Undesired signal canceller

Apparatus for cancelling undesired signals for a radient energy receiver having first and second antennas which are spaced a distance apart and including a phase compensation circuit connected to one of the antennas and a variable phase shift circuit connected to the second antenna with the variable phase shift circuit comprising a plurality of series connected inductors wherein adjacent inductors are magnetically coupled together and a plurality of voltage variable capacitors with adjacent ones of the voltage variable capacitors connected between ground and the junction points between adjacent ones of the series connected inductors and adding means connected to the outputs of the phase compensation circuit and the variable phase shift circuit and means for supplying a variable DC voltage to the variable voltage capacitors to control their capacitance.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention relates generally to an undesired signal canceller, and is 
directed more particularly to an undesired signal canceller for use in a 
ghost canceller and so on of a television receiver. 
2. Description of the Prior Art 
In the art, an undesired signal canceller shown in FIG. 1 is already 
proposed. In the example of FIG. 1, two antennas 1 and 2 are arranged 
apart with a distance d. The signal received by the antenna 1 is applied 
through a phase compensation circuit 3 with the phase shift of .phi.1 to a 
subtraction type circuit 4, while the signal received by the antenna 2 is 
supplied to the subtraction type circuit 4 through a variable phase 
shifter 5 with the phase shift of .phi.2. 
In this case, it is known that the total directivity coefficient D(.theta.) 
of the above antenna device or arrangement is expressed as follows: 
##EQU1## 
where k is a propagation constant and f(.theta.) is the directivity of 
each antenna. 
Accordingly, in order to make a null point or null direction (which is a 
point when the receiving sensitivity is 0) on the direction angle .theta., 
it is sufficient if the following equation (2) is established. 
##EQU2## 
In the above equation (2), if n=0 and .phi.2-.phi.1 is varied within the 
range of .+-.kd, the null point can be presented in a desired direction. 
Further, if .phi.2-.phi.1 is proportional to a frequency 
F(.phi.2-.phi.1=KF), k(=(2.pi.F)/(C), C is the velocity of light) is also 
proportional to the frequency F. Therefore, a null point .theta..sub.N is 
expressed as follows: 
##EQU3## 
Thus, the null point .theta..sub.N can be set in a constant direction 
irrespective of received signal frequencies. 
In this case, a coaxial cable is generally employed as the phase shifter 5 
whose phase shift amount is proportional to the frequency F. 
In this case, however, since the phase shift amount of the coaxial cable 
corresponds to its length, the phase adjustment thereof is very difficult. 
For this reason, a plurality of coaxial cables are prepared and they are 
switchably used. This adjustment, however, is rather complicated and it is 
also troublesome to change the null point after it has once been. Further, 
it is impossible to adjust the null point while a user stays near a 
television receiver. 
OBJECTS AND SUMMARY OF THE INVENTION 
Accordingly, an object of this invention is to provide a novel undesired 
signal canceller free from the defects inherent to the prior art. 
Another object of the invention is to provide an undesired signal canceller 
by which a null point can be easily adjusted. 
A further object of the invention is to provide an undesired signal 
canceller in which once a null point has been adjusted is not varied for 
received signal frequencies. 
According to an aspect of the present invention, an undesired signal 
canceller is provided which comprises: 
(A) a pair of antennas arranged in parallel for receiving an input signal 
within a predetermined frequency range, said input signal including 
desired and undesired signals; 
(B) a variable phase shifter connected to one of said pair of antennas, 
said variable phase shifter including a low pass filter network consisting 
of a plurality of L-C stages each of which includes a coil and a 
capacitor; and 
(C) an adding circuit connected between said variable phase shifter and the 
other of said pair of antennas, the degree of coupling between two 
neighboring coils of said low pass filter network is sequentially designed 
such that said undesired signal is effectively eliminated in the output of 
said adding circuit regardless of the frequency of said input signal 
within said predetermined frequency range. 
The other objects, features and advantages of the present invention will 
become apparent from the following description taken in conjunction with 
the accompanying drawings through which the like references designate the 
same elements and parts.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
This invention will be hereinafter described with reference to the attached 
drawings. 
In an example of the invention, a low pass filter type variable phase 
shifter is used as the phase shifter 5 in the prior art example of FIG. 1. 
An example of the low pass filter type variable phase shifter used in the 
invention will be now described with reference to FIG. 2. As shown in FIG. 
2, the variable phase shifter of this example is formed of a plurality of 
variable capacitance elements 11.sub.1 to 11.sub.n each of which is 
grounded at its one end and a plurality of coils 12.sub.1 to 12.sub.n-1 
which are respectively connected between other ends of adjacent variable 
capacitance elements. Further, coils 15a and 15b, each of which has an 
inductance which is one half of that of each of the coils 11.sub.1 to 
11.sub.n, are connected between an input terminal 13 and the first 
variable capacitance element 11.sub.1 at the input side and between the 
last variable capacitance element 11.sub.n at the output side and an 
output terminal 14, respectively. 
The amount of phase shift of this variable phase shifter is expressed as 
follows: 
##EQU4## 
where n is the number of L-C stages, C.sub.0 is the capacitance value of 
each of the variable capacitance elements 11.sub.1 to 11.sub.n and L.sub.0 
is the inductance value of each of the coils 12.sub.1 to 12.sub.n-1. 
Accordingly, if in the above variable phase shifter, for example, voltage 
controlled variable capacitance diodes are used as the variable 
capacitance elements 11.sub.1 to 11.sub.n and C.sub.0 thereof is varied, a 
certain desired amount of phase shift can be obtained. 
In the invention, a phase compensation circuit, which is formed to have a 
construction similar to that of the above variable phase shifter 5, is 
used as the phase compensation circuit 3 of the prior art. In this case, 
the capacitance value of the respective capacitance elements 11.sub.1 to 
11.sub.n shown in FIG. 2 is fixed at a desired value to form the 
capacitance elements 11.sub.1 to 11.sub.n as fixed capacitor. 
Turning to FIG. 3, an example of the antenna device or arrangement used in 
the invention, in which the above variable phase shifter and phase 
compensation circuit are used, will be described. 
In the example of FIG. 3, the signals from the antennas 1 and 2 are derived 
through a switching circuit 7. For example, the signal from the antenna 2 
through the switching circuit 7 is supplied to a subtraction type circuit 
4 through the variable phase shifter 5, while the signal from the antenna 
1 through the switching circuit 7 is supplied to the circuit 4 through the 
phase compensation circuit 3. The transmission line from the phase shifter 
5 and that from the phase compensation circuit 3 are connected together 
through a primary winding 31 of a transformer in the adding circuit 4. A 
secondary winding 32 of the transformer has one end grounded through a 
capacitor 33 and the other end is connected to a terminal 34. The windings 
31 and 32 are connected through a choke coil 35, and the middle point of 
the primary winding 31 is grounded through a capacitor 36 and a resistor 
37. 
Thus, in the antenna device of FIG. 3, the signals from the variable phase 
shifter 5 and phase compensation circuit 3 are combined and delivered to 
the terminal 34 and then to the antenna input terminal of a television 
receiver (not shown). When at the side of the television receiver a 
control DC voltage is superimposed on the transmission line connected to 
the terminal 34, this DC voltage is applied through the choke coil 35 and 
the primary winding 31 to the variable phase shifter 5 and the capacitive 
value of its variable capacitance diodes 11.sub.1 to 11.sub.n are 
controlled. 
In this case, the impedance value Z.sub.0 of the variable phase shifter 5 
is given by the following formula (5); 
##EQU5## 
And, if the variable range of the respective capacitive value C.sub.0 of 
the variable capacitance diodes 11.sub.1 to 11.sub.n is selected in the 
range from 3 to 12pF, the design value of C.sub.0 is selected as 6pF and 
the impedance value Z.sub.0 is selected as 75.OMEGA., the inductance value 
of each of the coils 12.sub.1 to 12.sub.n-1 becomes 33.8 nH. 
Further, when the number of L-C stages in the variable phase shifter 5 is 
selected as 10, its amount of phase shift .phi.2 becomes as follows: 
##EQU6## 
The amount of phase shift .phi.1 of the phase compensation circuit 3 is 
given by the following formula (7) when the capacitive value of each of 
the capacitors is selected as 7pF, the inductance value of each of the 
coils is selected as 36 nH and the number of L-C stages is selected to be 
4, 
##EQU7## 
Accordingly, the null point .theta..sub.N of the antenna device shown in 
FIG. 3 is given as follows: 
##EQU8## 
When d=135 cm, the expressions (6) and (7) are substituted into the 
expression (8) and the variable range of C.sub.0 is selected from 3 to 
12pF, the null point .theta..sub.N is varied as in the following table in 
which 1 to 12 channels show the Japanese television channels. 
TABLE 
______________________________________ 
C.sub.0 Received Channel 
.theta..sub.N 
______________________________________ 
3pF 1 13.034 
3pF 2 13.029 
3pF 3 13.025 
3pF 4 13.916 
3pF 5 13.981 
3pF 6 14.025 
3pF 7 14.047 
3pF 8 14.051 
3pF 9 14.047 
3pF 10 14.035 
3pF 11 14.019 
3pF 12 14.000 
6pF 1 32.130 
6pF 2 32.156 
6pF 3 32.184 
6pF 4 32.626 
6pF 5 32.676 
6pF 6 32.728 
6pF 7 32.782 
6pF 8 32.820 
6pF 9 32.878 
6pF 10 32.939 
6pF 11 33.003 
6pF 12 33.070 
9pF 1 49.726 
9pF 2 49.946 
9pF 3 50.115 
9pF 4 50.798 
9pF 5 51.050 
9pF 6 51.311 
9pF 7 51.603 
9pF 8 51.746 
9pF 9 51.920 
9pF 10 52.016 
9pF 11 52.054 
9pF 12 52.071 
12pF 1 76.985 
12pF 2 77.007 
12pF 3 77.001 
12pF 4 83.998 
12pF 5 85.000 
12pF 6 84.999 
12pF 7 83.999 
12pF 8 83.001 
12pF 9 82.998 
12pF 10 82.998 
12pF 11 84.001 
12pF 12 84.001 
______________________________________ 
According to the simple analysis of the antenna device shown in FIG. 3, as 
will be apparent from the above table, the null point .theta..sub.N varies 
in accordance with the received signal frequencies. 
In view of the above point, according to the invention, the coupling 
inductance or mutual inductance M between arbitrary adjacent coils of the 
phase compensation circuit 3 and the variable phase shifter 5 in the 
antenna device of FIG. 3 is selected to satisfy the following formula. 
EQU (M/L.sub.0 .perspectiveto.-0.2 
Thus, the non-linearity of the phase shifting amount can be remarkably 
improved. 
If a simple fundamental L-C phase shifting circuit shown in FIG. 4 is 
considered, the four-terminal parameters of this circuit become as 
follows: 
##EQU9## 
where z is j.omega. (L.sub.0 -M) and y is j.omega.C/(1-.omega..sup.2 CM). 
Further, the image impedance Z.sub.0 of this synmetrical four-terminal 
network becomes as follows: 
##EQU10## 
The phase constant .beta..sub.I of this circuit becomes as follows in the 
range (pass band) of A&lt;1: 
##EQU11## 
When the factor representing the degree of coupling between adjacent coils 
is taken as k(=M/L.sub.0) and the factor representing variation of 
capacitance is taken as K(=C/C.sub.0), the above formulae (10) and (11) 
can be respectively rewritten as follows: 
##EQU12## 
Accordingly, the amount of group delay d.beta..sub.I /d.omega. is expressed 
as follows: 
##EQU13## 
If the above equation (14) becomes constant for the received signal 
frequency of the VHF band, the phase characteristic of the circuit shown 
in FIG. 4 becomes proportional to the received signal frequency. 
If the equation (14) is differentiated again, the following equation is 
obtained. 
##EQU14## 
If the equation (15) is for the received signal frequency in the VHF band, 
it is sufficient. Accordingly, the following equation (16) is derived from 
the equation (15). 
EQU f(x,k)=B 5k+1-3a(1+k)x.sup.2 =5k+1-6Kx (16) 
The condition which makes the equation (16) zero is sufficient. 
The variation of f(x, k), when K=1 and k is taken as a parameter, is shown 
in the graph of FIG. 5. 
From the graph of FIG. 5 it will be apparent that if the degree of coupling 
k is selected as about (k=)-0.2, the amount of phase shift .beta..sub.I 
can be varied approximately linearly for the frequency. 
FIGS. 6A, 6B and 6C are graphs respectively showing the frequency to amount 
of phase shift .beta..sub.I characteristics when eight stages of L-C unit 
circuits, each being selected to have k=-0.2, are connected in cascade. In 
these graphs, a phase error .DELTA..beta..sub.I is the deviation from the 
straight line (error from linearity) connecting the respective amounts of 
phase shift at the frequencies of 0 and 90 MH.sub.z, which is calculated. 
As will be apparent from the graphs of FIGS. 6A to 6C, when it is selected 
that k=-0.2 the error from linearity will fall within .+-.5.degree. in the 
range of 0.5&lt;K&lt;2.0. 
In the above analysis, it is assumed that the circuit of FIG. 4 is formed 
of a lossless transmission line. In fact, however, there is a loss caused 
by the deterioration in Q of the L-C circuit formed by the coil and the 
variable capacitance diode, and this loss poses a problem when a plurality 
of the circuits, such as shown in FIG. 4, are connected in cascade. 
Therefore, a case where n-stages of the circuits shown in FIG. 4 are 
connected in cascade will be analized while the resistance R of its coils 
and the conductance G of its capacitors are taken into consideration. That 
is, since the phase characteristic is approximately linear as shown in the 
previous analysis, the circuit formed by connecting n-stages of the 
circuits of FIG. 4 in cascade may be considered equivalent to a 
distributed constant circuit. Accordingly, when a transmission signal is 
of high frequency, the propagation constant r=(.alpha.+j.beta.) is given 
by the following equations (17) and (18). 
##EQU15## 
where .omega..sub.L &gt;&gt;R and .omega..sub.C &gt;&gt;G. 
Thus, when the transmission line which has the characteristic impedance 
Z.sub.0 and the propagation constant r as shown in FIG. 7 is considered, 
the relationship of voltage V.sub.0 to, current I.sub.0 between input 
terminals 13a and 13b to voltage V.sub.l, to current I.sub.l between 
output terminals 14a and 14b becomes as follows: 
##EQU16## 
In this case, when the transmission line is terminated with the terminal 
resistance R.sub.L =R.sub.0, the actual transmission coefficient S.sub.B 
is given as follows: 
##EQU17## 
From the above S.sub.B, the actual amount of attenuation .alpha..sub.B is 
given as follows: 
EQU .alpha..sub.B =10 log .vertline.S.sub.B .vertline..sup.2 [dB](21) 
While, the propagation constant r.sub.n when n-stages of the circuits are 
connected in cascade is given as follows: 
EQU r.sub.n =n.alpha.+jn.beta..sub.I (22) 
The characteristic impedance Z.sub.0 and the phase constant .beta..sub.I 
are given by the equations (12) and (13), respectively. 
Accordingly, from the equations (19) and (20), the following equation (23) 
is derived. 
##EQU18## 
And, the actual phase .beta..sub.B becomes as follows: 
##EQU19## 
From the above equations (21) and (24), when the degree of coupling k=-0.2 
is selected and number of stages n=10, the inductance of the coil of each 
stage L=25.0 nH and that the voltage V.sub.c applied to the variable 
capacitance diode is taken as a parameter, the theoretical values of the 
frequency characteristic of errors .DELTA..alpha..sub.B and 
.DELTA..beta..sub.B from linearity of the actual attenuation degree 
.alpha..sub.B of the frequency characteristic and actual phase 
.beta..sub.B become as indicated by the solid lines in the graphs of FIGS. 
8 and 9, respectively. Further, the converting characteristic curve of the 
applied voltage V.sub.c to Q value of the variable capacitance diode 
becomes as shown in the graph of FIG. 10. 
As will be apparent from the above graphs, by selecting the degree of 
coupling k=-0.2, the errors .DELTA..alpha..sub.B and .DELTA..beta..sub.B 
from linearity of the actual degree of attenuation .alpha.HD B and the 
amount of phase shift .beta..sub.B can be greatly reduced. 
Therefore, if, for example, as shown in FIG. 11 conductive line patterns 
are formed or coated on both surfaces of a circuit board having a 
predetermined thickness to locate respective coils in close proximity, the 
coil arrangement can be formed in which the degree of coupling k between 
adjacent certain coils is -0.2. In detail, as shown in FIG. 11, on both or 
front and rear surfaces of the circuit board 51, respectively are formed 
printed coils 52a and 52b each formed of, for example, a spiral form of 
metal foil. In FIG. 11, the solid line represents the printed coil 52a on 
the front surface of the circuit board 51 and the broken line represents 
coil 52b on the rear surface thereof, respectively. In this case, the coil 
52b on the rear surface of the circuit board 51 is displaced from the coil 
52a on the front surface in the directions both to the right and down by 1 
mm and the unit of the numerals on the sheet is in millimeters. 
At the center portion 53 of the spiral of each of the printed coils 52a and 
52b, a bore 54 is formed through the circuit board 51, and the cathode 
lead of the variable capacitance diode 11 is inserted into the bore 54 and 
soldered to the printed coils 52a and 52b on the front surface and rear 
surface of the circuit board 51, respectively. 
Thus, the circuit portion shown in FIG. 11 which is surrounded by the 
one-dot chain line is equal to the fundamental circuit shown in FIG. 4. In 
this case, through the printed coils 52a and 52b, currents respectively 
flow in the direction indicated by the solid and broken lines arrows, in 
other words, through the opposing portions of printed coils 52a and 52b 
the currents flow in the same direction, so that the coupling therebetween 
is negative. By selecting the size and shape of the printed coils 52a and 
52b as indicated in FIG. 11, the inductance L of the coils can be set as, 
for example, 25.0 nH and the degree of coupling k between the coils can be 
set at -0.2. 
FIG. 12 is a graph showing the frequency characteristic of the printed 
coils 52a and 52b at Q values, and FIG. 13 is a graph showing the actually 
measured values of the frequency characteristic of the resistance values 
of the printed coils 52a and 52b, respectively. 
When the circuit surrounded by the one-dot chain line in FIG. 11 is 
connected in cascade with 10 stages, the actual measured values of the 
frequency characteristics of errors .DELTA..alpha..sub.B and 
.DELTA..beta..sub.B from linearity of the frequency characteristic of the 
actual degree of attenuation .alpha..sub.B and actual amount of phase 
shift .beta..sub.B respectively become as shown in the graphs of FIGS. 8 
and 9 by the broken lines which are approximately equal to the theoretical 
or calculated values. 
FIG. 14 is a graph showing an example of the directional characteristics of 
an undesired signal canceller which uses the above coil device, in which 
C.sub.0 =3p.sup.F. From this graph it will be apparent that the cancelling 
ratio of about -30 dB is obtained at the null point .theta..sub.N. 
The above example is the case where the undesired signal arrives from the 
right side relative to a desired signal, but in the case where the 
undesired signal arrives from the left side, by changing the switching 
circuit 7, the null direction can be formed at the left front. 
As described above, if the degree of coupling between the adjacent coils in 
the circuit of FIG. 3 is selected as about -0.2, it is apparent that the 
null direction can be stabilized regardless of the received signal 
frequencies. 
In the circuit of FIG. 3, the control bias voltage (V.sub.c) versus 
capacity (C.sub.0) characteristic of the variable capacitance diode is 
now-linear as shown in the graph of FIG. 15. As a result, an electric 
field of, for example, more than 90 dB is applied thereto, and beat 
interference may be generated in the circuit of FIG. 3. That is, when the 
signals of, for example, the first and third channels are mixed, a beat 
interference may be caused in the eighth channel. 
An improved phase shifter according to the invention, which will avoid the 
generation of the above beat interference, is shown in FIG. 16, the 
circuit elements corresponding to those of the phase shifter of FIG. 2 or 
3 are marked with the corresponding reference numerals and letters. 
In the example of FIG. 16, capacitors 17 and 18 are respectively connected 
to the input and output terminals 13 and 14, and a fixed bias is applied 
through a resistor 19 to the input coil 15a from a terminal 20. Further, 
the polarities of diodes 11.sub.1 to 11.sub.n are alternately made 
opposite, and the cold ends of the respective diodes are individually 
grounded through capacitors 22.sub.1, 22.sub.2 - - - 22.sub.n. The 
connection points between the anodes of every other diodes 11.sub.1, 
11.sub.3 - - - 11.sub.n-1 and the capacitors 22.sub.1, 22.sub.3 - - - 
22.sub.n-1 are connected together through resistors 21.sub.1, 21.sub.3 - - 
- 21.sub.n-1 to a bias terminal 23a, and the connection points between the 
cathodes of the diodes 11.sub.2, - - - 11.sub.n and the capacitors 
22.sub.2 - - - 22.sub.n are connected together through resistors 21.sub.2 
- - - 21.sub.n to another bias terminal 23b. 
In the circuit of FIG. 16, the bias voltage at the terminal 20 is fixed, 
and two control voltages, which vary oppositely with the fixed bias 
voltage as the center, are applied to the bias terminals 23a and 23b, 
respectively, so that the capacitance value C.sub.0 of the diodes 11.sub.1 
to 11.sub.n is controlled in a manner similar to the example of FIG. 3, 
whereby a desired amount of phase shift .phi. is obtained. 
The fact that the circuit of FIG. 16 greatly reduces the beat interference 
as compared with the circuit of FIG. 3 will be now explained in detail. 
The capacitance (C.sub.0)-DC bias voltage (V.sub.c) characteristic of the 
variable capacitance diodes is approximately expressed by the following 
equation (25). 
##EQU20## 
where 2 V.ltoreq.V.sub.c .ltoreq.13 V, a=3.6728.times.10.sup.10, 
b=1.7034.times.10.sup.10, c=-1.1924.times.10.sup.9 and 
d=8.0065.times.10.sup.7. 
Now, if it is assumed that when a signal .DELTA.e is applied to a variable 
capacitance diode which is biased with a DC voltage V.sub.c, its 
capacitance C(V.sub.c) changes by .DELTA.C and, the following equation is 
established. 
EQU C=A.sub.1 .DELTA.e+A.sub.2 (.DELTA.e).sup.2 +A.sub.3 (.DELTA.e).sup.3 + - - 
- (26) 
where 
##EQU21## 
Thus, since the relation of .DELTA.e to .DELTA.C becomes non-linear when, 
an input with a large amplitude is applied, higher harmonic components are 
generated. 
Now, consideration is given to the generation of how much harmonic 
distortion occurs when the above variable capacitance diode is applied to 
a phase shifter. Thus, such a circuit will be considered in which a 
fundamental unit circuit is connected in cascade with two stages as shown 
in FIG. 17. In this case, it is assumed that the load and the signal 
source resistances are each equal to the characteristic impedance Z.sub.l 
of the whole phase shifter. However, 
EQU Z.sub.1 =j.omega.(L-M) 
where 
M/L=-0.2 and 
##EQU22## 
where Z.sub.0 is the characteristic impedance of the phase shifter and 
C.sub.0 is the various capacitance by the DC bias value. 
When the signal source of e is applied to the above phase shifter, voltages 
e.sub.g1 and e.sub.g2 applied across the variable capacitance diodes of 
the circuits 1 and 2 in the phase shifter of FIG. 17 are given by the 
following equations. 
EQU e.sub.g1 =K.sub.107 e (27) 
EQU e.sub.g2 =g.sub.107 K.sub.107 e (28) 
where 
##EQU23## 
Under the assumption that when the voltages e.sub.g1 and e.sub.g2 are 
applied to the variable capacitance diiodes, harmonic distortion is 
generated, current sources for currents i.sub.1 and i.sub.2 which are 
shown in FIG. 18A and also given by the following equations will be 
considered: 
##EQU24## 
Further, since the amplitude of the harmonic components is small as 
compared with that of the fundamental wave component, it is considered 
that in the section other than that in question harmonic components are 
generated by only the fundamental wave component. The output e.sub.out of 
the circuit shown in FIG. 18A may be considered as follows. That is, the 
circuit of FIG. 18A is separated as shown in FIGS. 18B and 18C, and 
outputs e.sub.1 out and e.sub.2 out thereof are added to provide the 
output e.sub.out. The outputs e.sub.1 out and e.sub.2 out are given by the 
following equations (31) and (32), respectively. 
##EQU25## 
Therefore, 
##EQU26## 
From the above discussion, it will be considered that two sine waves which 
are different in frequency are simultaneously applied to the above 
non-linear circuit i.e. the signal e expressed by the following equation 
(34) is applied to the non-linear circut. 
EQU e=E.sub.1 cos .omega..sub.1 t+E.sub.2 cos .omega..sub.2 t (34) 
If the above equation (34) is substituted into the equations (27) and (28), 
the voltages e.sub.g1 and e.sub.g2 across the variable capacitance diodes 
respectively become as follows: 
EQU e.sub.g1 =K.sub.107 1 E.sub.1 cos .omega..sub.1 t+K.sub.107 2 E.sub.2 cos 
.omega..sub.2 t (35) 
EQU e.sub.g2 =g.sub.107 2 K.sub.107 1 E.sub.1 cos .omega..sub.1 t+g.sub.107 2 
K.sub.107 2 E.sub.2 cos .omega..sub.2 t (36) 
If the above equations (35) and (36) are substituted into the equations 
(29) and (30), respectively, the current sources of i.sub.1 and i.sub.2 
are obtained as follows: 
##EQU27## 
The above equations are developed, and considerating only the components of 
.omega..sub.1 +.omega..sub.2, they are substituted into the equations (31) 
and (32), respectively. Then, the outputs e.sub.1 out and e.sub.2 out can 
be expressed as follows: 
EQU e.sub.1 out =R.sub.107 1 B.sub..omega.1 sin.sub.107 1 t+R.sub.107 1 
+2B.sub.107 1 +.omega.2 sin (.omega.1+.omega.2)t (39) 
EQU e.sub.2 out =r.sub..omega.1 b.sub..omega.1 sin .omega.1t+r.sub..omega.1 
+.omega.2b.sub..omega.1 +.omega.2 sin (.omega.1+.omega.2) t (40) 
where 
##EQU28## 
Accordingly, the total output e.sub.out becomes as follows: 
##EQU29## 
From the above, the intermodulation product IP is given as follows: 
##EQU30## 
Next, a case will be considered in which the polarities of the variable 
capacitance diodes are alternately inverted. 
When the polarity of the DC bias to the variable capacitance diode is 
reversed, .DELTA.C is given from the equation (26) as follows: 
EQU .DELTA.C=A.sub.1 '.DELTA.e+A.sub.2 '(.DELTA.e).sup.2 +A.sub.3 
'(.DELTA.e).sup.3 + - - - (44) 
where 
##EQU31## 
and the dash shows the case where the polarity is inverted. 
The relation between A.sub.n in the equation (26) and A.sub.n ' in the 
equation (44) becomes as follows: 
EQU A.sub.n =(-1).sup.n A.sub.n ' (45) 
Accordingly, if the above relation is used, the intermodulation product can 
be similarly obtained even in the case where the polarity of the DC bias 
is alternately inverted. That is, when as shown in FIG. 19 the polarity of 
the rear stage of the variable capacitance diode in the fundamental 2 
stages is inverted, the following relation is established from the 
equation (45). 
##EQU32## 
So, from the equation (41), the following equation is established. 
##EQU33## 
As a result, the intermodulation products IP' when the polarity of the DC 
bias is inverted can be expressed as follows: 
##EQU34## 
From the above theoretical analysis, the intermodulation products in the 
case that the polarities of the DC biases to the variable capacitance 
diodes are not inverted but selected to be the same (refer to the equation 
(43)) and the intermodulation products in the case where the polarities 
are alternately inverted (refer to the equation (48)) are respectively 
obtained. 
When the equations (43) and (48) are compared, it is understood that in the 
case where the polarities of the DC biases to the variable capacitance 
diodes are inverted as in FIG. 19, the factors of .omega.1 (denominator) 
is the form of the sum and the same as that of the former case but the 
factor of .omega.1+.omega.2 (numerator) is expressed in the form of 
subtraction which is different from the former case. Therefore, if the 
following relation is established, the value of intermodulation products 
can be made zero. 
EQU R.sub..omega.1 +.omega.2B.sub..omega.1 +.omega.2=r.sub..omega.1 
+.omega.2b.sub..omega.1 +.omega.2 
Now, a case where the bias value ratio to the variable capacitance diode is 
worst is assumed, and then a practical value is applied for comparison. 
That is, under the condition that f.sub.1 =91.25 MH.sub.z, f.sub.2 =103.25 
MH.sub.z, E.sub.1 =E.sub.2 =45.times.10.sup.-3 V(90 dB.mu.), L=25.sub.n H, 
M=5.sub.n H and V.sub.c =+3 V(C.sub.0 =12.6.sub.p F), if it is calculated 
how much the intermodulation product is improved, the following results 
are obtained. 
1. When the polarities of the DC biases are all selected in the same 
direction, the intermodulation product IP is given from the equation (43) 
as follows: 
EQU IP=-45.02 dB 
2. When the polarities of the DC biases are varied alternately, the 
intermodulation product PI' is given from the equation (48) as follows: 
EQU IP'=-65.08 dB 
From the above, the degree of improvement of the intermodulation product 
becomes as follows: Degree of improvement of intermodulation 
product=-20.06 dB 
Now, a case where the L-C fundamental circuits of 10 stages are connected 
in cascade will be considered. At first, the intermodulation product 
IP.sub.n in the case where the L-C fundamental circuits of n stages are 
connected in cascade is approximately expressed as follows: 
EQU IP.sub.n =20 log {(1+intermodulation product of 1 stage).sup.n -1} 
Since the value is obtained under the assumption that 2 stages are one set, 
when the intermodulation product is calculated as n=5, the following 
results are obtained. 
1. When the polarities of the DC biases are all selected in the same 
direction, the intermodulation product IP becomes as follows: 
EQU IP=-30.96 dB 
2. When the polarities of the DC biases are varied alternately, the 
intermodulation product IP' becomes as follows: 
EQU IP'=-51.09 dB 
In general, if the intermodulation product is selected higher than -50 dB, 
no problem appears on a picture screen. Therefore, it will be understood 
that the phase shifter of the invention shown in FIG. 16 is sufficient for 
avoiding the harmonic distortion. 
Further, the merits of the circuit shown in FIG. 16 have been 
expedimentally proven. 
As described above, according to this invention, the coupling degree k 
between the coils forming the variable phase shifter is selected to be 
about or around -0.2, so that errors from linearity of the insertion loss 
and the phase shift can be reduced and hence a good characteristic is 
obtained. 
Further, in this invention when the respective coils of the variable phase 
shifter are made of printed coils, the coupling degree between the coils 
can be easily determined from the design thereof. Thus, the products can 
be made to have uniform characteristics and need not be adjusted. 
Further, in the invention when the shapes of the printed coils are selected 
to be synmetrical with respect to the front and rear sides, there is not 
generated any ununiform portions in the characteristic and the 
characteristics are improved. 
Also, in this invention when C.sub.0 is provided by the variable 
capacitance diode and can be varied by superimposing a DC voltage on the 
signal path, a user can change the null point while he stays near a 
television receiver. Thus, the adjustment becomes easy. At this time, 
since only one signal line is sufficient, the wiring and so on are simple. 
Although the present invention has been particularly shown and described 
with reference to certain preferred embodiments thereof, it will be 
readily apparent to those of ordinary skill in the art that various 
changes and modifications in form and details may be made without 
departing from the spirit and scope of the invention. Therefore, the 
spirits or scope of the invention should be determined by the appended 
claims.