Toner concentration measuring method, toner concentration measuring apparatus and image forming apparatus employing the same

A toner concentration measuring method and apparatus by which the concentration of toner in solvent can be detected accurately with a simple construction without being influenced by a variation of the conductivity caused by a variation of the amount of ions in the solvent. A stepped dc voltage is applied from a high dc voltage generation section between a pair of electrodes placed in solvent, and very weak current which flows in a circuit formed from the pair of electrodes is measured by a current measuring section. The solvent between the pair of electrodes is replaced into an equivalent circuit, and a capacitance of the equivalent circuit is calculated in accordance with a circuit equation to determine the amount of ions in the solvent. Further, in accordance with a function expression wherein the ion amount and a resistance of the equivalent circuit are used as parameters, a toner concentration from which an influence of a variation of the amount of ions in the solvent is eliminated is determined.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to a toner concentration measuring method, a 
toner concentration measuring apparatus and an image forming apparatus 
employing the same, and a toner concentration measuring method and a toner 
image measuring instrument by which the concentration of toner in solvent 
is detected and an image forming apparatus employing the same. 
2. Description of the Related Art 
Conventionally, a remaining ink amount detecting apparatus for liquid 
development is required only to detect presence or absence of ink and 
represent it in a binary value. Thus, a method wherein a voltage is 
applied to an electrode pair to measure a capacitance between the 
electrodes is disclosed, for example, in Japanese Patent Laid-Open No. Hei 
2-169259. Also another method wherein an amount of transmission light is 
measured using a light emitting diode and a light receiving element is 
disclosed, for example, in Japanese Patent Laid-Open No. Hei 6-241996. 
However, the conventional methods described above cannot be used to measure 
the concentration of toner in solvent. Also a method of measuring the 
concentration of toner in solvent is conventionally known. According to 
the conventional method, for example, a light emitting diode is used to 
read an analog variation of the amount of transmission light, and an ac 
voltage is applied to an electrode pair placed in solvent to measure the 
concentration of toner from a variation value of the capacitance between 
the electrodes. 
The conventional method just described has a problem of an accuracy of 
measurement. Where light is used to measure the concentration, the light 
is attenuated significantly while it passes through the solvent. 
Consequently, a light emitting element having a very large amount of light 
emission must be used. However, the amount of light which can be received 
by the light receiving element is still small, and consequently, the 
accuracy in measurement of the concentration based on the received mount 
of light is low. Further, if the light emitting element or the light 
receiving element is placed in the solvent, the light emitting face or the 
light receiving face must be kept clean, or if the light emitting element 
or the light receiving element is located outside a member in which the 
solvent is accommodated, then a light transmitting wall of the member must 
be kept clean. If a light transmitting portion becomes soiled, then the 
accuracy in measurement is further deteriorated. 
The conventional method has another problem in that a conventional 
technique for electric measurement cannot be applied as it is. This is 
because, even though the conductivity or the capacitance of the solvent 
can be measured, it is impossible to extract only the concentration of the 
toner from the measured conductivity or capacitance. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide a toner concentration 
measuring method and a toner concentration measuring apparatus by which 
the concentration of toner in solvent can be detected accurately with a 
simple construction without being influenced by a variation of the 
conductivity caused by a variation of the amount of ions in the solvent 
and an image forming apparatus which employs the toner concentration 
measuring apparatus. 
In order to attain the object described above, according to an aspect of 
the present invention, there is provided a toner concentration measuring 
method for measuring the concentration of toner in solvent, comprising the 
steps of applying a stepped dc voltage between a pair of electrodes in the 
solvent, measuring very weak current flowing in a circuit formed from the 
pair of electrodes, calculating, with the solvent between the pair of 
electrodes replaced into an equivalent circuit, a capacitance of the 
equivalent circuit in accordance with a circuit equation to determine the 
amount of ions in the solvent, and determining a toner concentration, from 
which an influence of a variation of the amount of ions in the solvent is 
eliminated, in accordance with a function expression in which the amount 
of ions and an impedance of the equivalent circuit are included as 
parameters. 
According to another aspect of the present invention, there is provided a 
toner concentration measuring apparatus for measuring the concentration of 
toner insolvent, comprising, a pair of electrodes placed in the solvent, 
voltage application means for applying a stepped dc voltage between the 
pair of electrodes, current measurement means for measuring very weak 
current flowing in a circuit formed between the pair of electrodes, ion 
amount calculation means for calculating, with the solvent between the 
pair of electrodes replaced into an equivalent circuit, a capacitance of 
the equivalent circuit in accordance with a circuit equation to determine 
the amount of ions in the solvent, and concentration calculation means for 
determining a toner concentration, from which an influence of a variation 
of the amount of ions in the solvent is eliminated, in accordance with a 
function expression in which the amount of ions and an impedance of the 
equivalent circuit are included as parameters. 
Preferably, the pair of electrodes are cylindrical electrodes extending in 
parallel to each other such that circumferential faces thereof are opposed 
to and located near to each other, and are rotated by power from the 
outside so that, when rotated, the circumferential surfaces thereof are 
cleaned by a cleaning member secured in the apparatus. 
The equivalent circuit can be determined in the following manner. 
It is assumed that the output current I.sub.total measured when the stepped 
dc voltage is applied to the solvent is composed of an output I.sub.toner 
originating from particles of the toner which have masses and originating 
from back plating of the particles of the toner and another output 
I.sub.ion originating from ions having little masses and originating from 
back plating of positive and negative ions such that an expression 
I.sub.total =I.sub.toner +I.sub.ion may be satisfied; the output value 
I.sub.ion which originates from the positive and negative ions is assumed 
as an output of an R.sub.1 C.sub.1 circuit of a resistance R.sub.1 and a 
capacitance C.sub.1 and represented by an attenuation function I.sub.ion 
=(V/R.sub.1)exp(-t/C.sub.1 R.sub.1)) where V is a voltage and t is time; 
also the output I.sub.toner originating from the toner is assumed as an 
output of an R.sub.2 C.sub.2 circuit of an impedance R.sub.2 and a 
capacitance C.sub.2 while oscillations of the output which are caused by 
the inertia of toner particles are considered to be a behavior of a 
second-order lag system, and the output I.sub.toner originating the toner 
is modeled with an R.sub.3 C.sub.3 L.sub.3 circuit of a resistance 
R.sub.3, a capacitance C.sub.3 and an inductance L.sub.3 to calculate the 
output I.sub.toner originating from the toner as a sum of an attenuation 
function I.sub.att and an attenuation oscillation function I.sub.osc in 
accordance with 
EQU I.sub.toner =I.sub.att +I.sub.osc 
EQU I.sub.toner-att =(V/R.sub.2)exp(-t/(C.sub.2 R.sub.2)) 
EQU I.sub.toner-osc =((.alpha..sup.2 +.omega..sup.2)/.omega.)C.sub.3 
exp(-.alpha.t)cos(.omega.t) 
where .alpha.=R.sub.3 /(2L.sub.3), 
.omega.=(1/(L.sub.3C3)-.alpha..sup.2).sup.1/2then, the output current 
value I.sub.total flowing in the equivalent circuit is represented by 
EQU I.sub.total =I.sub.ion +I.sub.toner-att +I.sub.toner-osc 
=(V/R.sub.1)exp(-t/C.sub.1 R.sub.1))+(V/R.sub.2)exp(-t/(C.sub.2 
R.sub.2))+((.alpha..sup.2 +.omega..sup.2)/.omega.)C.sub.3 
exp(-.alpha.t)cos(.omega.t) 
and then, the output current value I.sub.total is regarded as output 
current of the equivalent circuit composed of the R.sub.1 C.sub.1 series 
circuit, the R.sub.2 C.sub.2 series circuit and the R.sub.3 C.sub.3 
L.sub.3 series circuit connected in parallel. 
Alternatively, it is assumed that the capacitance component is uniform for 
simplified consideration of the equivalent circuit and consequently only 
one capacitance component is involved as represented by C.sub.1 
.ident.C.sub.2 .ident.C.sub.3 .ident.C, and the resistances R.sub.1 and 
R.sub.2 are composed and other variables are re-arranged to place 
EQU R.sub.att =1/(1/R.sub.1 +1/R.sub.2) 
EQU I.sub.att =I.sub.ion +I.sub.toner-att 
EQU R.sub.osc =R.sub.3 
EQU I.sub.osc =I.sub.toner-osc 
EQU L.sub.osc =L.sub.3 
and then, the composed circuit is determined as the equivalent circuit 
including the R.sub.osc L.sub.osc series circuit and the resistance 
R.sub.att are connected in parallel. 
In this instance, the current amount I.sub.total which flows through the 
circuit when the switch of the equivalent circuit is closed is represented 
by 
EQU I.sub.total =((.alpha..sup.2 
+.omega..sup.2)/.omega.)CVexp(-.alpha.t)cos(.omega.t)+(V/R.sub.att)exp(-t/ 
CR.sub.att) 
where .alpha.=R.sub.osc /(2L.sub.osc) and .omega.=(1/(L.sub.osc 
C)-.alpha..sup.2)-.alpha..sup.2).sup.1/2, and the behavior I of toner 
particles and ions in the solvent is defined as represented by 
EQU I=P.sub.1 (-P.sub.2 t)cos(P.sub.3 t)+P.sub.4 exp(-P.sub.5 t) 
where 
EQU P.sub.1 =1/(L.sub.osc /C-(R.sub.osc /2).sup.2).sup.1/2 
EQU P.sub.2 =(R.sub.osc /(2L.sub.osc)) 
EQU P.sub.3 =(1/(L.sub.osc C)+R.sub.osc /(2L.sub.osc)).sup.2).sup.1/2 
EQU P.sub.4 =V/R.sub.osc 
EQU P.sub.5 =1/(R.sub.att C) 
and, from the expression above, the RCL components of the equivalent 
circuit is determined as 
EQU C=P.sub.4 /(P.sub.5 V) 
EQU L.sub.osc =1/(C(P.sub.2 2+P.sub.3 2) 
EQU R.sub.osc =2L.sub.osc P.sub.2 
EQU R.sub.att =V/P.sub.4 
and then, toner concentration information F.sub.1 () is determined in 
accordance with a function expression represented by F.sub.1 ()=R.sub.att 
C.sup.K, where K is a coefficient which depends upon the temperature, the 
viscosity of the solvent used or the amount of charge of the toner. 
The toner concentration measuring method and apparatus is advantageous in 
that the toner concentration can normally be measured accurately. The 
reason is that, while an electric concentration sensor cannot normally 
perform accurate concentration measurement in a process in which charge is 
exchanged frequently as in an electrophotographic printer because the 
electric concentration sensor is normally influenced by ions in the 
solvent, the toner concentration measuring method and apparatus is not 
influenced by ions in the solvent. 
The toner concentration measuring method and apparatus is advantageous also 
in that concentration measurement can be realized with a simple structure. 
The reason is that, while, where an electric concentration sensor is 
employed, in order to measure the amount of ions, it is necessary to 
correct a result of detection of the electric concentration sensor using a 
separate conductivity sensor because the electric concentration sensor is 
normally influenced by ions in the solvent as described above, the toner 
concentration measuring method and apparatus is not influenced by ions in 
the solvent. 
The above and other objects, features and advantages of the present 
invention will become apparent from the following description and the 
appended claims, taken in conjunction with the accompanying drawings in 
which like parts or elements are denoted by like reference symbols.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
Referring to FIG. 1, there is shown a toner concentration measuring 
apparatus to which the present invention is applied. The toner 
concentration measuring apparatus shown includes an electrode pair 1 
placed in solvent 6, a cleaning member 2 for cleaning the electrode pair 
1, a high dc voltage generation section 3 for applying a stepped high dc 
voltage to the electrode pair 1, a current measuring section 4 for 
measuring current of a circuit formed between the electrodes of the 
electrode pair 1, and a data processing section 5 for processing 
measurement data. 
The electrode pair 1 includes a pair of cylindrical electrodes extending in 
parallel to each other such that circumferential faces thereof are 
positioned in an opposing and neighboring relationship to each other. The 
electrodes of the electrode pair 1 are rotated by power transmitted 
thereto from the outside such that, before and after measurement of the 
concentration, the surfaces thereof are cleaned by the cleaning member 2 
secured in the apparatus when the electrodes of the electrode pair 1 are 
rotated. The high dc voltage generation section 3 has a function of 
generating a stepped electric field of 2 MV/m to 4 MV/m between the 
electrodes of the electrode pair 1. The current measuring section 4 has, 
for example, a sampling frequency of 1 kHz or more and a resolution of 8 
bits or more. 
FIG. 2 shows an example of a waveform measured by the current measuring 
section 4. Operation of the toner concentration measuring apparatus of the 
present embodiment is described with reference to FIGS. 1 and 2. 
A stepped high dc voltage generated by the high dc voltage generation 
section 3 is applied between the electrodes of the electrode pair 1. 
Thereupon, a strong electric field of approximately 2 MV/m to 4 MV/m is 
generated momentarily in the proximity of and between the electrodes of 
the electrode pair 1. By the strong electric field, toner particles in the 
solvent are charged and begin to migrate toward the polarities opposite to 
the polarities of the toner particles themselves. Also positive and 
negative ions in the solvent similarly move in the directions toward the 
polarities opposite to the polarities of the ions themselves. If the ions 
arrive at the surfaces of the electrode pair 1, then they lose their 
charge, and toner particles are polarized and stick to the surfaces of the 
electrode pair 1. Consequently, very weak current flows in the circuit 
formed between the electrodes of the electrode pair 1. Since the toner 
particles sticking to the surfaces of the electrode pair 1 lower the 
absolute values of the potentials on the surfaces of the electrode pair 1, 
after the high voltage is applied, the current value which first exhibits 
a peak value decreases gradually and finally to zero. 
Since the current value measured in this instance varies depending upon the 
amount of ions and the concentration of the toner in the solvent, if the 
amount of ions has little variation, then the magnitude of the measured 
current value represents information of the toner concentration. However, 
this is impossible in an environment wherein ions are added or removed in 
the process of development. In the environment described, an output value 
which only depends upon the toner concentration must be extracted from the 
measurement value. 
Thus, the phenomenon described above is modeled into an electric equivalent 
circuit, and output current of the electric equivalent circuit is defined 
with a mathematical expression. When a stepped voltage is applied to the 
solvent 6, output current measured by the current measuring section 4 
attenuates while oscillating as seen in FIG. 2. It is estimated that such 
oscillations arise from some lag in electric migration of toner particles 
in the solvent 6 caused by inertial forces of them because they themselves 
have masses. Therefore, the output current I.sub.total measured as a total 
value is decomposed into an output I.sub.toner originating from toner 
particles having masses (an output based on back plating of toner 
particles) and another output I.sub.ion originating from ions having 
little masses (an output based on back plating of positive and negative 
ions) as given by the following expression (1): 
EQU I.sub.total =I.sub.toner +I.sub.ion (1) 
Of the components, the output value I.sub.ion originating from positive and 
negative ions can be regarded an output of an RC circuit composed of a 
resistance (R) and a capacitance (C) and can be represented by such an 
attenuation function as given by the following expression (2): 
EQU I.sub.ion =(V/R.sub.1)exp(-t/C.sub.1 R.sub.1)) (2) 
Meanwhile, as regards the output I.sub.toner originating from the toner, 
oscillations of the output which are caused by the inertia of toner 
particles are considered to be a behavior of a second-order lag system of 
a circuit having a resistance (R) and a capacitance (C) similarly as 
described above. Thus, the output I.sub.toner originating the toner is 
modeled with an RCL circuit including an inductance (L) in addition to the 
resistance (R) and the capacitance (C) and can thus be represented as a 
sum of an attenuation function I.sub.att and an attenuation oscillation 
function I.sub.osc as given in the following expression (3): 
EQU I.sub.toner =I.sub.att +I.sub.osc (3) 
EQU I.sub.toner-att =(V/R.sub.2)exp(-t/(C.sub.2 R.sub.2)) (4) 
EQU I.sub.toner-osc =((.alpha..sup.2 +.omega..sup.2)/.omega.)C.sub.3 
exp(-.alpha.t)cos(.omega.t) (5) 
where 
EQU .alpha.=R.sub.3 /(2L.sub.3), .omega.=(1/(L.sub.3 
C.sub.3)-.alpha..sup.2).sup.1/2 
Consequently, the output current value I.sub.total flowing in the 
equivalent circuit can be represented by the following expression (6): 
EQU I.sub.total =I.sub.ion +I.sub.toner-att +I.sub.toner-osc 
=(V/R.sub.1)exp(-t/C.sub.1 R.sub.1))+(V/R.sub.2)exp(-t/(C.sub.2 
R.sub.2))+((.alpha..sup.2 +.omega..sup.2)/.omega.)C.sub.3 
exp(-.alpha.t)cos(.omega.t) (6) 
This can be represented in such a circuit diagram as shown in FIG. 3 which 
includes a switch S. Here, although it is estimated that the capacitance 
component (C) in actual ink has a distribution due to some local presence 
of toner particles and/or ions in the fluid and so forth, it is assumed 
that the capacitance component is uniform for simplified consideration on 
the equivalent circuit. Further, since timings at which the electrodes 
actually become saturated by toner particles and ions sticking thereto are 
equal, it is assumed that only one capacitance component is involved. 
Consequently, 
EQU C.sub.1 .ident.C.sub.2 .ident.C.sub.3 .ident.C (7) 
Further, in order to simplify the circuit configuration, the resistances 
R.sub.1 and R.sub.2 are composed and other variables are re-arranged to 
place 
EQU R.sub.att =1/(1/R.sub.1 +1/R.sub.2) (8) 
EQU I.sub.att =I.sub.ion +I.sub.toner-att 
EQU R.sub.osc =R.sub.3 
EQU I.sub.osc =I.sub.toner-osc 
EQU L.sub.osc =L.sub.3 
Thus, the composed circuit has such a configuration as shown in FIG. 4. The 
current amount I.sub.total which flows through the circuit when the switch 
S of the circuit is closed is represented by the following expression (9): 
EQU I.sub.total =((.alpha..sup.2 
+.omega..sup.2)/.omega.)CVexp(-.alpha.t)cos(.omega.t)+(V/R.sub.att)exp(-t/ 
CR.sub.att) (9) 
where 
EQU .alpha.=R.sub.osc /(2L.sub.osc) and .omega.=(1/(L.sub.osc 
C)-.alpha..sup.2).sup.1/2. 
From the foregoing, in the present specification, the behavior I of toner 
particles and ions in the insulating solvent is defined as given by the 
following expression (10): 
EQU I=P.sub.1 (-P.sub.2 t)cos(P.sub.3 t)+P.sub.4 exp(-P.sub.5 t) (10) 
where 
EQU P.sub.1 -1/(L.sub.osc /C-(R.sub.osc /2).sup.2).sup.1/2 
EQU P.sub.2 =(R.sub.osc /(2L.sub.osc)) 
EQU P.sub.3 =(1/(L.sub.osc C)+R.sub.osc /(2L.sub.osc)).sup.2).sup.1/2 
EQU P.sub.4 =V/R.sub.osc 
EQU P.sub.5 =1/(R.sub.att C) 
From the expression (10), the RCL components of the equivalent circuit can 
be determined in the following manner: 
EQU C=P.sub.4 /(P.sub.5 V) (11) 
EQU L.sub.osc =1/(C(P.sub.2 2+P.sub.3 2) (12) 
EQU R.sub.osc =2L.sub.osc P.sub.2 (13) 
EQU R.sub.att =V/P.sub.4 (14) 
The C component in the expression (11) above represents the capacitance 
component of the equivalent circuit. This is information indicative of the 
capacitance of the solvent and is not influenced very much by the toner 
concentration value. Further, R.sub.att in the expression (14) is 
information representative of the conductivity of the entire toner 
particles and ions. Consequently, by setting the following function 
expression based on the information given above, toner concentration 
information F.sub.1 () from which an influence of ions in the solvent is 
removed can be determined: 
EQU F.sub.1 ()=R.sub.att C.sup.K (15) 
where K is a coefficient which depends upon the temperature, the viscosity 
of the solvent used or the amount of charge of the toner. 
In the following, an example is described. Measurement and calculation were 
performed for four different solvents having different ion amounts from 
one another using the electrode pair structure described above. It was 
proved that, with the information of a peak value simply indicative of the 
maximum value of a waveform, a correlation is found between the 
concentration and the measured and calculated value only in the same 
solvent, but no correlation between them is found between the solvents 
having different ion amounts as seen from FIG. 5. FIG. 5 is a graph 
indicating maximum current values (peak values) of current-time data 
obtained by the measurement in regard to the peak value-concentration 
value for the inks having different concentrations. It can be seen that 
even the same ink of the same concentration exhibits different peak values 
before and after it is used. 
On the other hand, all values obtained by the measurement and calculation 
based on the expression (15) which is a function expression of R.sub.att 
and C of the modeled equivalent circuit represent concentration 
information accurately as seen from FIG. 6 irrespective of a variation of 
the ion amount. FIG. 6 is a graph indicating F.sub.1 () values of 
current-time data obtained by the measurement in regard to the F.sub.1 () 
value-concentration value for the individual inks of different 
concentrations. Different from FIG. 5, it can be seen from FIG. 6 that the 
same ink of the same concentration exhibits an equal F.sub.1 () value 
before and after the ink is used. 
In the toner concentration measuring apparatus of the present embodiment, 
the parameters in the expressions are determined so that the measured 
current waveform may be approximated to that provided by the expression 
(10). To this end, a non-linear optimization technique is required. Where 
a non-linear optimization technique is used, a high load is applied to an 
arithmetic section in the apparatus, and in the worst case, calculation 
diverges and this disables searching out of an optimum value. Thus, 
attention is paid to the function expression (15), and this expression is 
compared with the expression (10). From the comparison, it can be found 
that only the second term of the expression (10), that is, only the value 
of the attenuation term, is used as the parameter in the expression. Thus, 
if it is assumed that, at a time when oscillations of the measured 
waveform are reduced sufficiently, the second term on the right side of 
the expression (10) is almost equal to zero, then the expression (10) can 
be represented as 
EQU I.apprxeq.P.sub.4 exp(-P.sub.5 t). 
Since this expression is a function expression of the variable value I and 
t, it can be determined in accordance with the following simultaneous 
expressions (16) if data (t.sub.1, I.sub.1) and (t.sub.2, I.sub.2) at two 
measurement points when sufficient time passes from the time of the 
current-time data obtained by the measurement and oscillations are reduced 
sufficiently: 
EQU I.sub.1 =P.sub.4 exp(-P.sub.5 t.sub.1) 
EQU I.sub.2 =P.sub.4 exp(-P.sub.5 t.sub.2) (16) 
By solving the same, 
EQU F.sub.1 ()=R.sub.att C.sup.K =(V/P.sub.4)(P.sub.4 /(P.sub.5 V)).sup.K 
=(I.sub.1 /exp(-.lambda.t.sub.1)).sup.K-1 .lambda..sup.-K V.sup.1-K (17) 
where 
EQU .lambda.=(log.sub.e (I.sub.1 /I.sub.2))/(I.sub.2 -I.sub.1). 
From this, an F.sub.1 () value can be determined using the function 
expression from measurement values of two points without using a 
complicated optimization technique which is based on measurement a 
waveform. The F.sub.1 () value is not influenced by the ion amount in the 
solvent, and concentration values can be detected accurately and values 
corresponding in a one by one corresponding relationship to the 
concentration values are outputted. The output values are peculiar values 
depending upon the solvent, the type of the toner, the air temperature and 
the configuration of the circuit for measurement. Consequently, the 
correspondence between the concentration values and the F.sub.1 () values 
is converted into a table by evaluation in advance so that an F.sub.1 () 
value obtained by measurement with an actual apparatus can be converted 
directly into a concentration value by comparing the F.sub.1 () value with 
the data of the table prepared in the apparatus. 
While a preferred embodiment of the present invention has been described 
using specific terms, such description is for illustrative purposes only, 
and it is to be understood that changes and variations may be made without 
departing from the spirit or scope of the following claims.