MATHEMATICAL REPRESENTATIONS
3.5 IMPEDANCE AND ADMITTANCE MODELS
3.5.1 System Impedance Models and Terminal Constraints
The concept of impedance can be applied to the analysis of systems with multiple inputs and outputs. The expression for a system impedance, equation (3.105), can be written in matrix notation as
φ1
... φn
=
Z11 · · · Z1n
... . .. ... Zn1 · · · Znn
ψ1
... ψn
. (3.118)
The system-level impedance model is a convenient framework for analyzing the re- lationships between forces and fluxes in a dynamic system. When a terminal constraint exists between a force and a flux, the impedance model expressed in equation (3.118) can be modified to determine the effect of this terminal constraint on the remain- ing force and flux terms. We define a terminal constraint as an explicit relationship between a single force–flux term,
φk= −Zcψk, (3.119)
where the constraint is expressed as an impedance−Zc. The negative sign is chosen for convenience. Expanding equation (3.118) to include the terminal constraint, we have
φ1= Z11ψ1+ · · · +Z1kψk+ · · · +Z1nψn
...
φk= Zk1ψ1+ · · · +Zkkψk+ · · · +Zknψn (3.120) ...
φn = Zn1ψ1+ · · · +Znkψk+ · · · +Znnψn.
Substituting equation (3.119) into equation (3.121) and solving thekth equation for ψkyields
ψk= − Zk1
Zc+Zkk
ψ1− · · · − Zkn
Zc+Zkk
ψn. (3.121)
Substituting this expression into the remainingn−1 equations produces φ1=
Z11− Z1kZk1
Zc+Zkk
ψ1+ · · · +
Z1n− Z1kZkn
Zc+Zkk
ψn (3.122) ...
φn =
Zn1− ZnkZk1 Zc+Zkk
ψ1+ · · · +
Znn− ZnkZkn
Zc+Zkk
ψn. (3.123)
Examining equation (3.122), we see that a general expression for theith force when a constraint exists at thekth location is
φi = n m=1
Zi m− Zi kZkm
Zc+Zkk
ψm m=k. (3.124)
This result indicates that, in general, every force–flux relationship is affected by a terminal constraint at a single location in the system.
Two particular types of terminal constraints arezero-force constraintsandzero- flux constraints. A zero-force constraint can be determined by setting Zc=0 in equation (3.119) and rewriting the remainingn−1 transduction equations. The result is
φiφk = n
j=1
Zi j− Zi kZk j
Zkk
ψj j =k. (3.125)
The superscript on the force term indicates that a zero-force constraint exists at loca- tionk. We can rewrite equation (3.125) as
φiφk = n m=1
Zi j
1− Zi kZk j
Zi jZkk
ψj j=k (3.126)
and denote
Zi kZk j
Zi jZkk
=K (3.127)
as the generalized coupling coefficient. With this definition we can rewrite equa- tion (3.126) as
φiφk = n
j=1
Zi j(1−K)ψj j =k. (3.128)
This definition makes it clear that the coupling coefficient describes how much the impedance function changes upon the introduction of a zero-force constraint at loca- tionk.
Zero-flux constraints are imposed by letting Zc→ ∞and substituting the result into equation (3.124). Letting the constraint impedance approach infinity produces the result
φiψk = n
j=1
Zi jψj j =k. (3.129)
v1 i1
v2 i2 R1
R2
Figure 3.4 Voltage divider with input voltage and current and output voltage and current.
This result demonstrates that the impedance terms at the remainingn−1 locations are not changed by the introduction of a zero-flux constraint at locationk.
Another interpretation of the coupling coefficient is related to the change in the impedance from a zero-force to a zero-flux constraint at location k. The coupling coefficient is also equal to the difference in the zero-flux and zero-force impedance divided by the original impedance,
φiψk−φiφk
Zi m
= Zi m−Zi m(1−K) Zi m
=K. (3.130)
We will find in future chapters that the concept of coupling coefficients is related directly both to material properties and to the interaction of an active material with the external system.
Example 3.6 Consider the voltage divider shown in Figure 3.4 with input voltage, v1, and input current, i1, and output voltage and current, v2 andi2, respectively.
Determine the impedance model of this circuit.
Solution The voltage divider has two forces and two flux terms; therefore, the impedance model will be a 2×2 system. The individual impedance terms can be ob- tained by setting one of the flux terms equal to zero and determining the corresponding force terms.
The impedance termsZ11andZ21can be obtained by setting the currenti2equal to zero and determining the voltage as a function of the currenti1. Wheni2=0, the voltage terms are
v1 =(R1+R2)i1 v2 =R2i1 and the impedance terms can be written as
Z11= R1+R2
Z12= R2.
Setting the currenti1equal to zero produces a zero voltage drop overR1and the two voltages are equivalent. Thus,
v1 =R2i2 v2 =R2i2
and
Z12= R2
Z22= R2
Combining the impedance terms yields the matrix
Z =
R1+R2 R2
R2 R2
. (3.131)
Example 3.7 Determine (a) the coupling coefficients of the system introduced in Example 3.5 and (b) the impedance Z11 when there is a zero-force constraint at terminal 2.
Solution (a) The voltage divider discussed in Example 3.5, has two force–flux terms; therefore,n=2. Fori=1,k=2, andm=1, equation (3.127) is reduced to
K = Z12Z21
Z11Z22
= R22
R2(R1+R2) = R2
R1+R2
. (3.132)
The coupling coefficient wheni =2,k=1, andm=2 is K = Z21Z12
Z22Z11
= R22
R2(R1+R2) = R2
R1+R2
. (3.133)
This is consistent with the result that there aren−1 coupling coefficients for a system that hasnterminals.
(b) The impedance Z11 when a zero-force constraint exists at terminal 2 can be computed from equation (3.128):
Z11φ2= Z11(1−K)
=(R1+R2)
1− R2
R1+R2
=(R1+R2) R1
R1+R2
= R1. (3.134)
v1 i1
v2 = 0 R1
R2
Figure 3.5 Voltage divider with a zero-force (voltage) constraint at terminal 2.
This result is reasonable because a zero-force constraint at terminal 2 is equivalent to connecting the terminal to ground. Therefore, the impedance of the resulting circuit is simply the resistanceR1(see Figure 3.5).
Example 3.8 Terminal 2 of the voltage divider is connected to a digital voltmeter that draws no current, as shown in Figure 3.6. Determine the ratio of the output voltage to the input voltage with this constraint at terminal 2.
Solution The digital voltmeter is assumed to draw no current; therefore, it imposes a zero-flux constraint at terminal 2. With a zero-flux constraint at the terminal, the transduction equations reduce to
v1=(R1+R2)i1
v2=R2i1 (3.135)
Solving forψ1from the first expression and substituting it into the second expression yields
v2= R2
R1+R2v1, (3.136)
i1= 0
v2 i2 = 0 R1
R2 v1
digital volt- meter
Figure 3.6 Voltage divider with a measurement device that draws zero flux (current).
which can be solved to yield the ratio v2
v1
= R2
R1+R2. (3.137)
This is the common result for a voltage divider, except that this analysis highlights the fact that the typical equation for a voltage divider implicitly assumes that zero current is being drawn at the output terminal.