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Worksheet Impedance Scaling

1. (a) Impedance scaling or magnitude scaling by a factor Km

:

Znew

(s) KmZ

old(s) .

(b) Znew

(s) KmZ

old(s) is achieved by impedance scaling the individual circuit

elements by Km

: Rnew

KmR

old,

1

Cnews

Km

Colds

1Cold

Kms

, and Lnews (KmLold)s .

2. Review Test: (a) Cnew _________ Cold.

(b) Lnew

_________Lold

(c) Rnew

_________Rold

3. Consider the series RLC circuit below.

(a) Zin

(s) ___________________

s2 s

s

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2 R. A. DeCarlo

(b) Impedance scale each circuit element by Km

. Then

Zin,new(s) ___________________

s

2

s

s

Zold

(s)

(c) Compute Transfer function without impedance scaling:

H(s)Vout

(s)

Vin (s)

__________________________________

(d) Compute Transfer function with impedance scaling:

Hnew

(s) Vout

(s)

Vin(s)

________________________________

Hold

(s)

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Worksheet on Frequency Scaling

1. (a) Frequency scaling by a factor Kf

: ss

Kf

.

2. ss

Kf

achieved by frequency scaling each of the circuit elements:

(a) IfZC

(s) 1

Csthen frequency scaling by K

fmeans:

ZC,new(s) ZC,old

s

Kf

implies Cnew _________Cold.

(b) IfZL

(s) Ls then frequency scaling by Kf

means:

ZL,new(s) ZL,old sKf

implies Lnew _________Lold.

(c) IfZR

(s) R then frequency scaling by Kf

means:

ZR,new

(s) ZR,old

s

Kf

means R

new

_________Rold

.

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3. Consider the series RLC circuit below.

(a) Zin

(s) ___________________

s2 s

s

(b) Frequency scale each circuit element by Kf

. Then using the frequency scaled

parameter values to obtain:

Zin

(s)

s2 s

s

s

2

s

s

(c) Without any scaling

H(s) Vout

(s)

Vin (s)

__________________________________

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(d) With individual element frequency scaling

Hnew

(s) Vout

(s)

Vin (s)

________________________________

(e) With frequency scaling by ss

Kf

Hnew

(s) Hold

s

Kf

________________________________

SHOW THAT YOUR ANSWERS TO (d) AND (e) ARE THE SAME.

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WorksheetCombined Magnitude and Frequency Scaling

For each question below, write down the "correct" answer.

1. IfZC

(s) 1

Csthen frequency scaling by K

fand magnitude scaling by K

mmeans

that Cnew

_________ Cold

.

2. IfZL

(s) Ls then frequency scaling by Kf

and magnitude scaling by Km

means that

Lnew _________Lold.

3. IfZR

(s) R then frequency scaling by Kf

and magnitude scaling by Km

means that

Rnew

_________Rold

.

True-False: Impedance scaling does not change current ratios: _________________

True-False: Impedance scaling does not change voltage ratios: _________________

True-False: Frequency scaling does not change s-dependent voltage ratios:

_______________

True-False: Impedance scaling does not change transfer functions: _______________

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TEST YOURSELF:

In the left hand column below there is an approximate gain magnitude responseassociated with a transfer function H(s). In the right hand columns there is a new gainmagnitude response whose transfer function is related to H(s) by scalaing. Properly labelthe horizontal axis of the graph on the right.

(a)

(b)