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Lecture 5. Impedance spectroscopy

MIT Student

Last time we talked about the non-linear response to DC voltage, and today we will

discuss about the AC linear response at frequency . A simplified model of voltage

response of galvanic cell can be written as

V(I,Q)=V0(Q)-(I,Q)

If we consider small perturbations around a reference state (constant current), voltage,

current and charge can be expressed as

V=V-Vref

I=I-Iref

Q=Q-Qref

Since we are assuming small changes (perturbation), we can use taylor expansion to

express

V

voltage,

!

!!"#

where

+ R!"#I (1)

!

!!"#

=

!!

!!!!"#,!!"#

=

!!

!!

!

!!"#,!!"#

!!

!!!!"#,!!"#

, C!"# = differencialcapacitance

R!"# =!!

!! !!"#,!!"#

=

!!

!! !!"#,!!"#

,R!"# = differencialresistance

I =Qt

Consider alternating-current (AC) sinusoidal perturbations at frequency and introduce

complex amplitudes for the perturbations (to induce phase lag):

V=Re(Veit)= Vcos(t) ; chose t to fit real number forV, V:real number)

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iI= Re(Ie t)=Icos(t+)

i tQ= Re(Qe ) , where I =iQ since I =!!

!!

************************************************************************

************

Note: ! = ! + !" = !!!" !!" = !"#$ + !"!#$ ,! =

!! + !!=z, ! = arg(!)

************************************************************************

***********

From equation (1), we have our series RC circuit and for the same , time dependence

disappears

! !

:

C!"#

+ R!"#i! =!

iC!

To characterize AC linear response, we use following variables,

+ R!!

Impedance, ! =!

!=

!

!e!!!, (= angle of phase lag of current behind voltage) this

expression is complex generalization of resistance for linear AC current.

Admittance,! = !!=

!

!this expression is the generalization of conductance.

Complex capacitance, ! =!

!

To express the total impedance of circuit, it is importance to separate the circuit into

parallel part and series part. To start with simple examples, lets consider two simple

extremes of circuit, parallel and series.

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(1)Series RC circuit

! = R!"#+

!

!!!!"# !

= 1+

!

!!

! =!!!

!!!!!

!"#

!"#!!"#

! =!!

!!!!

! =!

!!!!!

!"#

!"#!!"#

! =!

!!!!

R

(dimensionless form : = !"#C!"#,! =!

!!"#

, ! = YR!"#, ! = !!!"#

)

There are two ways to represent impedance spectra. First one is Nyquist Plot,

representing Z() in complex plane. Second one is Bode plot and it is beneficial

to scale view.

(a)Nyquist plot (series RC circuit)

! =!

!e!!!

, ! = 1+!

!!= 1 !

!

!

(b)Bode plot (series RC circuit)! = !! = 1+

1

i1

1

i= 1+!!

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! = arg( !) = !"#!!!"!

!"!= !"#

!!1

(2)Parallel RC circuit ! = !

!!!!

! = 1+ i

! = 1+ i

(a)Nyquist plot (parallel RC circuit)

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(b)Bode plot (parallel RC circuit)

************************************************************************

************

Note: Using Kirchoffs circuit rules, it is easy to see that impedances can be combinedjust like resistances.

! = !! + !! dominated by larger impedance

!

!=

!

!!

+!

!!

dominated by smaller impedanceLater, we will see that electrochemical double layers can be modeled as parallelRC circuit elements if Faradaic reactions occur.

************************************************************************************

By combining impedances in series and parallel, we can analyze more complicated

circuits.

(1)

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(2)

C2= geometrical capacitance (dielectric response of dipoles)

in series with the circuit above, representing the response of

mobile ions. In nyquist plot, there are two separated semicircles if there is a separation of

RC time scales R2C2 (fast electrode double layer relaxation)

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In the limit dx0, this yields the Warburg impedance,!

!"! =

!

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10.626 / 10.462 Electrochemical Energy Systems

Spring 2011

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