Handbook of Electrochemical Impedance Spectroscopy Im Z u c T u c T Re Z CIRCUITS made of RESISTORS and INDUCTORS ER@SE/LEPMI J.-P. Diard, B. Le Gorrec, C. Montella Hosted by Bio-Logic @ www.bio-logic.info September 6, 7
Contents Circuits containing one L 5. Circuit (R+L)............................. 5.. Impedance.......................... 5.. Reduced impedance..................... 5. Circuit (R/L)............................. 6.. Impedance.......................... 6.. Reduced impedance..................... 6.3 Circuit (R +( /L ))........................ 7.3. Impedance.......................... 7.3. Reduced impedance..................... 7.4 Circuit (( +L )/R )........................ 8.4. Impedance.......................... 8.4. Reduced impedance..................... 8.5 Transformation formulae (R+(R/L)) ((R+L)/R)..................... 8.5. Transformation formulae (R+(R/L)) ((R+L)/R)... 8.5. Transformation formulae ((R+L)/R) (R+(R/L))... 9.6 Circuits containing L vs. circuits containing C........... 9.6. Transformation formulae circuit circuit and circuit 3 circuit 4.................. 9.6. Transformation formulae circuit 3 circuit....... 9.6.3 Transformation formulae circuit circuit 3....... 9.6.4 Transformation formulae circuit 4 circuit....... 9.6.5 Transformation formulae circuit circuit 4........6.6 Transformation formulae circuit 3 circuit........6.7 Transformation formulae circuit circuit 3........6.8 Transformation formulae circuit 4 circuit........6.9 Transformation formulae circuit circuit 4........7 Modified inductance......................... Circuits made of one R and two Ls. Circuit (( /L )+L ).......................... Impedance............................ Time constants..........................3 Reduced impedance...................... Circuit (( +L )/L ).......................... Impedance.......................... 3 3
4 CONTENTS.. Time constants........................ 3..3 Reduced impedance..................... 3.3 Transformation formulae ((R/L)+L) ((R+L)/L)........ 3.3. Transformation formulae ((R/L)+L) ((R+L)/L).... 3.3. Transformation formulae ((R+L)/L) ((R/L)+L).... 3
Chapter Circuits containing one L. Circuit (R+L) R L Figure.: Circuit (R+L)... Impedance Z(ω) = R + i L ω, Re Z(ω) = R, Im Z(ω) = L ω.. Reduced impedance Z (u) = Z/R = + i u, u = ω τ, τ = L/R Re Z (u) =, Im Z (u) = u - Im Z * - u c = Im Y * u c = Re Z * Re Y * Figure.: Nyquist diagrams of reduced impedance and admittance (Y = R Y ) for the (R+L) circuit. 5
6 CHAPTE. CIRCUITS CONTAINING ONE L. Circuit (R/L) L R Figure.3: Circuit (R/L)... Impedance Z(ω) = i ω L + R = i L R ω R + i L ω Re Z(ω) = L R ω R + L, Im Z(ω) = L ω.. Reduced impedance Z (u) = Z R = Re Z (u) = R ω R + L ω i u + i u, u = ω τ, τ = L/R u + u, Im Z (u) = u + u - Im Z * -.5 u c = Im Y * - u c = Re Z * Re Y * Figure.4: Nyquist diagrams of reduced impedance and admittance (Y = R Y ) for the (R/L) circuit.
.3. CIRCUIT (R +( /L )) 7.3 Circuit (R +( /L )) L R Figure.5: Circuit (R +(/L ))..3. Impedance Z(ω) = R + + = i ω L ( R + i ω L ) ( + R ) R + i ω L Re Z(ω) = ω L ω L + R + R, Im Z(ω) = ω L ω L +.3. Reduced impedance Z (u) = Z/R = + i T u + i u, u = τ ω, τ = L / (.) T = ( + R )/R = + /R > Re Z (u) = + T u + u, Im Z ( + T) u (u) = + u lim Re u Z (u) =, lim Re u Z (u) = T T Im Z u c T u c Im Y.5 u c u c T T Re Z Re Y Figure.6: Nyquist diagram of reduced impedance and admittance (Y = R Y ) for the (R +(/L )) circuit, plotted for T = 3.
8 CHAPTE. CIRCUITS CONTAINING ONE L.4 Circuit (( +L )/R ) R L Figure.7: Circuit ((+L )/R )..4. Impedance Z(ω) = (i ω L + ) R i ω L + + R = ( R + i ω L ) ( + R ) ( + i ω L ) + R Re Z(ω) = R ( ω L + ( + R ) ) ω L R ω L + ( + R ), Im Z(ω) = ω L + ( + R ).4. Reduced impedance Z (u) = Z ( + R ) R = + i T u + i u, u = τ ω, τ = L /( + R ) T = τ /τ, τ = L /, T = + R / > cf..3., Eq. (.) and Fig..6..5 Transformation formulae (R+(R/L)) ((R+L)/R) L R R L Figure.8: The (R+(R/L)) and ((R+L)/R) circuits are non-distinguishable..5. Transformation formulae (R+(R/L)) ((R+L)/R) R = + R, L = L, = R + R
.6. CIRCUITS CONTAINING L VS. CIRCUITS CONTAINING C 9.5. Transformation formulae ((R+L)/R) (R+(R/L)) L = L R R, =, R = R + R + R + R.6 Circuits containing L vs. circuits containing C L R R L Circuit : R R L Circuit : R L R C 3 R 4 R 3 C 4 3 4 Circuit 3: R R C Circuit 4: R C R Figure.9: (R+(R/L)), ((R+L)/R), (R+(R/C)), ((R+C)/R) circuits are nondistinguishable, taking account of negative values of parameters..6. Transformation formulae circuit circuit and circuit 3 circuit 4 cf..5 and Handbook of EIS : Circuits made of Rs and Cs..6. Transformation formulae circuit 3 circuit L = C 3 3, R = 3 + R 3, = 3.6.3 Transformation formulae circuit circuit 3 C 3 = L, R 3 = + R, 3 =.6.4 Transformation formulae circuit 4 circuit R 4 L = C 4 R 4, R = R 4, = 4 + R 4
CHAPTE. CIRCUITS CONTAINING ONE L.6.5 Transformation formulae circuit circuit 4 R 4 = R, 4 = R ( + R ), C 4 = L R.6.6 Transformation formulae circuit 3 circuit L = C 3 3, R = R 3, = R 3 (3 + R 3 ) 3.6.7 Transformation formulae circuit circuit 3 C 3 = L ( + R ), R 3 = R, 3 = R R + R.6.8 Transformation formulae circuit 4 circuit L = C 4 (4 + R 4 ), R = 4 R 4 4 + R 4, = 4.6.9 Transformation formulae circuit circuit 4 4 =, R 4 = R + R, C 4 = L ( + R ).7 Modified inductance Z = L α (i ω) α, Re Z = L α ω α c α, Im Z = L α ω α s α c α = cos( π α ), s α = sin( π α ) Z = L α ω α, φ Z = π α = cte The L α unit (H cm s α ) depends on α. Α Π Α Π Im Z Im Y Re Z Re Y Figure.: Nyquist diagrams of the impedance and admittance for the modified inductance L α, plotted for α =.8.
Chapter Circuits made of one R and two Ls. Circuit (( /L )+L ) L L Figure.: Circuit ((/L )+L )... Impedance i ω (L + L ) Z(ω) = + + iω L = i ω L.. Time constants ( + i ω L ) L (L + L ) + i ω L Z(ω) = (L + L ) i ω ( + i ω τ ) + i ω τ, τ = L, τ = L L (L + L ) Re Z(ω) = ω (L + L ) (τ τ ), Im Z(ω) = ω (L ( ) + L ) + ω τ τ + ω τ + ω τ lim Re Z(ω) = (L + L ) (τ τ ) = R ω τ
CHAPTER. CIRCUITS MADE OF ONE R AND TWO LS..3 Reduced impedance Z (u) = Z(ω)/ = i u ( + i T u) T + i u (.) u = ω τ, T = τ τ = L L + L < Re Z (u) = u + u, Im Z (u) = u ( ) + T u ( + T) ( + u ), lim Re Z(u) = u Im Z Re Z Figure.: Nyquit diagrams of reduced impedance for the ((/L )+L ) circuit (Eq. (.), Fig.., T = /4, /9, /9, the line thickness inceases with increasing T). Horizontal tangent for T /9 (L /L 8). Dots: reduced caracteristic angular frequency: u c = ; circles: reduced caracteristic angular frequency u c = /T.. Circuit (( +L )/L ) L L Figure.3: Circuit ((+L )/L ).
.3. TRANSFORMATION FORMULAE ((R/L)+L) ((R+L)/L) 3.. Impedance ( Z(ω) = i ω L i ω L + i ω L ) (i ω L + ) = i ω L + i ω L + + i ω (L + L ).. Time constants Z(ω) = i ω L ( + i ω τ ), τ = L + L, τ = L + i ω τ Re Z(ω) = ω L (τ τ ) + ω, Im Z(ω) = ω L ( ) + ω τ τ τ + ω τ lim Re Z(ω) = L (τ τ ) = L ω τ (L + L )..3 Reduced impedance Z (u) = Z(ω)/ = ( T) i u ( + i T u) + iu (.) u = ω τ, T = τ τ = L L + L < Re Z (u) = ( + T) u + u, Im Z ( + T) u (u) = + u (Fig..4) lim Re Z(u) = L u (L + L ) ( + T u ).3 Transformation formulae ((R/L)+L) ((R+L)/L).3. Transformation formulae ((R/L)+L) ((R+L)/L) = (L + L ), L = L + L, L = L + L L L.3. Transformation formulae ((R+L)/L) ((R/L)+L) L = L L, L = L, = L L + L L + L (L + L )
4 CHAPTER. CIRCUITS MADE OF ONE R AND TWO LS Im Z Re Z Figure.4: Nyquit diagrams of the reduced impedance for the ((+L )/L ) circuit (Eq. (.), Fig..3, T = /4, /9, /9, the line thickness inceases with increasing T). Horizontal tangent for T /9 (L /L 8). Dots: reduced caracteristic angular frequency: u c = ; circles: reduced caracteristic angular frequency u c = /T. L L L L Figure.5: The ((R/L)+L) and ((R+L)/L) circuits are non-distinguishable.