎦sIin

5. V2 = ±?? MsI1 ±?? L2sI2 = ⎡

⎣⎤⎦sIin

6. Vin = ⎡

⎣⎤⎦sIin

7. Zin(s) = ______________

lecture 31: basics of magnetically coupled circuits—part 1 ...ee202/lecture/s18/lec 31...

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