Impedance Modeling – multi circuit elements (Complete)
Title:
Impedance Model – multi circuit elements (Complete)
Keyword:Impedance, |Z|-θ f-characteristics, Impedance analyzer, LCR meter, modeling.
Purpose: This article provides the general |Z|-θ f-characteristics for several useful impedance models. By comparing the gain-phase response
obtained from impedance analyzer, engineer will able to select the corresponding equivalent circuit for modeling purpose.
Detail: There are 4 equivalent circuits which used to model different devices. The 4 equivalent circuits can be expressed as 2-elements or
3-elements models, depend on the engineering requirement. In this article, the 2-element and 3-element equivalent circuit which used to
model the same device are in same chart for comparison. A MATLAB program is written and user can simulate the impedance analyzer
output with different modeling method.
How to use the chart to select equivalent model of device under test (DUT)?
•
•
Using an impedance analyzer, measure the DUT with |Z|-θ over the whole frequency range.
Observe the gain-phase relationship, in general, we have following guideline
o General guideline: Observe the phase at low, resistor gives 0o, inductor gives 90o and capacitor gives -90o.
o Resistor guideline: Resistor normally gives 0o at low frequency, depend on its type, it can be inductive or capacitive at high frequency.
o Inductor guideline: Inductor may have 0o to 90o at low frequency, which depend on the winding resistance. Stray capacitance may
effect at high frequency end and change the phase angle to negative.
o Capacitor guideline: Capacitor gives -90o at low frequency. ESR (equivalent series resistor) reduces its capacitive effect when
frequency increases. Lead inductance can even force the phase angle to 90o at high frequency.
•
Because resistor, inductor and capacitor are combined with all 3 types of elements practically, Rs-Ls, Rp-Lp, Rs-Cs and Rs-Cp models are
ONLY true for a certain frequency region. If only LCR meter is available, although it is possible to use two 2-element models to obtain
3-element parameters, engineer must able to determine the exact circuit model before doing that.
•
Always keep in mind that when measuring a component with 2-element models (Rs-Ls, Rp-Lp, Rs-Cs or Rs-Cp), you are forcing your
component look like that model, that why you can obtain negative inductance or capacitance when frequency is not in operation range.
•
Always keep in mind that when phase angle is 0o, it is not saying that your component is a resistor, it just represents your component look like a
resistor at that frequency. Same rule applies for inductor and capacitor.
Dr. Kelvin K. S. Leung
www.kskelvin.net
P.1
Impedance Modeling – multi circuit elements (Complete)
Model ID
Equivalent Circuit
Impedance Analyzer
Type of DUTs
Simulation
Value
|Z|-θ f-characteristics
|Z| and theta plot
1000
100
‘2_1’
500
0
3
10
Low-value resistor
General Inductor
0
4
10
5
6
10
10
frequency (Hz)
7
10
theta (degree)
|Z|
50
Ra = 1e-3
Ca = 1e-9
La = 1e-6
8
10
100
450
80
400
60
350
40
300
20
250
0
200
-20
150
-40
100
-60
50
-80
0
3
10
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4
10
5
6
10
10
frequency (Hz)
7
10
theta (degree)
‘3_1’
|Z|
|Z| and theta plot
500
-100
8
10
P.2
Impedance Modeling – multi circuit elements (Complete)
Model ID
Equivalent Circuit
Impedance Analyzer
Type of DUTs
Simulation
Value
|Z|-θ f-characteristics
|Z| and theta plot
100
100
80
80
60
20
|Z|
60
‘2_2’
0
40
theta (degree)
40
20
0
3
10
Inductor with
high-core loss
6
7
10
Ra = 10e1
Ca = 1e-9
La = 1e-6
8
10
|Z| and theta plot
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0
4
10
5
6
10
10
frequency (Hz)
7
10
theta (degree)
100
50
0
3
10
Dr. Kelvin K. S. Leung
5
10
10
frequency (Hz)
100
|Z|
‘3_2’
4
10
-100
8
10
P.3
Impedance Modeling – multi circuit elements (Complete)
Model ID
Equivalent Circuit
Impedance Analyzer
Type of DUTs
Simulation
Value
|Z|-θ f-characteristics
|Z| and theta plot
‘2_3’
100
0
theta (degree)
|Z|
200
-50
0
3
10
Capacitor
4
10
5
6
10
10
frequency (Hz)
7
10
-100
8
10
Ra = 100e-3
Ca = 1e-6
La = 1e-9
100
150
50
100
0
50
0
3
10
Dr. Kelvin K. S. Leung
www.kskelvin.net
theta (degree)
‘3_3’
|Z|
|Z| and theta plot
200
-50
4
10
5
6
10
10
frequency (Hz)
7
10
-100
8
10
P.4
Impedance Modeling – multi circuit elements (Complete)
Model ID
Equivalent Circuit
Impedance Analyzer
Type of DUTs
Simulation
Value
|Z|-θ f-characteristics
|Z| and theta plot
500
|Z|
300
‘2_4’
0
200
-20
theta (degree)
400
-40
100
-60
-80
0
3
10
High-value Resistor
4
10
5
6
10
10
frequency (Hz)
7
10
-100
8
10
Ra = 50
Ca = 1e-9
La = 100e-9
100
450
80
400
60
350
40
300
20
250
0
200
-20
150
-40
100
-60
50
-80
0
3
10
Dr. Kelvin K. S. Leung
www.kskelvin.net
4
10
5
6
10
10
frequency (Hz)
7
10
theta (degree)
‘3_4’
|Z|
|Z| and theta plot
500
-100
8
10
P.5
Impedance Modeling – multi circuit elements (Complete)
Following models are not commonly used as equivalent model for electronic components.
Model ID
Equivalent Circuit
Model ID
‘2_5’
‘2_6’
‘3_5’
‘3_7’
‘3_6’
‘3_8’
Dr. Kelvin K. S. Leung
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Equivalent Circuit
P.6
Impedance Modeling – multi circuit elements (Complete)
Appendix
MATLAB program simulate the output from impedance analyzer
case '2_5'
Z = XLa + XCa
case '2_6'
Z = 1./(1./XLa+1./XCa);
case '3_1'
Z = 1./(1./XCa+1./(Ra+XLa));
case '3_2'
Z = 1./(1./Ra + 1./XLa + 1./XCa);
case '3_3'
Z = Ra + XLa + XCa;
case '3_4'
Z = XLa + 1./(1./Ra+1./XCa);
case '3_5'
Z = 1./(1./Ra+1./(XLa+XCa));
case '3_6'
Z= 1./(1./XLa+1./XCa) + Ra;
case '3_7'
Z = XCa + 1./(1./Ra+1./XLa);
case '3_8'
Z = 1./(1./XLa + 1./(Ra+XCa));
case '4_1'
Z = 1./(1./XCb + 1./(Ra+XCa+XLa));
otherwise
Z = 1;
% written by: Dr. Kelvin K. S. Leung
% www.kskelvin.net
%
% Program environment: MATLAB
% Purpose: Show the Gain-Phase response of a passive device by defining its
% R, L and C. Rs-Ls, Rp-Lp, Rs-Cs and Rp-Cp will also generate and help
% the user to understand how an impedance analyzer operate.
clc;
clear;
close all;
% user input parameter
freq_start = 1e3; % start frequency of plot
freq_stop = 100e6; % stop frequency of plot
Ra = 100e-3;
% resistance
Ca = 10e-9;
% capacitance
La = 100e-6;
% inductance
Cb = 100e-12; % for 4 resonator ONLY
model_select = '3_1';
% model select refer to the article
% basic calculation and conversion
freq = logspace(log10(freq_start),log10(freq_stop),100);
omega = 2*pi*freq;
end
XLa = j*omega*La;
XCa = 1./j./omega./Ca;
XCb = 1./j./omega./Cb;
% impedance equation
switch model_select
case '2_1'
Z = Ra + XLa;
case '2_2'
Z = 1./(1./Ra+1./XLa);
case '2_3'
Z = Ra + XCa;
case '2_4'
Z = 1./(1./Ra+1./XCa);
Dr. Kelvin K. S. Leung
figure;
% |Z| and theta plot
subplot(221);
[AX,H1,H2] = plotyy(freq,abs(Z),freq,angle(Z)*180/pi,'semilogx');
ylabel(AX(1),'|Z|');ylabel(AX(2),'theta (degree)');
xlim(AX(1),[freq_start freq_stop]);xlim(AX(2),[freq_start freq_stop]);
%ylim(AX(2),[-100 100]);
grid off; xlabel('frequency (Hz)');title('|Z| and theta plot');
% R+jX plot
subplot(222);
[AX,H1,H2] = plotyy(freq,real(Z),freq,imag(Z),'semilogx');
ylabel(AX(1),'R (real)');ylabel(AX(2),'X (imagine)');
xlim(AX(1),[freq_start freq_stop]);xlim(AX(2),[freq_start freq_stop]);
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P.7
Impedance Modeling – multi circuit elements (Complete)
grid off; xlabel('frequency (Hz)');title('R+jX plot');
figure;
% Rs-Cs plot
subplot(221);
[AX,H1,H2] =
plotyy(freq,abs(Z).*cos(angle(Z)),freq,-1./omega./abs(Z)./sin(angle(Z)),'semilogx');
ylabel(AX(1),'Rs (\Omega)');ylabel(AX(2),'Cs (F)');
xlim(AX(1),[freq_start freq_stop]);xlim(AX(2),[freq_start freq_stop]);
grid off; xlabel('frequency (Hz)');title('Rs-Cs plot');
% Rs-Ls plot
subplot(222);
[AX,H1,H2] =
plotyy(freq,abs(Z).*cos(angle(Z)),freq,1./omega.*abs(Z).*sin(angle(Z)),'semilogx');
ylabel(AX(1),'Rs (\Omega)');ylabel(AX(2),'Ls (H)');
xlim(AX(1),[freq_start freq_stop]);xlim(AX(2),[freq_start freq_stop]);
grid off; xlabel('frequency (Hz)');title('Rs-Ls plot');
% Rp-Cp plot
subplot(223);
[AX,H1,H2] =
plotyy(freq,abs(Z)./cos(angle(Z)),freq,-1*sin(angle(Z))./omega./abs(Z),'semilogx');
ylabel(AX(1),'Rp (\Omega)');ylabel(AX(2),'Cp (F)');
xlim(AX(1),[freq_start freq_stop]);xlim(AX(2),[freq_start freq_stop]);
grid off; xlabel('frequency (Hz)');title('Rp-Cp plot');
% Rp-Lp plot
subplot(224);
[AX,H1,H2] =
plotyy(freq,abs(Z)./cos(angle(Z)),freq,abs(Z)./omega./sin(angle(Z)),'semilogx');
ylabel(AX(1),'Rp (\Omega)');ylabel(AX(2),'Lp (H)');
xlim(AX(1),[freq_start freq_stop]);xlim(AX(2),[freq_start freq_stop]);
grid off; xlabel('frequency (Hz)');title('Rp-Lp plot');
Dr. Kelvin K. S. Leung
www.kskelvin.net
P.8