Variable impedance circuit

A variable impedance circuit includes a first differential amplifier circuit having an input terminal pair, an output terminal pair and a capacitive element connected between the emitters of a transistor pair. The variable impedance circuit further inclludes a second differential amplifier circuit having an input terminal pair and an output terminal pair. The output terminal pair of the first differential amplifier circuit is connected to the input terminal pair of the second differential amplifier circuit. Furthermore, the output terminal pair of the second differential amplifier circuit is connected to the input terminal pair of the first differential amplifier circuit.

BACKGROUND OF THE INVENTION 
This invention relates to a variable impedance circuit capable of 
electrically increasing or decreasing the impedance, such as resistance 
and capacitance. 
Recently, the function of semiconductor integrated circuits has been 
greatly improved, and the filter circuit has come to be incorporated in a 
semiconductor chip as an integrated circuit. Generally a filter circuit is 
composed of a resistance element and a capacitance element. To change the 
filtering characteristic of the filter circuit, it is necessary to vary 
the value of constituent elements, that is, the capacitance element or the 
resistance element. Accordingly, hitherto, a variable impedance circuit 
has been used, which can vary the value of capacitance element or 
resistance element built in a semiconductor chip. 
FIG. 12 shows a conventional variable capacitance circuit used for such a 
purpose, expressed only by an AC circuit. In FIG. 13, the variable 
capacitance circuit in FIG. 12 is expressed by both an AC circuit and DC 
circuit. 
In FIG. 12 and FIG. 13, a differential amplifier circuit 15 is composed of 
transistors Q.sub.5, Q.sub.6, and a resistance element 10 connected 
between their emitters. A differential amplifier circuit 16 is composed of 
transistors Q.sub.7, Q.sub.8, and a resistance element 11 connected 
between their emitters. A differential amplifier circuit 17 is composed of 
transistors Q.sub.1, Q.sub.2, and a resistance element 12 connected 
between their emitters. A differential amplifier circuit 18 is composed of 
transistors Q.sub.3, Q.sub.4, and a resistance element 13 connected 
between their emitters. As is clear from FIG. 12 and FIG. 13, the 
differential amplifier circuits 15, 16 are so connected that the input 
terminals of one differential amplifier circuit are connected to the 
output terminals of the other differential amplifier circuit. The 
differential amplifier circuits 17, 18 are also connected in a similar 
relation. These differential amplifier circuits 15, 16, 17 and 18 are 
connected as shown in FIG. 12, FIG. 13, and a capacitance element 14 is 
connected between two output terminals of the differential amplifier 
circuit 17. The transistors Q.sub.1 to Q.sub.8 for composing the 
differential amplifier circuits 15, 16, 17 and 18 are supplied with biases 
from voltage sources V.sub.0, V.sub.1, and constant current source I.sub.0 
as shown in FIG. 13. 
The voltage-current conversion factors of the differential amplifier 
circuits 15, 16, 17 and 18 are determined respectively by the 
characteristics of the resistance elements 10, 11, 12 and 13. 
The operation is explained below. 
The relation 
EQU i.sub.2 =g.sub.1 .multidot.V.sub.1 ( 1) 
is established between voltage V.sub.1 and output current i.sub.2 across 
input terminals of the differential amplifier circuit 17. In this equation 
g.sub.1 denotes the voltage-current conversion factor of the differential 
amplifier circuit 17, and supposing the emitter resistance value of 
transistors Q.sub.1, Q.sub.2 to be r.sub.el, and the value of resistance 
element 12 between emitters to be R.sub.1, g.sub.1 it is expressed as 
follows. 
##EQU1## 
The characteristics of voltage V.sub.2 and current i.sub.2 occurring at 
both ends of the capacitance element 14 of capacitance value C.sub.0 are 
obtained as follows. 
EQU i.sub.2 =-j.omega.C.sub.0 .multidot.V.sub.2 ( 3) 
Similarly, the characteristics of voltage V.sub.2 across input terminals 
and output current i.sub.1 of the differential amplifier circuit 18 
composed of transistors Q.sub.3, Q.sub.4 are obtained as follows: 
EQU i.sub.1 =-g.sub.2 .multidot.V.sub.2 ( 4) 
where g.sub.2 denotes the voltage-current conversion factor of the 
differential amplifier circuit 18, and supposing the resistance value of 
each emitter of transistors Q.sub.3, Q.sub.4 to be r.sub.e2, and the value 
of the resistance element 13 between the emitters to be R.sub.2, g.sub.2 
is expressed as follows. 
##EQU2## 
Solving V.sub.1 and i.sub.1 from equations (1), (3) and (4), it is known 
from equations (1), (3) that 
##EQU3## 
and from equations (4), (6) that 
##EQU4## 
That is, 
##EQU5## 
Here, the inductance L is given as follows. 
##EQU6## 
Furthermore, from equations (2), (5), it follows that 
EQU L=(R.sub.1 +2r.sub.el)(R.sub.2 +2r.sub.e2)C.sub.0 ( 10) 
Usually, the resistance values can be set so as to establish the relation 
of R.sub.1 &gt;r.sub.el, R.sub.2 &gt;r.sub.e2, and hence the inductance L can be 
approximated as 
EQU L=R.sub.1 .multidot.R.sub.2 .multidot.C.sub.0 
Likewise, the characteristics of input terminal voltage V.sub.3 and output 
current i.sub.4 of the differential amplifier circuit 15 composed of 
transistors Q.sub.5, Q.sub.6 may be expressed as follows, supposing the 
voltage-current conversion factor of the differential amplifier circuit 15 
to be g.sub.3 : 
EQU i.sub.4 =g.sub.3 .multidot.V.sub.3 ( 11) 
Supposing the voltage-current conversion factor of the differential 
amplifier circuit 16 to be g.sub.4, the characteristics of input terminal 
voltage V.sub.4 and output current i.sub.3 of the differential amplifier 
16 composed of transistors Q.sub.7, Q.sub.8 are as follows. 
EQU i.sub.3 =-g.sub.4 .multidot.V.sub.4 ( 12) 
The circuit systems respectively composed of the differential amplifier 
circuits 15, 16, and the differential amplifier circuits 17, 18 represent 
the conventionally used phase conversion circuits. The circuit system 
composed of the differential amplifiers 17, 18 is designed to apply from 
capacitance characteristics to inductance characteristics, in terms of 
circuitry. 
In FIG. 12 and FIG. 13, the relations 
EQU V.sub.4 =V.sub.1 ( 13) 
EQU i.sub.4 =-i.sub.1 ( 14) 
are established between the voltages V.sub.1, V.sub.4, and currents 
i.sub.4, i.sub.1, respectively. Hence, equation (8) is rewritten as 
##EQU7## 
and further by eliminating i.sub.4 from equation (11), it results in 
##EQU8## 
and from equations (16) and (12), it follows that 
##EQU9## 
The capacitance value C applied between the voltage V.sub.3 and current 
i.sub.3 is given as 
##EQU10## 
That is, by properly selecting the values for the voltage-current 
conversion factors g.sub.1, g.sub.2, g.sub.3 and g.sub.4, the capacity 
value C is newly created electrically. 
However, in the conventional variable capacitance circuit shown in FIG. 12, 
FIG. 13, at least four differential amplifier circuits are needed in order 
to obtain a new capacitance value by electrically increasing or decreasing 
the capacitance. Accordingly, the circuit composition is complicated, the 
number of required elements increases, and the chip area increases. 
A conventional variable resistance circuit incorporated in a semiconductor 
chip is explained below while referring to FIG. 14. 
In FIG. 14, transistors 41, 42 are connected between a constant voltage 
source 40 and the grounding potential. A constant voltage source 43 is 
connected to the base of the transistor 41. A variable voltage source 44 
is connected to the base of the transistor 42. An output terminal 45 is 
connected to the connecting points of the transistors 41, 42. 
In the structure in FIG. 14, the resistance value as seen from the output 
terminal 45 is equal to the differential emitter resistance of the 
transistor 41 (that is, the impedance of the transistor 41 seen from its 
emitter), and it is given in the following formula. 
##EQU11## 
where k is Boltzmann constant, T is absolute temperature, q is electric 
charge quantity of an electron, and I.sub.0 is emitter current flowing in 
the transistor 41. When the voltage of the variable voltage source 44 is 
varied, the current I.sub.0 changes, and hence the resistance value as 
seen from the output terminal 45 varies. Therefore, by controlling the 
voltage of the variable voltage source 44, a variable resistance may be 
obtained. 
However, in the conventional variable resistance circuit shown in FIG. 14, 
since the differential emitter resistance itself of the transistor 41 is 
used as a variable resistance component, the variable range of the 
resistance value is narrow. 
SUMMARY OF THE INVENTION 
It is hence a first object of the invention to present a variable impedance 
circuit capable of varying the impedance, such as capacitance value and 
resistance value, by using two differential amplifier circuits. 
It is a second object of the invention to present a variable impedance 
circuit capable of widening the variable range of impedance, such as 
capacitance value and resistance value. 
The invention is, in short, intended to obtain a new impedance from the 
voltage and current characteristics caused between the pair of input 
terminals of a first differential amplifier circuit, by connecting the 
output terminal pair of the first differential amplifier circuit to the 
input terminal pair of a second differential amplifier circuit, connecting 
the output terminal pair of the second differential amplifier cirucuit to 
the input terminal pair of the first differential amplifier circuit, and 
connecting an impedance element such as capacitance element or resistance 
element between the emitters of the transistor pair composing the first 
differential amplifier circuit. 
In this way, the variable impedance circuit is composed of two differential 
amplifier circuits, and the circuit composition may be simplified, and the 
number of required elements is smaller. Therefore, when this variable 
impedance circuit is incorporated in a semiconductor chip, the required 
chip area may become small. Moreover, as compared with the prior art in 
which differential emitter resistance is directly used as variable 
resistance component, the variable range of the impedance may become 
greater.

DETAILED DESCRIPTION OF THE INVENTION 
Referring now to FIG. 1 and FIG. 2, a first embodiment of the invention is 
described in detail below. 
In FIG. 1, FIG. 2, a differential amplifier 1 is composed of transistors 
Q.sub.1, Q.sub.2, and a capacitance element 3 connected between their 
emitters. A differential amplifier circuit 2 is composed of transistors 
Q.sub.3, Q.sub.4, and a resistance element 5 connected between their 
emitters. The output terminal pair of the differential amplifier circuit 1 
is connected to the input terminal pair of the differential amplifier 
circuit 2. The output terminal pair of the differential amplifier circuit 
2 is connected to the input terminal pair of the differential amplifier 
circuit 1. A resistance element 6 is connected between the input terminals 
of the differential amplifier circuit 2. Biases are supplied to the 
transistors Q.sub.1 to Q.sub.4 from voltage source V.sub.1, and constant 
current sources I.sub.0, I.sub.1. 
The operation is explained below. 
Supposing the input terminal AC voltage of the differential amplifier 
circuit 1 to be V.sub.1, and the voltage-current conversion factor of the 
differential amplifier circuit 1 to be g.sub.m1, the output AC current 
i.sub.2 is given as 
EQU i.sub.2 =g.sub.m1 .multidot.V.sub.1 (19) 
The voltage V.sub.2 produced between the both ends of the resistance 
element 6 of which resistance value is R.sub.3 is given as 
EQU V.sub.2 =-i.sub.2 .multidot.R.sub.3 (20) 
Supposing the voltage-current conversion factor of the differential 
amplifier circuit 2 to be g.sub.m2, the output current i.sub.1 flowing due 
to the voltage V.sub.2 applied between the input terminals of the 
differential amplifier circuit 2 is given as 
EQU i.sub.1 =-g.sub.m2 .multidot.V.sub.2 (21) 
From equations (19), (20), it follows that 
EQU V.sub.2 =-g.sub.m1 .multidot.R.sub.3 .multidot.V.sub.1 (22) 
From equations (22), (21), it follows that 
EQU i.sub.1 =g.sub.m1 .multidot.g.sub.m2 .multidot.R.sub.3 .multidot.V.sub.1( 
23) 
Here, the voltage-current conversion factor g.sub.m2 of the differential 
amplifier circuit 2 is given by the resistance characteristic. In other 
words, the voltage-current conversion factor g.sub.m2 is determined by the 
emitter resistance value r.sub.e2 of transistors Q.sub.3, Q.sub.4 and the 
resistance value R.sub.2 of the resistance element 5, and is obtained as 
follows. 
##EQU12## 
Supposing here r.sub.e2 &lt;&lt;R.sub.2, it follows that 
##EQU13## 
Next, the voltage-current conversion factor g.sub.m1 of the differential 
amplifier circuit 1 is given by the characteristics of the capacitance 
element 3. Here, the voltage-current conversion factor g.sub.m1 is given 
by the impedance of the capacitance element 3 
##EQU14## 
and the emitter resistance value r.sub.e1 of the transistors Q.sub.1, 
Q.sub.2, and it becomes 
##EQU15## 
Here, if selecting as 2r.sub.e1 &lt;&lt;.vertline.Z.vertline., it follows that 
##EQU16## 
Putting equations (24), (26) into equation (23) yields 
##EQU17## 
As a result, between the voltage V.sub.1 and current i.sub.1, a new 
capacitance value C is given as 
##EQU18## 
By selecting the ratio of the resistance value R.sub.3 and R.sub.2, 
another new capacitance value C may be obtained from the original 
capacitance value C.sub.1. 
Furthermore, as seen from equation (23), it is enough when the capacitance 
characteristic is given to either one of the voltage-current conversion 
factor g.sub.m1 or g.sub.m2. 
FIG. 3 and FIG. 4 relate to a second embodiment of the invention. The 
variable impedance circuit shown in FIG. 3, FIG. 4 comprises differential 
amplifier circuits 8, 9, and a resistance element 7 is connected between 
the emitters of transistors Q.sub.1, Q.sub.2 for composing the 
differential amplifier circuit 8, and a capacitance element 4 is connected 
between the emitters of transistors Q.sub.3, Q.sub.4 for composing the 
differential amplifier circuit 9. The remaining composition is the same as 
in the first embodiment shown in FIG. 1, FIG. 2. 
Now, assuming the capacitance value of the capacitance element 4 to be 
C.sub.2 and the resistance value of the reistance element 7 to be R.sub.1, 
equations (26), (24) in the first embodiment are respectively rewritten as 
##EQU19## 
and the current i.sub.1 becomes 
##EQU20## 
That is, same as in the first embodiment, a new capacitance value C is 
applied between the voltage V.sub.1 and current i.sub.1 as 
##EQU21## 
Therefore, by selecting the ratio of the resistance value R.sub.3 and 
R.sub.1, a new capacitance value C may be obtained from the initial 
capacity value C.sub.2. 
FIG. 5 and FIG. 6 relate to a third embodiment of the invention. In this 
embodiment, as clearly seen from FIG. 5, the output terminal pair of the 
differential amplifier circuit 2 is connected to the input terminal pair 
of the differential amplifier circuit 1 by way of a current-current 
conversion circuit 20. The current-current conversion circuit 20 may be 
realized, for example as shown in FIG. 6, by a known Gilbert 
multiplication circuit composed of transistors Q.sub.5 to Q.sub.8, and 
current source 2I.sub.2. Meanwhile, to the transistors Q.sub.7, Q.sub.8 of 
the current-current conversion circuit 20, a base bias is applied from a 
voltage source V.sub.2. The remaining composition is substantially the 
same as shown in FIG. 2, FIG. 4. 
When such current-current conversion circuit 20 is used, the current value 
of the differential amplifier circuit 2 may be varied independently of the 
composition of the differential amplifier circuit 2. In other words, 
supposing there is a coefficient K for converting current i.sub.1 into 
other current i.sub.1 ', the relation is 
EQU i.sub.1 '=K.multidot.i.sub.1 (32) 
Putting equation (32) into equation (27), the converted current i.sub.1 ' 
is expressed as 
##EQU22## 
Therefore, the capacitance value C' after conversion is 
##EQU23## 
As known from equation (34), the new capacitance value C' may be increased 
or decreased by the quantity given as the product of K and R.sub.3 
/R.sub.2. Since K is determined by I.sub.2 /I.sub.0, equation (34) is 
finally rewritten as 
##EQU24## 
That is, according to the embodiment shown in FIG. 6, the capacitance 
value can be changed not only by the ratio of resistance components 
(R.sub.3 /R.sub.2), but also by the ratio of current components (I.sub.2 
/I.sub.0). Therefore, a greater capacitance value may be obtained. 
FIG. 7 shows a fourth embodiment of the invention, having a current-current 
conversion circuit 20 added to the embodiment in FIG. 3. In FIG. 7, the 
current value of the differential amplifier circuit 9 may also be varied 
independently of the composition of the differential amplifier circuit 9. 
In FIG. 7, the capacitance value C' after conversion is given as 
##EQU25## 
When the current-current conversion circuit 20 is used as shown in the 
embodiments in FIG. 5 to FIG. 7, the following advantages are brought 
about. Since the resistance element and capacitance element of the 
variable impedance circuit of the invention are formed in a semiconductor 
chip, their values are fixed. However, when actually composing a filter 
circuit by using this variable impedance circuit, it may be sometimes 
desired to vary the frequency characteristics of the filter only very 
slightly. In such a case, by using the current-current conversion circuit 
20, the filter circuit may be furnished with a kind of a variable resistor 
function, so that a desired frequency characteristic may be realized quite 
easily. 
FIG. 8 shows a fifth embodiment of the invention modifying the embodiment 
shown in FIG. 6. That is, instead of the linear resistance elements 6, 6 
in FIG. 6, nonlinear characteristic resistances 21, 21 composed of 
transistors Q.sub.9, Q.sub.10 are used, and a base bias is applied to 
these transistors Q.sub.9, Q.sub.10 from a voltage source V.sub.2. 
Meanwhile, in the composition in FIG. 8, a resistance element is not 
connected between the emitters of the transistors Q.sub.3, Q.sub.4 for 
composing a differential amplifier circuit 22. 
In the embodiment shown in FIG. 8, the transistors Q.sub.9, Q.sub.10, the 
transistors Q.sub.3, Q.sub.4 for composing the differential amplifier 
circuit 22, and the current source 2I.sub.0 fulfill substantially an 
equivalent current-current converting function as the Gilbert 
multiplication circuit 20 shown in FIG. 6. Accordingly, the 
current-current converting function is realized in a simple circuit 
structure. Moreover, when composed as shown in FIG. 8, since the base 
potentials of the transistors Q.sub.9, Q.sub.10 are fixed by the potential 
of the voltage source V.sub.2, the emitter potentials of the transistors 
Q.sub.3, Q.sub.4 for composing the differential amplifier circuit 22 are 
also fixed nearly at specific potentials. Therefore, particularly when the 
supply voltage is small, the circuit design becomes quite easy, and the 
dynamic range of the transistors Q.sub.1, Q.sub.2 for composing the 
differential amplifier circuit 1 may be increased. 
FIG. 9 shows a sixth embodiment shown of the invention, improving the 
embodiment in FIG. 2. Instead of the transistor Q.sub.1 in FIG. 2, an 
operational amplifier 24 is used, and instead of transistor Q.sub.2 in 
FIG. 2, an operational amplifier 25 is used. The operational amplifier 24 
is composed of transistors Q.sub.11 to Q.sub.14, Q.sub.19, and a current 
source I.sub.3, while the operational amplifier 25 is composed of 
transistors Q.sub.15 to Q.sub.18, Q.sub.20, and a current source I.sub.3. 
By thus composing, the emitter resistances of the transistors Q.sub.19, 
Q.sub.20 become smaller in reverse proportion to the open gain of the 
operational amplifiers 24, 25. As a result, the resistance component added 
serially to the capacitance element 3 is decreased. 
FIG. 10 shows an embodiment shown of the invention applied to a variable 
resistance circuit. 
The embodiment in FIG. 10 is a modified version of the composition of FIG. 
8, and the same parts are identified with the same reference numbers and 
repeated explanations are omitted. A transistor Q.sub.21 and a variable 
voltage source V.sub.3 make up a current source, and an electric current 
I.sub.x is supplied into the differential amplifier circuit 2. On the 
other hand, resistance element 26, 27 are connected in series between the 
emitters of the transistors Q.sub.1, Q.sub.2 for composing the 
differential amplifier circuit 1, and a constant current source 28 is 
connected between their connecting point and the reference potential. 
Supposing the current flowing in the constant current source 28 to be 
I.sub.0, the differential emitter resistance r.sub.eN of the transistros 
Q.sub.1, Q.sub.2, Q.sub.9 and Q.sub.10 is an expressed as follows: 
EQU r.sub.eN =(KT/q)/(I.sub.0 /2) (36) 
where K is Boltzmann's constant, T is absolute temperature, and q is 
electric charge quantity of an electron. The potentials applied to input 
terminals 29 and 30 are supposed to be V.sub.4 and V.sub.5. When the 
potentials V.sub.4 and V.sub.5 are changed, the emitter current flowing in 
the transistors Q.sub.1 and Q.sub.2 varies. At the same time, the emitter 
current flowing in the transistors Q.sub.9 and Q.sub.10 varies, and 
accordingly the potentials V.sub.6 and V.sub.7 of the emitters of the 
transistors Q.sub.9 and Q.sub.10 are also changed. At this time, the rate 
of change of voltage (V.sub.7 -V.sub.6) to the voltage (V.sub.4 -V.sub.5), 
that is, the voltage amplification factor G.sub.N is given in the 
following equation: 
EQU G.sub.N =d(V.sub.6 -V.sub.7)/d(V.sub.4 -V.sub.5)=r.sub.eN /(r.sub.eN 
+R)(37) 
where R is the resistance value of resistance elements 26 and 27, and it 
functions to increase the emitter series resistance of the transistors 
Q.sub.9 and Q.sub.10. Supposing the collector current of the transistor 
Q.sub.21 to be I.sub.x, each emitter current and collector current of the 
transistors Q.sub.3 and Q.sub.4 are I.sub.x /2. Therefore, the 
differential emitter resistance r.sub.ep of the transistors Q.sub.3 and 
Q.sub.4 is given as follows: 
EQU r.sub.er =(KT/q)/(I.sub.x /2) (38) 
At this time, when the current value flowing in the constant current 
sources 31 and 32 is set at I.sub.x /2, the current flowing in from the 
input terminals 29 and 30 becomes zero, which is very convenient. The rate 
of change of collector currents of the transistors Q.sub.3 and Q.sub.4 by 
the change of the voltage (V.sub.7 -V.sub.6) generated between the bases 
of the transistors Q.sub.3 and Q.sub.4, that is, the voltage-current 
conversion factor (mutual conductance) g.sub.mp is given as follows. 
EQU g.sub.mp =1/r.sub.ep (39) 
The change of the collector current of transistors Q.sub.3 and Q.sub.4 is 
equal to the change of the currents I.sub.1 and I.sub.2 flowing in from 
the input terminals 29 and 30. Therefore, the rate of change of currents 
I.sub.1 and I.sub.2 by the changes of the potentials V.sub.4 and V.sub.5, 
d(I.sub.1 -I.sub.2)/d(V.sub.4 -V.sub.5), is given as follows. 
##EQU26## 
Futhermore, since the change of current I.sub.1 and change of current 
i.sub.2 are equal to each other, the resistance value seen from the input 
terminals 29 and 30, d(V.sub.4 -V.sub.5)/dI.sub.1 is expressed as follows. 
##EQU27## 
As known from the equation (41), the value of the resistance is equal to 
the product of the differential emitter resistance r.sub.ep of transistors 
Q.sub.3 and Q.sub.4 multiplied by the coefficient {2(r.sub.eN +R)/r.sub.eN 
}. Here, since the resistance value R of the resistances 26 and 27 is a 
positive value, the coefficient {2(r.sub.eN +R)/r.sub.eN } is greater than 
2. To change the resistance value seen from the input terminals 29 and 30, 
meanwhile, the voltage value of the variable voltage source V.sub.3 is 
changed and the collector current I.sub.x of the transistor Q.sub.21 is 
changed, thereby varying vary the differential emitter resistance 
r.sub.ep. 
According to the embodiment shown in FIG. 10, as compared with the 
variation width of the differential emitter resistance r.sub.ep, the 
resistance value can be varied in a by more than twice the range. 
In the embodiment shown in FIG. 10, meanwhile, the collector of the 
transistor Q.sub.3 is connected to the base of the transistor Q.sub.2, and 
the collector of the transistor Q.sub.4 is connected to the base of the 
transistor Q.sub.1, but instead the collectors of the transistors Q.sub.3 
and Q.sub.4 may be mutually exchanged. In this case, as the resistance 
value seen from the input terminals 29 and 30, a negative resistance may 
be realized. Moreover, in the embodiment in FIG. 10, the transistors 
Q.sub.9 and Q.sub.10 are used as the resistive load of the transistors 
Q.sub.1 and Q.sub.2, but it is also possible, needless to say, to use an 
ordinary resistance element as shown in FIG. 2. 
Thus, according to the embodiment shown in FIG. 10, the resistance value 
can be changed in a broader range than the variation of the differential 
emitter resistance of a transistor, and the variable range of the 
resistance value can be freely changed by changing the current value of 
the current source connected to the transistor pair of the differential 
amplifier circuit. 
FIG. 11 relates to an eighth embodiment of the invention. In FIG. 11, 
variable capacitance circuits 100, 101, substantially equivalent to the 
composition shown in FIG. 2, are connected in cascade. A capacitance 
element 33 is connected between the emitters of the transistors Q.sub.1, 
Q.sub.2 for composing the differential amplifier circuit 1 of the variable 
capacitance circuit 101. The capacitance value of this capacitance element 
33 is supposed to be C.sub.2. The resistance value of the resistance 
element 34 connected to the base of the transistor Q.sub.1 of the variable 
capacitance circuit 100 is supposed to be R.sub.34, and the resistance 
value of the resistance element 35 connected between the reference 
potential and the base of the transistor Q.sub.2 of the variable 
capacitance circuit 100 is supposed to be R.sub.35. The rest of the 
composition is the same as in FIG. 2. 
This embodiment is intended to obtain a large (infinite, theoretically) 
capacitance on the basis of the following principle. 
Supposing the capacitance values of two capacitance elements to be 
-C.sub.A, C.sub.B, and the synthetic capacitance by connecting them in 
series to be C, the following relation is obtained. 
##EQU28## 
Therefore, it follows that 
##EQU29## 
and when C.sub.A =C.sub.B, C is infinite. Generally, in a semiconductor 
integrated circuit, since the relative precision of elements is extremely 
high, when the circuit is designed so that, for example, C.sub.A =1.01 
C.sub.B, C is 
##EQU30## 
and a capacitance value of over 100 times will be easily capacitance 
obtained. By the same principle, a negative capacitance may be realized. 
Next, according to the composition shown in FIG. 11, its operation is 
explained below. 
Supposing the base AC voltage of the transistor Q.sub.1 of the variable 
capacitance circuit 100 to be V.sub.o, and the base AC voltage of the 
transistor Q.sub.2 to be V.sub.b, they are 
##EQU31## 
Here, assuming the two variable capacitance circuits 100, 101 to be 
symmetrical, and R.sub.2 =R.sub.3, from equations (45), (46), it follows 
that 
##EQU32## 
Putting equation (47) into equation (45) yields 
##EQU33## 
From equation (48), it is found that 
##EQU34## 
Therefore, when C.sub.1 =C.sub.2, Q is infinite, and a secondary low pass 
filter with an extremely high selectivity will be realized. 
On the other hand, when the resistance value R.sub.35 is an extremely large 
value, it results in 
##EQU35## 
Hence, when C.sub.1 =C.sub.2, a primary low pass filter having 35 
equivalently an extremly large capacitance value 
##EQU36## 
may be realized. 
Along with the advancement in the degree of integration of the 
semiconductor integrated circuit, it is demanded to incorporate external 
parts, in particular, capacitance elements with large capacitance value 
into a semiconductor chip. Generally, in a semiconductor integrated 
circuit, the relative precision of elements, such as capacitors, resistors 
and transistors is extremely high, and therefore it is quite easy to set 
the capacitance values C.sub.1, C.sub.2 of the two capacitance elements 3 
and 33 in FIG. 11 nearly equal to each other (for example, C.sub.1 =1.01 
C.sub.2). Therefore, hitherto, within the semiconductor chip, it was bound 
to realize a capacitance value of only about 100 pF from an economical 
point of view, while, by contrast, in the composition as shown in FIG. 11, 
an extremely large capacitance value may be realized in a semiconductor 
integrated circuit.