THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA

171 Moultrie Street, Charleston, SC 29409

ELEC 425 – Interference Control in Electronics

Lecture 1(c)

Review of Decibels

& Decibel Arithmetic

Dr. Gregory J. Mazzaro

Fall 2017

2

Logarithms

logx

bN b N x where N = positive number (“linear value”)

b = the base of the logarithm

x = the exponent of the logarithm

-- a way to easily write/compare numbers

that are very large and/or very small,

simultaneously

-- an alternative to scientific notation

0.000001 6

0.001 3

1 0

1,000 3

1,000,000 6

N x

using b = 10

(“base-10”)…

Typical electric fields range from 1 mV/m

to 200 V/m 8 orders of magnitude.

The decibel scale compresses this data to a

narrower range of numbers.

3

Gain & Decibels

Gain (A) refers to the ratio of output-to-input

voltage, current, field, power, etc.

out outv p

in in

PA A

P

V

V

Ap,1 Ap,2PoutPin

p p,1 p,2A A A

dB dB dB

p p,1 p,2A A A

Decibels are a convenient format

used to express very high/low gain

(up/down to very high/low

values of voltage, field, power).

dB dBout outv 10 p 10

in in

20log 10logP

A AP

V

V

ApPoutPin

out in pP P A

dBW dBW dB

out in pP P A

4

Decibels: dB

dB

10 v v20 log A A

The decibel scale is a logarithmic

scale that uses base b = 10 . 10log N x

By convention,

gain in decibels is

v 1 2A

dB

v 6 dBA

Gai

n (

V/V

)G

ain

(d

B)

dB

v v

3

2

2

3

10 60

10 40

10 20

2 6

1 0

0.5 6

0.1 20

10 40

10 60

A A

dB

10 p p10 log A A

5

Decibels: dBm

1010 log1 mW

P

The decibel scale is a logarithmic

scale that uses base b = 10 . 10log N x

By convention, power in decibels

(referenced to 1 milliwatt) is

dBm

3

3

6

9

12

W

10 60

10 40

1 30

0.1 20

0.01 10

10 0

10 30

10 60

10 90

P P

1 2 mWP

dBm 3 dBmP

Po

wer

(m

W)

–10

–20

–30

Po

wer

(d

Bm

)

6

Decibels: dBmV/m

1020 log1μV m

E

The decibel scale is a logarithmic

scale that uses base b = 10 . 10log N x

By convention, electric field in decibels

(referenced to 1 microvolt per meter) is

1 2μV/mE

dB

μV/m 6 dBμV/mE

Fie

ld (m

V/m

)F

ield

(d

Bm

V/m

)

dB

μV/m μV/m

3

2

2

3

10 60

10 40

10 20

2 6

1 0

0.5 6

0.1 20

10 40

10 60

E E

7

Decibels: Circuits & Gain

outv

in

outi

in

outp

in

VA

V

IA

I

PA

P

2

inin

in

2

outout

L

VP

R

VP

R

dB outv 10

in

dB outi 10

in

dB outp 10

in

20log

20log

10log

VA

V

IA

I

PA

P

where Vin, Vout, Iin, Iout are

assumed to be RMS values.

Iin Iout

8

Decibels: Voltage, Current, Power, Field

where V, I, E are assumed

to be RMS values.

dB

mV 10

dB

μV 10

dB

mA 10

dB

μA 10

20log1 mV

20log1μV

20log1 mA

20log1μA

VV

VV

II

II

dB

mV/m 10

dB

μV/m 10

dB

mW 10

20log1 mV/m

20log1μV/m

10log1 mW

EE

EE

PP

Iin Iout

9

Examples: Decibel Conversions

Express the ratios of the following quantities in decibels:

Convert the following quantities to the specified decibel units:

(a) a power of 20 W

to a power of 1 mW

(b) a current of 2 mARMS

to a current of 0.5 ARMS

(c) 20 mV/m

to dBmV/m

(d) 300 mW

to dBm

THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA

171 Moultrie Street, Charleston, SC 29409

ELEC 425 – Interference Control in Electronics

Lecture 1(d)

Cable Losses &

High-Frequency Signal Sources

Dr. Gregory J. Mazzaro

Fall 2017

12

Transmission-Line Theory

The general solution to this wave equation is

The ratio of voltage-to-current for one of the

waves is the characteristic impedance, ZC…

which is a pair of waves: one travels from source to measurer ;

the other travels from measurer to source .

0

0

C

V r j lZ

I g j c

A complete mathematical model for

the transmission-line circuit yields

(in Section 1.5.1)…

22

20

d VV

dz

j r j l g j c

0 0 0 0,z z z zV z V e V e I z I e I e

0

zV e

0

zV e

13

Circuit Models for Instruments

At a particular frequency (or within a narrow band of frequencies)…

14

Matched System (50 W)

When all impedances are matched, the solution is

0

zV e

which is one wave, traveling from source to measurer .

A system is matched when the

Thevenin impedance of the source,

the characteristic impedance, and the

Thevenin impedance of the measurer

are equal.

Our industry standard is 50 W.

0 0,z zV z V e I z I e For a mismatched load/cable, the signal

source output may be determined using

15

Cable (Power) Loss

When all impedances are matched, the solution is

0 0, ,z zV z V e I z I e j

2

0 2

avg

1 1Re cos

2 2 C

z

Z

C

VP z V z I z e

Z

which carries the time-average power

attenuation with distance

into the cable

z

z = 0 z = L

Cable loss (in dB) is calculated from

2

dB 1010log 8.7L e LL where is the attenuation constant

in Nepers per meter (Np/m).

The system is assumed to be

matched (usually to 50 W).

16

Example: Cable Loss

A 50-W source is attached to a 50-W signal measurer with 300 ft of RG58U coaxial

cable. The source is tuned to a frequency of 100 MHz, and the dial indicates an

output of –15 dBm. At this frequency, the cable loss is 4.5 dB / 100 ft.

Determine the (RMS) voltage at the input to the signal measurer in dBmV.

dB

mW 10

dB

μV 10

10log1 mW

20log1μV

PP

vV

18

Example: Signal Source Output

A 50-W source is attached to a measurer whose input impedance is 25 W.

The dial on the signal generator indicates that it is outputting a level of –20 dBm.

Determine the (RMS) voltage at the input to the signal measurer in dBmV.