ECE 4445 Audio Engineering Formula Sheet

dF

dt→ jωF

ZFdt→ F

jωω = 2πf j =

√−1 S = Area V = Volume

c =

sγP0ρ0

= 345m/ s = 1131 ft s−1 γ = 1.4 P0 = 1.013× 105 Pa ρ0 = 1.18 kgm−3 f2

f1= 21/q

fn = 1000× 2n/q fc =pf fu p (t) = P (t)− P0 SPL = 20 log

µprms

pref

¶dB pref = 2× 10−5 Pa

SPLtotal = 10 log

"Xi

10SPLi/10

#dB S = 2(P−40)/10 Ptotal = 10 log

ÃXi

10Pi/10

!

Stotal =1

16

ÃXi

10Pi/10

!log 2S (f) =

V 20

fV/Hz V 2

rms = V 20

Z f2

f1

df

f= V 2

0 ln

µf2f1

¶V

p (z) = p0e−jkz k =

ω

cuz (z) =

p (z)

ρ0cZs =

p (z)

uz (z)= ρ0c = 407mks rayls

Iave =p2rms

ρ0c= u2rmsρ0c PAR = I × S λ =

c

f=2πc

ω=2π

kξ =

u

jω=

p

jωρ0cp (r) = K

e−jkr

r

ur (r) =1

jωρ0

µ1

r+ jk

¶Ke−jkr

rZs =

p (r)

ur (r)=

ρ0c

1 + c/jωrU =

dV

dt= SD

dx

dt= SDu

p (r) =jωρ0U

1 + jωr1/c

e−jk(r−r1)

4πrp (r) ' jωρ0U

e−jkr

4πrPAR = U2rmsRAR RAR =

ω2ρ04πc

or=

ω2ρ02πc

p (r, θ) = jωρ0U

·2J1 (ka sin θ)

ka sin θ

¸e−jkr

2πrp (r, 0) = jωρ0U

e−jkr

2πrr ≥ 8a

2

λ=2D2

λRA =

ρ0c

S

MA =ρ0 eff

Seff = + f + uf

or= + 2 f

or= + 2 uf f = 0.8488

rS

πuf = 0.6132

rS

π

CA = S2CM =V

ρ0c2

p = p1 − p2 = UZA ZA = RA ZA = jωMA ZA =1

jωCAf0 =

1

2π√MACA

u = jωx f = jωMMu f = RM (u1 − u2) f =u1 − u2jωCM

e = B u f = B i f = τq − x

CM

e = −τx+ q

CEe =

i

CE0s+

uE

x0sf = − qQ

x0CE0− x

CM= − iE

x0s− u

CMsHprox (s) = 1 +

c

sr

f = S (pF − pB) = Sp U = Su MAD =MMD

S2DRAS =

RMS

S2DCAS = S2DCMS RAE =

(B )2

S2DREMA1 =

8ρ03π2a

MAS =MAD + 2MA1 =MMD

S2D+ 2

8ρ03π2a

CAE =S2DLE (ω)

(B )2R0AE =

(B )2

S2DR0E (ω)

MMS = S2DMAS UD =SDegB

RAE

RAT

RATCASs

MASCASs2 +RATCASs+ 1=

SDegB

RAE

RAT

(1/QTS) (s/ωS)

(s/ωS)2 + (1/QTS) (s/ωS) + 1

RAT = RAE +RAS =(B )

2

S2DRE+

RMS

S2DωS = 2πfS =

1√MASCAS

=1√

MMSCMS

VAS = ρ0c2S2DCMS = ρ0c

2CAS

QTS =QMSQES

QMS +QESQMS =

1

RAS

rMAS

CAS=

1

RMS

rMMS

CMSQES =

1

RAE

rMAS

CAS=

RE

(B )2

rMMS

CMS

p =ρ02π

sUD =ρ02π

B egSDREMAS

G (s)Tu1 (s) G (s) =(s/ωS)

2

(s/ωS)2+ (1/QTS) (s/ωS) + 1

Tu1 (s) =1

1 + s/ωu1

1

ωu1 = 2πfu1 =MAS

MADRAECAE=

REMAS

MADLE=

MMSRE

MMDLEQ0ES =

µ1 +

Rg

RE

¶QES

f 0u1 =ω0u12π

=(RE +Rg)MAS

2πLEMAD=

µ1 +

Rg

RE

¶fu1 f = fS,C

hX +

pX2 + 1

i1/2X =

Ã1

2Q2TS,TC− 1!

ZV C (s) = RE + Ze (jω) +RES(1/QMS) (s/ωS)

(s/ωS)2 + (1/QMS) (s/ωS) + 1

RES = REQMS

QESZe (jω) = (jω)

nLe n =

1

90arctan

·Im (Ze)

Re (Ze)

¸Le =

|Ze|ωn

p1Vsens =ρ02π

B SDREMMS

=

√2πρ0c

f3/2S

µVAS

REQES

¶1/2SPL1Vsens = 20 log

µp1Vsenspref

¶SPL1Wsens = 20 log

µp1Vsens

√RE

pref

¶η0 =

PARPE

=4π2

c3f3SVASQES

xD = eg

µVAS

ρ0c2S2DREωSQES

¶1/21

(s/ωS)2 + (1/QTS) (s/ωS) + 1

CAB =VABρ0c

2MAB =

Bρ

πaMAC =MAD +MAB +MA1 RAC = RAS +RAB CAT =

CAS

1 + α=

αCAB

1 + α

α =CAS

CAB=

VASVAB

VAT =VAS1 + α

=αVAB1 + α

QTCMAC=MAS=

1

RATC

rMAS

CAT

RATC = RAE +RAC ωC = 2πfCMAC=MAS=

1√MASCAT

=√1 + αωS QTC =

QMCQEC

QMC +QEC

QMCMAC=MAS=

1

RAC

rMAS

CAT=

RAS

RAS +RAB

√1 + αQMS QEC

MAC=MAS=1

RAE

rMAS

CAT=√1 + αQES

p =ρ02π

B egSDREMAC

GC (s) GC (s) =(s/ωC)

2

(s/ωC)2+ (1/QTC) (s/ωC) + 1

Tu1 (s) =1

1 + s/ωu1ωu1 =

REMAC

LEMAD

ωB = 2πfB =1√

MAPCAB

p =ρ02π

B egSDREMAS

GV (s) GV (s) =(s/ω0)

4

(s/ω0)4 + a3 (s/ω0)

3 + a2 (s/ω0)2 + a1 (s/ω0) + 1

|GV (j2πf)|2 = (f/f0)8

(f/f0)8 +A3 (f/f0)

6 +A2 (f/f0)4 +A1 (f/f0)

2 + 1LP =

µc

2πfB

¶2SPVAB

− 1.463r

SPπ

h =fBfS

q =f

fSLw =

REw

2πfwCm =

1

2πfm1REmLm =

REm

2πfm2Ct =

1

2πftREt

Lw =REw

2πfwQwCw =

Qw

2πfwREwLm1 =

REm

2πfm1Qm1Cm1 =

Qm1

2πfm1REm

C1 =1

3πfcREC2 = 3C1 L =

3RE

8πfcL2 =

RE

4πfcL1 = 3L2 C =

2

3πfcRE

R2 =RinRE

(RE/kpad)−RinR1 = Rin −R2kRE

R1 = RE C1 =Le

(2π)(1−n)

R2E

hfn1 f

(2+n)2

i (1−n)2(1+n)

C2 =Le

(2π)(1−n)

R2E

hf(2+n)1 fn2

i (1−n)2(1+n)

− C1

R2 =1

2πf

1(1+n)1 f

n(1+n)2 C2

R3 = RE

µ1 +

QEC

QMC

¶L1 =

REQEC

2πfCC3 =

1

2πfCREQEC

PL(ave) =V 2P

2RL=

V 2o(rms)

RLPL(ave) = PL(peak) =

V 2P

RL

vOvI=

A

1 + bA' 1

b

vO =A1A21 + bA

vI +A2

1 + bAvN Rout =

Ro

1 + bADF =

RL

Routϕm = 180

◦ + ϕ (ωx) ωx = bAω1

A0 =1

b

ω2ω1

ω01 =ω2bA

ω02 = bAω1

2