Handbook of Electrochemical Impedance Spectroscopy A B D Im Y * Im Y * Im Y * Im Y * e Y * e Y * e Y * e Y * IUITS made of ESISTOS, INDUTOS and APAITOS E@SE/EPMI J.-P. Diard, B. e Gorrec,. Montella Hosted by Bio-ogic @ www.bio-logic.info August 3, 2
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ontents ircuits made of, and 5. +/ circuit.......................... 5.. ircuit............................. 5..2 Impedance.......................... 5..3 educed impedance..................... 5.2 ++/ circuit....................... 6.2. Impedance.......................... 7.2.2 educed impedance..................... 7.3 +/ circuit.......................... 7.3. ircuit............................. 7.3.2 Impedance.......................... 7.3.3 educed impedance..................... 8.4 /+/ circuit......................... 9.4. Impedance.......................... 9.4.2 educed impedance......................4.3 Nyquist impedance diagrams.................4.4 Inductive and capacitive Nyquist diagrams.........4.5 ρ = / 2 =........................ 2.4.6 Array of Nyquist impedance diagrams........... 2.5 parallel circuit......................... 2.5. ircuit............................. 2.5.2 Admittance.......................... 3.5.3 educed admittance..................... 4.5.4 Impedance.......................... 4.5.5 educed impedance..................... 4.6 serie circuit........................... 4.6. ircuit............................. 4.6.2 Impedance.......................... 5.6.3 educed impedance..................... 5.6.4 Admittance.......................... 5.6.5 educed admittance..................... 6.7 + parallel circuit...................... 6.7. ircuit............................. 6.7.2 Impedance.......................... 6.8 Transformation formulae / + / 2 r + parallel 7 3
4 ONTENTS 2 Quartz resonator 9 2. BVD equivalent circuit........................ 9 2.2 Admittance.............................. 9 2.3 educed admittance......................... 9 2.3. haracteristic frequencies.................. 2
hapter ircuits made of, and. +/ circuit.. ircuit Fig... Figure.: ircuit +/...2 Impedance e Zω = Zω = i ω + + i ω 2 2 ω 2, Im Zω = ω +..3 educed impedance 2 2 2 ω 2 + Z u = Zu = i T u + + i u, u = ω, T = e Z u = u 2 +, Im Z u = u T u 2 + educed characteristic angular frequency u c = with: T < : e Zu c = /2, Im Zu c = T /2 5 2.
6 HAPTE. IUITS MADE OF, AND u Im Z= = T T, e Zu Im Z= = T. reduced angular frequency at the apex : u a = 2T + 8T + 2 T with: e Zu a = 4 8T + + Im Zu a = T 8T + 3 2T + 8T + 2 8T + u c u a u Im Z.5.5 Figure.2: Nyquist diagram of the reduced impedance for the +/ circuit Fig.., Eq..2 plotted for T =.2 left and T =.,.2,.5,,.5 right. The line thickness increases with increasing T. Dots: reduced characteristic angular frequency u c = right..2 ++/ circuit Fig..3. Figure.3: ircuit ++/.
.3. +/ IUIT 7.2. Impedance e Zω = +.2.2 educed impedance Zω = + i ω + + i ω 2 2 ω 2, Im Zω = ω + 2 2 2 ω 2 + Z u = Zu = ρ + i T u + + i u, u = ω, ρ =, T = e Z u = ρ + u 2 +, Im Z u = u educed characteristic angular frequency u c = with: T < : u Im Z= = T u 2 + e Zu c = ρ + /2, Im Zu c = T /2 T T, e Zu Im Z= = ρ + T. reduced angular frequency at the apex : u a = 2T + 8T + 2 T 2.2 with: e Zu a = ρ + 4 8T + + Im Zu a = T 8T + 3 2T + 8T + 2 8T +.3 +/ circuit.3. ircuit Fig..5..3.2 Impedance Zω = i ω + i ω + i ω e Zω = 2 ω 2 2 ω 2, Im Zω = 2 ω + 2 2 ω 2 + 2 ω
8 HAPTE. IUITS MADE OF, AND u c u a u Im Z Ρ Ρ 2 Ρ Ρ Ρ 2 Ρ Figure.4: Nyquist diagram of the reduced impedance for the ++/ circuit Fig.., Eq..2 plotted ρ =.2 and T =.2 left and T =.,.2,.5,,.5 right. The line thickness increases with increasing T. Dots: reduced characteristic angular frequency u c = right. Figure.5: ircuit +/..3.3 educed impedance Z u = Zu = i T u + i u + i u, u = ω, T = 2 e Z u = u2 u 2 +, Im Z u = u u 2 + T u educed characteristic angular frequency u c = with:.3 T > : u Im Z= = e Zu c = /2, Im Zu c = /2 /T T, e Zu Im Z= = T. reduced angular frequency at the apex : u a = 2 T + T + 8 T + 2 T
.4. /+/ IUIT 9 u Im Z u c ua.5.5 Figure.6: Nyquist diagram of the reduced impedance for the +/ circuit Fig..5, Eq..3 plotted for T = 5 left and T =.66,, 2, 5, right. The line thickness increases with increasing T. Dots: reduced characteristic angular frequency u c = right. with: Im Zu a = T e Zu a = + 4 T T +8 T 2T + T + 8 3 T + T + T + 8 T +8 T +2 T.4 /+/ circuit Fig..7. 2 2 Figure.7: ircuit /+/..4. Impedance Zω = i ω 2 + i ω + 2 2 i ω + = τ i ω + τ i ω + 2 + τ 2 i ω, τ =, τ 2 = 2 2.4
HAPTE. IUITS MADE OF, AND e Zω = τ 2 + ω2 + lim Zω = 2, ω.4.2 educed impedance Z u = Zω 2 e Z u = = ρ 2 τ2 2, Im Zω = ω2 + lim ω Zω = τ ω τ 2 ω2 + i u + i u + + T i u, u = ω τ, ρ =, T = τ 2 2 τ u2 ρ u 2 + + T 2 u 2 +, Im Z u =.4.3 Nyquist impedance diagrams T >, Fig..8. uρ u 2 + T u T 2 u 2 + 2τ 2 ω τ 2 2 ω2 + T Ρ T Ρ Ρ Ρ Figure.8: T >. Nyquist diagrams of the impedance for the /+/ circuit Fig..7, Eq..4 plotted for : top : ρ < ρ =.5, bottom : ρ > ρ =.5. T T = 2 left and increasing values of T right. The line thickness increases with increasing T. T <, Fig..9. T =, Fig... T = Z u = + ρ i u + i u
.4. /+/ IUIT T Ρ T Ρ Ρ Ρ Figure.9: T <. Nyquist diagrams of the impedance for the /+/ circuit Fig..7, Eq..4 plotted for : top : ρ < ρ =.5, bottom : ρ > ρ =.5. T T = 2 left and increasing values of T right. The line thickness increases with increasing T. Ρ Ρ Ρ Figure.: T =. Nyquist diagrams of the impedance for the /+/ circuit. eft: ρ <, middle: ρ =, right: ρ >..4.4 Inductive and capacitive Nyquist diagrams T > and T < ρ < T or T < and T < ρ < T T ρ u Im= =, e Im = = + ρ T T ρ + T Fig...
2 HAPTE. IUITS MADE OF, AND T T u Im u Im Ρ Ρ T Ρ Ρ T Figure.: Inductive and capacitive Nyquist diagrams. eft : T > and T < ρ < T, right : T < and T < ρ < T..4.5 ρ = / 2 = Fig..2. Z u = i u + i u + + T i u T T u u 2 T 2 T Figure.2: ρ = / 2 =. Nyquist diagram: full circle. eft: T >, right T <..4.6 Array of Nyquist impedance diagrams Fig..3..5 parallel circuit.5. ircuit Fig..4.
.5. PAAE IUIT 3 logρ log T Figure.3: Array of Nyquist impedance diagrams for the /+/ circuit. T = ρ = Z u =, u. Figure.4: ircuit //..5.2 Admittance Y ω = i ω + i ω + i ω + + i ω2 = i ω e Y ω =, Im Y ω = ω + ω e Y ω is constant, lim ω Im Y ω =, lim ω Im Y ω = Nyquist diagram of Y ω is a vertical straight line.
4 HAPTE. IUITS MADE OF, AND.5.3 educed admittance i u + i u Y u = Y u = + Λ, u = ω, Λ = e Y u =, Im Y u = Λ u u u c3 Im Y u c2 u c e Y Figure.5: Nyquist diagram of the // circuit reduced admittance. u c = + + 4 Λ 2 /2 Λ, u c2 =, u c3 = + + 4 Λ 2 /2 Λ..5.4 Impedance Zω = Y ω = i ω + i ω + 2 ω 2 e Zω = 2 ω 2 + ω 2 = i ω i ω + + i ω 2 2 ω ω 2 2, Im Zω = 2 ω 2 + 2 + ω 2 2 The Nyquist diagram of Y ω is a vertical straight line the Nyquist diagram of Zω is a full circle..5.5 educed impedance Z u = Zu = i u Λ + i u + Λ i u 2, u = ω, Λ = e Z u = u 2 u 2 + Λ 2 u 2 2, Im Λ u u 2 Z u = u 2 + Λ 2 u 2 2.6 serie circuit.6. ircuit Fig..7.
.6. SEIE IUIT 5.5 u c3 u u c2.5 u c Figure.6: Nyquist diagram of the // circuit reduced impedance. u c = + + 4 Λ 2 /2 Λ, u c2 = u r =, u c3 = + + 4 Λ 2 /2 Λ u c3 u c = Λ. Figure.7: ircuit ++..6.2 Impedance Zω = + i ω + + i ω + i ω2 = i ω i ω e Zω =, Im Zω = ω + ω e Zω is constant, lim ω Im Zω =, lim ω Im Zω = Nyquist diagram of Zω is a vertical straight line..6.3 educed impedance Z u = Zu = + Λ i u + i u e Z u =, Im Z u = Λ, u = ω, Λ = u u.6.4 Admittance Y ω = Zω = i ω + i ω + i ω 2 2 ω 2 ω ω 2 e Y ω = 2 2 ω 2 + + ω 2 2, Im Y ω = + ω 2 2 + 2 + ω 2 The Nyquist diagram of Zω is a vertical straight line the Nyquist diagram of Y ω is a full circle.
6 HAPTE. IUITS MADE OF, AND u c u c2 u c3 Figure.8: Nyquist diagram of the ++ circuit reduced impedance. u c = Λ + 4 + Λ 2 /2, u c2 =, u c3 = Λ + 4 + Λ 2 /2..6.5 educed admittance Y Λ i u u = Y u = + Λ i u + i u 2, u = ω, Λ = e Y u = u 2 Λ 2 + u 4 + u 2 2 + Λ 2, Im Y u Λ u 2 u = + u 4 + u 2 2 + Λ 2.5 u c Im Y u u c2.5 u c3 e Y Figure.9: Nyquist diagram of the ++ circuit reduced admittance. u c = Λ + 4 + Λ 2 /2, u c2 = u r =, u c3 = Λ + 4 + Λ 2 /2, u c3 u c = Λ..7 + parallel circuit.7. ircuit Fig..2..7.2 Impedance Zω = + i ω i ω + + i ω 2.5
.8. TANSFOMATION FOMUAE / + / 2 + PAAE 7 Figure.2: + parallel circuit. Z u = Zu = ρ + i u Λ + i u + Λ i u 2, ρ =, u = ω, Λ =.5 u c3 u u c2.5 u c Ρ Ρ Figure.2: Nyquist reduced impedance diagram of the + parallel circuit. u c = + + 4 Λ 2 /2 Λ, u c2 = u r =, u c3 = + + 4 Λ 2 /2 Λ u c3 u c = Λ..8 Transformation formulae / + / 2 r + parallel r + r 2 /l 2 /c 2 parallel circuit is not-distinguishable from / + / 2 circuit for 2 2 / 2 > Fig..22. 2 p 2 + 2p + Zp =.6 2 p 2 + p 2 2+ + r c 2 l 2 p 2 + l2pr+r2 r r 2 + zp = c 2 l 2 p 2 + l2p.7 r 2 + 2 c 2 = 2 2, l 2 = 2, 2 2 r 2 = 2 2 +.8
8 HAPTE. IUITS MADE OF, AND c 2 2 r l 2 r 2 Figure.22: / + / 2 / + 2 / 2 circuit with = 2 and r + r 2 /l 2 /c 2 parallel circuit.
hapter 2 Quartz resonator 2. BVD equivalent circuit Fig. 2., [3, 5, 6, ]. Figure 2.: BVD Butterworth-van Dyke-equivalent circuit of a quartz resonator. 2.2 Admittance Y ω = + i ω + + i ω = i ω i ω 2 ω 2 e Y ω = 2 2 ω 2 + + ω 2 2, Im Y ω = ω + i ω i ω + + ω 2 + ω 2 2 + 2 + ω 2 + 2.3 educed admittance Y Λ i u u = Y u = + Λ i u + i u 2 + γ i u, u = ω, Λ =, γ = e Y u 2 Λ 2 u = + u 4 + u 2 2 + Λ 2, Im Y u Λ u 2 u = u γ + + u 4 + u 2 2 + Λ 2 9
2 HAPTE 2. QUATZ ESONATO B u Im Y * u m u p u s u r u 2 e Y * Figure 2.2: Definitions for u, u m, u r, u s, u 2 and u p. A B Im Y * Im Y * e Y * e Y * D Im Y * Im Y * e Y * e Y * Figure 2.3: hange of admittance diagram with γ. Λ =, γ = 2 A, 2 B, /3, /2 D. 2.3. haracteristic frequencies Maximum of the real part of Y for: u r = e Y u r =, Im Y u r = γ
2.3. EDUED ADMITTANE 2 Zero-phase reduced angular frequencies: u s and u p defined for γ < /2 + Λ :. γ < 2 + Λ Λ γ 2 + Λ u s = 2 Λ 2 γ Λ + γ 2 4 + Λ 2, 2 γ e Y u s = + γλ + γλ 2 γλ + 2 2 2 γ + Λ γ Λ u p = 2 + Λ 2 γ Λ + γ 2 4 + Λ 2, 2 γ e Y u p = + γλ γλ 2 γλ + 2 2 2. γ = 2 + Λ u s = u p = γλ2 + 2γ + Λ e Y u s = e Y u p = + γλ Fig. 2.3. 2 3. γ > 2 + Λ no zero-phase reduced angular frequency Fig. 2.3D. eal quartz : 2 F, 4 F, Ω, 2 H Λ 4 and γ 2 [4, 2]. 2γ
22 HAPTE 2. QUATZ ESONATO
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