Alternating & Direct Currents

P. Piot, PHYS 375 – Spring 2008

Alternating & Direct Currents

– AC versus DC signals– AC characterization– Mathematical tools:

• Complex number• Complex representation of an AC signal

– Resistor in an AC circuit– Capacitors– Reactance and Impedance– RC circuits– High and low-pass filters


Alternating Current (AC) versus Direct Current (DC)

• With AC it is possible to build electric generators, motors and power distribution systems that are far more effcient than DC.

• AC is used predominately across the world of high power


Alternating Current (AC): waveforms

)()( tSTtS =+• AC signal are periodic:

periodπω2

1==

Tf

frequency pulsation

Heinrich Rudolf Hertz (1857-1894)

UNITS:f: in Hertz (Hz)ω: in rad.s-1


• Can an AC waveform be characterized by a few parameters?

• Peak-to-peak (PP)

• Peak

• Average

• Practical Average

• Root-mean-square

Alternating Current (AC): characterization

)max(SP =

∫+

=Tt

t

dttST

S )(1

∫+

=Tt

t

dttST

AVG )(1

2/1

222 )(1

=−≡ ∫

+Tt

t

dttST

SSRMS ∫+

=Tt

t

nn dttST

S )(1where

)min()max( SSPP −=



22)(

21)(sin11)(sin1)(

2

0

222

=⇒

=== ∫∫+

RMS

dT

dttT

RMSTt

t

π

ϑϑω

ω

• For some analytical waveform, there exits relation between the different parameters

• Take a sinusoidal waveform with amplitude 1 then

πϑϑ

π

ϑϑω

ω

π

π

2)sin(212

|)sin(|11|)sin(|1)(

0

2

0

==

==

∫

∫∫+

d

dT

dttT

AVGTt

t



• It matters what waveform is considered

• For instance for the same peak value, a square waveform will result in higher power than a triangular waveform.


• In the following we will consider sinusoidal-type waveform (in principle any waveform can be synthesized as a series of sine wave (Fourier)

• We will write (in real notation)

• It often better to use complex notation:

• And will often do calculation in complex notation and at the end recall that our physical signal is the real part of the complex results

• We can associate a vector in the complex plane to this complex number

Alternating Current (AC): mathematical tools

][)( )(0

φω +ℜ= tieStS

V

Real

Imag

inar

y

φ)cos()( 0 φω += tStS


Resistor in an AC circuit

VTVR

VRIR

Real

Imag

inar

y

RR RIV =

• R is a real number. So in the complex plane, all quantities are along real axis

• Current and Voltage are said to be in phase

• When instantaneous value of current is zero corresponding instantaneous value of voltage is zero

• Note power > 0 at all time resistor always dissipates energy

PR

IRVR


Capacitors: voltage versus current relation

tP

tE

tDJ

∂∂

+∂∂

=∂∂

= ε

CQ

AQLV

AQEQdSE

≡=⇒

=⇒=∫

0

00

.

ε

εε

• Current induced by electric displacement:.

• Assume a simple model of two plate separated by a small distance. Gauss’s law gives:

capacitance

tVCII

CAIdJd

tV

tEJ

∂∂

=⇔===∂∂

⇒∂∂

=1

000 εεε


Capacitors: technical aspects

M. Faraday (1791-1867)

• Unit for Capacitance is Farad (in honor to Faraday)

• Capacitor symbol:

•

• Real world capacitors also introduce a resistance (we will ignore this effect)


Capacitor

• A capacitor either acts as a load or as a source

• A capacitor can therefore store energy.


Capacitor in an AC circuits

VT

VC

ICReal

Imag

inar

y

IC

VC• Capacitors do not behave the same as resistors

• Resistors allow a flow of e- proportional to the voltage drop

• Capacitors oppose change by drawing or supplying current as they charge or discharge.

CC

C CVidt

dVCI ω==

PC


• The general linear relation between V and I is of the form

Z is called impedance.

• For a resistor Z=R is a real number.

• For a capacitor is an imaginary number

• Generally Z will be a complex number (if V and I are written in their complex forms)

• For instance if a circuit has both capacitor(s) and resistor(s) we expect Z to generally be a complex number

• For a capacitor the quantity is called reactance and is in Ohm (Ω)

Reactance and Impedance

IVZ /≡

CiZ

ω−

=

CX C ω

1−=


Z

VT

Example: Impedance of a series RC Circuits

C

RCT

iXRCiRZ

ICiRRII

CiVVV

+=−=⇒

−=+−

=+=

ω

ωω)(

VT

• Let’s compute the total impedance of the RC circuit:

• The impedance can be written as:

• NA: Z=5-26.52i or |Z|=29.99 and Ξ=-79.325 degreeRC

witheC

RZ i

ωω1tan,1

222 −=Ξ+= Ξ

VC

IVR

XC

R


XC

IR

Example: Impedance of a parallel RC Circuits

VT

I

RXR

CiZ

VR

CiIII

C

RC

1111

1

1

+=

+=⇒

+=+=

−

ω

ω

• Let’s compute the total impedance of the RC circuit:

• NA: Z=4.83-0.91i or |Z|=4.91 and Ξ=-10.68 degree

IC I

I

R


General Analysis of an RC series circuits

( )tieVtVV ωω 00 cos ℜ==

V

dtdV

RI

RCdtdI

dtCIRIVVV CR

11=+⇔

+=+= ∫• Let’s write the ODE for the current

• How do we solve?


Solving the differential equation for the RC series circuit

• Previous equation is of the form:

• First find the solution for the homogeneous equation

• Then find a particular solution of the inhomogeneous equation

• The general solution is of the form

• So finally we have



t

CRCR

Vt

CRRVe

CRRVtI RC

t

ωω

ωωω

ω

ω

ω sin1cos111)(22

220

222

20

222

20

+−

++

+−=

−

• Applying previous results to RC series circuits gives:

• Or in real notations:

−

+

++=

+−111)(

222

2

0 RCtti

RCt

e

CR

RCi

eRVtI

ω

ω

ωω



I/(V/

R)

Time (sec)

f=200 Hz

1,100

10,100 1000,100

(R[Ω],C[µF])


RC series circuits as frequency filters: low pass

VCR

iRCV

ZV

CiI

CiV

c

c

22211

ωωωω

+−

=⇒

−=−=

)arctan(;1

1||

11

222

222

ωω

ωω

RCCR

A

CRiRC

VVA C

−=Θ+

=⇒

+−

==

C

RCT

XRCiRZ

ICiRVVV

+=−=⇒

−=+=

ω

ω)( • The voltage across capacitor is

• The gain A is defined as:

0lim;90lim

1||lim;0||lim

/1/1

/1/1

=Θ−=Θ

==

<<>>

<<>>

RCRC

RCRCAA

ωω

ωω

• Note the limits

Θ= ieAA ||


RC series circuits as frequency filters: low passG

ain

Phas

e (r

d)ωRC

ωRC

−π/4

0.707

• Signal with frequencies below1/RC are unaltered,

• Signal with frequency above1/RC are attenuated


RC series circuits as frequency filters: high pass

=Θ

+=⇒

++

==

ωωω

ωωω

RCCRRCA

CRiRCCR

VVA C

1arctan;1

||

1

222

222

222

• The gain A is defined as:

90lim;0lim

||lim;1||lim

/1/1

/1/1

=Θ=Θ

==

<<>>

<<>>

RCRC

RCRCRCAA

ωω

ωωω

• Note the limits

Θ= ieAA ||

V

CiR

RV

ZVRRIV

c

R

ω−

=⇒

==

1

• The voltage across capacitor is


RC series circuits as frequency filters: high passG

ain

Phas

e (r

d)ωRC

ωRC

π/4

0.707

• Signal with frequencies above1/RC are unaltered,

• Signal with frequency below1/RC are attenuated

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Alternating & Direct Currents