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2006 Keith W. Whites

Lecture 15: Transformer ShuntInductance. Tuned Transformers.

In the last lecture, we derived the transformer equations p

p ss

N V V

N = (6.21)

and

m t

p s p s

p p

I I

V N I I

j L N = + (6.22)

where I m = magnetization current and I t = transformer current.An equivalent electrical circuit for such a nonideal transformercan be constructed from these two equations (Fig. 6.4a):

Lp

Np : N s

T = Idealtransformer

VsVp

Ip

Im

It

+

-

+

-

Is

In particular, we have a shunt inductor that appears at the primary terminals of an ideal transformer .

With the shunt inductance in the model, the high-pass nature ofa physical transformer is properly accounted for, since at DC the primary terminals of T will be shorted.

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Without the shunt L, the ideal transformer appears to transformvoltages and currents equally well for all frequency, whichcannot be true (by Faradays Law).

Secondary Inductance

Equivalently, the shunt inductance can also have beenincorporated from the secondary.

To do this, we begin again with Fig. 6.2:

I p+

-V p N p

m

Ns

Is+

-Vs

mAl

and (6.12): m p l p s l s N A I N A I = (1)

Solving for I s p ms ps s l

N I I

N N A = (2)

Now, from (6.10) s s mV N j = or sms

V

N j

= (3)

Substituting (3) into (2) leaves

2

s

p ss p

s s l

L

N V I I

N j N A = (4)

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or

t m

p ss p

s s

I I

N V I I

N j L = (5)

The equivalent electrical circuit for (6.21) and this lastexpression is (Fig. 6.4b):

Ls

Np : N s

Ip

Im

It

Vs

+

-

Vp

+

-

Is

To check the directions for the current shown in this figure, wecan apply KCL at the secondary:

t m s I I I = + or s t m I I I = (6)

The minus sign here agrees with (5).

To summarize this work so far, whether the magnetizationcurrent effect is included on the primary side or the secondaryside of the transformer is immaterial: they are equivalent.

Actually, we can develop this latter secondary inductanceequivalent circuit simply from the impedance transformation property of the ideal transformer!

Begin with

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Lp

Np : N s

Ideal T

VsVp

Ip

+

-

+

-

Is

Weve seen previously that for an ideal transformer

2

ss p

p

N Z Z

N

=

(7), (6.19)

Here p p Z j L = and 2 p l p L A N = (8), (6.23)

so that ( )2

2 22

s

ss l p l s

p L

N Z j A N j A N

N = =

or s s Z j L = which we can model simply as an ideal transformer with a shunt Ls as shown on the previous page (Fig. 6.4b).

Tuned Transformers

Achieving impedance match between the various subsystems ina multistage communications circuit is very important.Otherwise, precious signal is needlessly wasted.

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Transformers specifically ideal transformers can be used asmatching networks since, as weve already seen,

2 p

p ss

N Z Z N

= (6.19)

We can choose N p/ N s to change (or transform) Z p to a desiredvalue for matching.

Note that (6.19) is valid only for ideal transformers. One way tonegate the effects of the magnetization current I m in a practicaltransformer (so that the ideal T equations apply) is to use atuning capacitor :

L p

N p : N s

Ideal T

V s

I p

+

-

I s

V p

+

-

C

We can adjust C to resonate out the effects of L p at the desiredfrequency of operation. That is, suppose the transformer isdesigned to operate at f = f 0. For an LC resonance at

( )0 1/ 2 f LC = , then adjust C such that2 2

0

14

C Lf

= F (9)

Consequently, now at this operating frequency f 0

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|| p p

p

c L

c Lc L

Z Z Z Z

Z Z = =

+

and the equivalent circuit for this tuned transformer circuit becomes

L p

N p : N s

Ideal T

V s

I p

+

-

I s

V p

+

-

C

no effect at resonance

which is simply an ideal transformer. Very cool!

This resonant method is only a narrow-band solution, but it can be extremely useful. Capacitive transformer tuning effectively

makes a band-pass filter from a high-pass filter.

Examples

The two tuned transformers in the NorCal 40A are T2 (RF

Filter) and T3 (matching between RF Mixer and IF Filter). Letsconsider both of these quickly once again in the light of ourexpanded understanding of transformers.

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1. T3 (between RF Mixer and IF Filter): This transformer wasalso considered in the previous lecture. T3 is used to transformthe output impedance from the RF Mixer to match the inputimpedance of the IF Filter (200 ).

From the data sheet for the SA602AN IC, the output impedanceis 2 1500 3000 = . Using (7):

2 26 3000 204.223

ss p

p

N Z Z

N

= = =

which is very close to the desired 200 for the IF Filter!

2. T2 (RF Filter): Consider once again the second orderButterworth bandpass filter example we discussed earlier inLecture 12:

L1

6.0

nF

C 1

50

50

V s86.2

nH

+

-

L1 and C1 are components that are soldered onto your PCB.Where do the 86.2 nH and 6.0 nF components come from? As

we mentioned earlier in Lecture 12, they both come from T2!

To see this explicitly for T2, 2 9 2 266 10 H/turn 1 p l p L A N = = =

66 nH, which is close to the 86.2 nH shown above that is neededfor a second-order bandpass filter.

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What about the C ? That comes from C2 and the impedancetransforming properties of T2:

22 p s

p s p ss p

N N Z Z C C N N

= =

With N s = 20, N p = 1 and C s = 15 pF, then220 15 pF 6 nF

1 pC

= =

which is exactly the value needed for the second order

Butterworth bandpass filter! Very cool.