7.6 LCR circuits

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7.6 LCR circuits	147

7.6 LCR circuits

LCR deﬁnitions

Current

I =

(7.139)

Ohm’s law

V = IR

(7.140)

Ohm’s law (ﬁeld

J = σE

(7.141)

form)

Resistivity

ρ =

(7.142)

Capacitance

C =

(7.143)

Current through

I = C

(7.144)

capacitor

Self-inductance

L =

(7.145)

Voltage across

V = −L

(7.146)

inductor

Mutual

L12 =

Φ1

= L21

(7.147)

inductance

L12

= k

(7.148)

Coe cient of

L1L2

coupling

(

Linked magnetic

Φ = Nφ

(7.149)

ﬂux through a coil

Icurrent

Qcharge

Rresistance

Vpotential di erence over R

Icurrent through R

Jcurrent density

Eelectric ﬁeld

σconductivity

ρresistivity

Aarea of face (I is normal to face)

llength

Ccapacitance

Vpotential di erence across C

Icurrent through C

ttime

Φtotal linked ﬂux

Icurrent through inductor

Vpotential di erence over L

Φ1	total ﬂux from loop 2
	linked by loop 1
L12	mutual inductance
I2	current through loop 2

kcoupling coe cient

between L1 and L2 (≤ 1)

Φlinked ﬂux

Nnumber of turns around φ

φﬂux through area of turns


		l
		A
		A

148	Electromagnetism

Resonant LCR circuits

ω0

resonant

Phase

1/LC

(series)

angular

frequency

resonant

ω0 = "1/LC

−

/L2

(parallel)

inductance

frequencya

(7.150)

capacitance

resistance

δω

half-power

Tuningb

δω

(7.151)

bandwidth

ω0

ω0L

quality

factor

Quality

Q = 2π

stored energy

(7.152)

factor

energy lost per cycle

aAt which the impedance is purely real.

bAssuming the capacitor is purely reactive. If L and R are parallel, then 1/Q = ω0L/R.

series

RL C

parallel

Energy in capacitors, inductors, and resistors

stored energy

Energy stored in a

1 Q2

capacitance

capacitor

U =

2 =

QV =

(7.153)

charge

2 C

potential di erence

inductance

Energy stored in an

1 Φ2

U =

ΦI =

(7.154)

linked magnetic ﬂux

inductor

current

Power dissipated in

V 2

power dissipated

a resistora (Joule’s

W = IV = I

(7.155)

R = R

resistance

law)

relaxation time

0 r

Relaxation time

τ =

(7.156)

relative permittivity

conductivity

aThis is d.c., or instantaneous a.c., power.

Electrical impedance

Impedances in series	Z tot =	n		Z n		−1	(7.157)
Impedances in parallel	Z tot = n				Z n−1	−1	(7.158)
Impedance of capacitance			i
Impedance of capacitance	Z C = − ωC						(7.159)
	Z C = − ωC						(7.159)
Impedance of inductance	Z L = iωL						(7.160)

Impedance: Z	Capacitance: C
Inductance: L	Resistance: R = Re[Z ]
Conductance: G = 1/R	Reactance: X = Im[Z ]
Admittance: Y = 1/Z	Susceptance: S = 1/X

7.6 LCR circuits	149

Kirchho ’s laws

			Ii	currents impinging
Current law	Ii = 0	(7.161)		on node
	node			on node
	node

			Vi	potential di erences
Voltage law	Vi = 0	(7.162)		around loop
	loop			around loop
	loop

Transformersa

turns ratio

number of primary turns

number of secondary turns

Z 1

primary voltage

secondary voltage

Z 2

primary current

secondary current

Z out

output impedance

Z in

input impedance

Z 1

source impedance

Z 2

load impedance

Turns ratio

n = N2/N1

(7.163)

Transformation of voltage and current

V2 = nV1

(7.164)

I2 = I1/n

(7.165)

Output impedance (seen by Z 2)

Z out = n2Z 1

(7.166)

Input impedance (seen by Z 1)

Z in = Z 2/n2

(7.167)

aIdeal, with a coupling constant of 1 between loss-free windings.

Star–delta transformation

1		1
‘Star’			‘Delta’

Z1	Z12		Z13

2	Z2	Z3	3	2	3
					Z23

Star	Zi =	Zij Zik	(7.168)
impedances		Zij + Zik + Zjk

Delta	1		1		1
impedances	Zij = ZiZj		+		+		(7.169)
		Zi		Zj		Zk

i,j,k node indices (1,2, or 3)

Zi impedance on node i

Zij impedance connecting nodes i and j

150	Electromagnetism

7.7 Transmission lines and waveguides

Transmission line relations

∂V

∂I

Loss-free

= −L

(7.170)

∂x

∂t

transmission line

∂I

∂V

equations

∂x = −C ∂t

(7.171)

Wave equation for a

1 ∂2V

∂2V

(7.172)

LC ∂x2

∂t2

lossless transmission

1 ∂2I

∂2I

line

LC ∂x2 = ∂t2

(7.173)

Characteristic

impedance of

Zc = C

(7.174)

lossless line

Characteristic

R + iωL

impedance of lossy

Z c = G+ iωC

(7.175)

line

Wave speed along a

lossless line

vp = vg =

√

(7.176)

Input impedance of

Z in = Zc

Z t coskl − iZc sinkl

(7.177)

a terminated lossless

2Zc coskl − iZ t sinkl

line

= Zc

/Z t if l = λ/4

(7.178)

Reﬂection coe cient

Z t − Z c

from a terminated

r =

(7.179)

line

Z t + Z c

Line voltage

vswr =

1 + |r|

(7.180)

standing wave ratio

1 − |r|

Vpotential di erence across line

Icurrent in line

Linductance per unit length

Ccapacitance per unit length

xdistance along line

ttime

Zc characteristic impedance

Rresistance per unit length of conductor

Gconductance per unit length of insulator

ωangular frequency

vp phase speed vg group speed

Zin (complex) input impedance

Zt (complex) terminating

impedance

kwavenumber (= 2π/λ)

ldistance from termination

r(complex) voltage reﬂection coe cient

Transmission line impedancesa

Coaxial line

Zc =

(7.181)

4π2

√

Open wire feeder

Zc =

120

(7.182)

π2

√

Paired strip

Zc =

µ d

377 d

(7.183)

√

Microstrip line

Zc √

377

(7.184)

[(w/h) + 2]

Zc characteristic impedance (Ω)

aradius of inner conductor

bradius of outer conductor

permittivity (= 0 r)

µpermeability (= µ0µr)

rradius of wires

ldistance between wires ( r)

dstrip separation

wstrip width ( d)

hheight above earth plane ( w)

aFor lossless lines.

7.7 Transmission lines and waveguides	151

Waveguidesa

wavenumber in guide

angular frequency

Waveguide

ω2

m2π2

n2π2

guide height

kg2 =

−

(7.185)

guide width

equation

m,n

mode indices with respect to

a and b (integers)

speed of light

νc = c

Guide cuto

νc

cuto frequency

frequency

(7.186)

ωc

2πνc

Phase velocity

vp =

(7.187)

phase velocity

above cuto

(2 1 − (νc/ν

)

frequency

Group velocity

above cuto

vg = c /vp = c 1 − (νc/ν)

(7.188)

group velocity

ZTM

wave impedance for

ZTM = Z0

(νc/ν)2

(7.189)

transverse magnetic modes

Wave

wave impedance for

−

impedancesb

transverse electric modes

(νc/ν)2

(7.190)

ZTE = Z0/

impedance of free space

−

(= (

)

µ0/ 0

Field solutions for TEmn modesc
Bx =	ikgc2 ∂Bz				iωc2 ∂Bz
Bx =	ωc2 ∂x			Ex =		2	∂y
	ikgc2 ∂Bz				ωc		∂y
By =	ikgc2 ∂Bz			Ey = −i		ωc2 ∂B		z	(7.191)
By =	ωc2	∂y		Ey = −i		2		z
		mπx	nπy		ωc		∂x		b	7
		mπx	nπy	Ez =0
Bz =B0 cos			cos b
Bz =B0 cos		a	cos b
Field solutions for TMmn modesc								z	a
								x
Ex =	ikgc2 ∂Ez				−iω ∂Ez			y
Ex =	2	∂x		Bx =	−iω ∂Ez			y
	ωc	∂x			2		∂y
	ikgc2 ∂Ez				ωc		∂y
	ikgc2 ∂Ez				iω ∂Ez				(7.192)
Ey = ωc2				By =	iω ∂Ez				(7.192)
Ey = ωc2		∂y		By =	ω2	∂x
Ez =E0 sin mπx sin nπy					c
Ez =E0 sin mπx sin nπy				Bz =0
		a	b
aEquations are for lossless waveguides with rectangular cross sections and no dielectric.
bThe ratio of the electric ﬁeld to the magnetic ﬁeld strength in the xy plane.
cBoth TE and TM modes propagate in the z direction with a further factor of exp[i(kgz − ωt)] on all components.
B0 and E0 are the amplitudes of the z components of magnetic ﬂux density and electric ﬁeld respectively.

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