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VII. SINUSOIDAL STEADY-STATE ANALYSIS

Any steady-state (SS) voltage or current in a linear time-invariant (LTI) circuit with a sinusoidal input source is

sinusoidal with the same frequency. Only the magnitude and phase (relative to the source) may be different.

Phasors vectors (i.e., complex numbers) that represent sinusoids. Since all V,I in the circuit

are sinusoids with the same frequency, only magnitude & phase are needed to describe any V,I.

sinusoids: v(t) = Vcos(t+) = Re[Vej(t+)] = Re[Vejet] phasor: Vej= Vv(t) = Vsin(t+) = Vcos(t+-/2)phasor: V(-/2)

For convenience, define phasors in terms of cosine (i.e., the real part of a complex exponential)

Eulers Identity: )sin()cos( xjxejx += , ( )jxjx eex +=

21)cos( , ( jxjx

j eex

=21)sin(

Differentiation/integration become algebraic operations w/ phasors (i.e., complex exponentials)

jdtd

jdt 1 Ex: ( ) )()( ++ = tjtjdtd eje

Capacitor Impedance:

CjZC

1= ICE Current (I) LEADS Voltage (EMF) by 90

Inductor Impedance: LjZL = ELI Voltage (EMF) LEADS Current (I) by 90

Complex Impedance/Generalized Ohms Law:I

VZ=

allows for easy nodal analysis (no differential equations); series/parallel resistor laws apply

Maximum Average Power Transfer Theorem

power transferred to load impedance ZL

is maximized when ZL=Zth*

Decibel (dB) unit of measure for ratios of power, voltage, and current levels (often used to

express gain). Power: 1dB=10log10(P1/P2); V,I: 1dB=20log10(V1/V2)=20log10(I1/I2)

Frequency Response systems inputoutput transfer function vs. frequency (given

sinusoidal input). Both magnitude and phase plots are needed (output freq = input freq)

General transfer function can be written as a product of poles and zeroes

( )

+

+

+

+

=

21

21

11

11

)(

pp

zznj

jj

jj

jAeH

Break point frequencyBP poles and zeros are break point freqsat a zero frequency, the magnitude is +3dB (=2) and the phase is +45at a pole frequency, the magnitude is -3dB (=1/2) and the phase is -45

Bode Plot logarithmic plots for frequency response

Aej

j 1/j (1+j/z) 1/(1+j/p)

to draw Bode plot for general transfer function, add individual pole and zero plots

z z10

10z

dB40

dB0

dB20dec

dB20+

zeroes roots of the numeratorpoles roots of the denominator

z z10

10z

2+

04+

p p1010p

dB40

dB0dB20

decdB20

p p1010p

24

0

)( jH

)(jH

)(jH

11

dec

dB20+decdB20

2

002+

A

dB0 dB0

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Filters

Lowpass Filter (LPF) VCin RC circuit / VRin RL circuit / VCand RLC circuit(for current output, switch from series to parallel and switch L and C)

RLjV

V

in

outH

+

==1

1)( RCjV

V

in

outH

+

==1

1)(( ) LCjRCjV

V

in

outH2

1

1)(

++

==

Highpass Filter (HPF) VLin RL circuit / VRin RC circuit / VLin RLC circuit(for current output, switch from series to parallel and switch L and C)

RLj

RLj

V

V

in

outH

+==

1)(

RCj

RCj

V

V

in

outH

+==

1)( ( )

( ) LCjRCjLCj

V

V

in

outH2

2

1)(

++==

Bandpass Filter (BPF) VR, IRin RLC circuit

( ) LCjRCjRCj

V

V

in

outH2

1)(

++==

at low freq, cap. impedanceCjC

Z1= dominates inoutinZ

V

Z

VRCVjIRVCVjI

C

in

tot

in === ,

at high freq, ind. impedance LjZL = dominatesRLj

VoutLj

V

Z

V

Z

Vinin

L

in

tot

in IRVI

=== ,

Resonant FrequencyLC

o1=

At o, oCL

CjC jZjZ

o

===1 , oC

LoL jZjLjZ +=+== inout VV =

(capacitor and inductor impedances are equal in magnitude, opposite in sign)

Characteristic Impedance: CLZo =

BPF Bandwidth = 2= difference between half-power frequencies

Quality Factor Q (1) measure of peakiness or filter selectivity (high Qlow bandwidth)(2) measure of energy stored vs. energy dissipated (high Qlow loss)

21

2 ===

ooQ series RLC:

R

CL

R

ZoQ == parallel RLC:

CL

RZR

o

Q ==

Tradeoffs: Bandwidth/selectivity/speed/energy loss

(e.g., high Qlow (high selectivity)low slow transients e-t)