TUNABLE GROUNDED POSITIVE AND NEGATIVE IMPEDANCE MULTIPLIER

Abstract

A tunable impedance multiplier with high multiplication factor is described. A single externally connected resistor is used and the multiplier is free of passive elements. The circuit can realize a positive or a negative impedance multiplier. Applications of the design to low and high pass filters are also presented. The simulation and experimental results show that the new design enjoys a multiplication factor above 400 at 2 Hz-to 7 MHz.

Claims

1. A tunable impedance multiplier, comprising: a current feedback operational amplifier, CFOA, having a current input Y, a current input X, a voltage output W and a voltage output P; a first operational transconductance amplifier, OTA.sub.1, having a gain g.sub.m, a first positive voltage input V+, a first negative voltage input V−, a first bias current input I.sub.B1, a positive supply voltage, a negative supply voltage, and a current output I.sub.o; a resistance, R.sub.o, wherein the resistance R.sub.o is connected to the voltage output P at a first end and is connected to ground at a second end; an impedance, Z, wherein the impedance Z is connected to the input X at a first end and is connected to ground at a second end; wherein the voltage output W is connected to the voltage input V of OTA.sub.1; and wherein the output I0 is connected as feedback to the current input Y of the CFOA.

2. The tunable impedance multiplier of claim 1, wherein the impedance Z is a resistor or a capacitor and the equivalent input impedance, Z.sub.in, of the impedance multiplier is given by the equation $\mathrm{Zin}=\frac{Z}{\mathrm{gm}XR0}.$

3. The tunable impedance multiplier of claim 2, wherein g.sub.m=20×I.sub.B.sub.1 and the equivalent input impedance, Z.sub.in, of the impedance multiplier is given by the equation $\mathrm{Zin}=\frac{Z}{20X\mathrm{IB}1XR0}.$

4. The tunable impedance multiplier of claim 3, wherein the impedance Z is a capacitor, C, and the equivalent capacitance of the impedance multiplier is given by
C.sub.eq=(20×I.sub.B1×R.sub.o) C.

5. The tunable impedance multiplier of claim 3, where the impedance Z is a resistor, g.sub.m=20 ×I.sub.B1 and the equivalent resistance of the impedance multiplier is given by $\mathrm{Req}=\frac{R}{20X\mathrm{IB}1XR0}.$

6. The tunable impedance multiplier of claim 1, where the resistance R.sub.o is provided by a second operational transconductance amplifier, OTA.sub.2, where the second operational transconductance amplifier includes an inverting voltage input, a non-inverting voltage input, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the inverting voltage input is connected to the voltage output P and the non-inverting voltage input is connected to ground.

7. The tunable impedance multiplier of claim 6, wherein the impedance Z is a resistor or a capacitor and the equivalent input impedance, Z.sub.in, of the impedance multiplier is given by the equation $\mathrm{Zin}=Z\frac{\mathrm{IB}2}{\mathrm{IB}1}.$

8. The tunable impedance multiplier of claim 1, wherein the resistance R.sub.o is provided by a second operational transconductance amplifier, OTA.sub.2, where the second operational transconductance amplifier includes an inverting voltage input, a non-inverting voltage input, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the non-inverting input is connected to the voltage output P and the inverting input is connected to ground.

9. The tunable impedance multiplier of claim 8, wherein the impedance Z is a resistor or a capacitor and the equivalent input impedance, Z.sub.in, of the impedance multiplier is given by the equation $\mathrm{Zin}=-Z\frac{\mathrm{IB}2}{\mathrm{IB}1}.$

10. The tunable impedance multiplier of claim 4, further comprising a low pass filter circuit including: a resistor, R.sub.1, connected at a first end to the current input Y; a voltage output, V.sub.o, connected to the first end; a first end of an alternating voltage source, V.sub.in, connected to a second end of the resistor, R1; a second end of the alternating voltage source, V.sub.in, connected to ground.

11. The tunable impedance multiplier of claim 10, where the resistor R.sub.o is provided by a second operational transconductance amplifier, OTA.sub.2, where the second operational transconductance amplifier includes an inverting voltage input, a non-inverting voltage input, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the inverting voltage input is connected to the voltage output P and the non-inverting voltage input is connected to ground.

12. The tunable impedance multiplier of claim 10, wherein the resistor R.sub.0 is provided by a second operational transconductance amplifier, OTA.sub.2, where the second operational transconductance amplifier includes an inverting voltage input, a non-inverting voltage input, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the non-inverting input is connected to the voltage output P and the inverting input is connected to ground.

13. The tunable impedance multiplier of claim 5, further comprising a high pass filter circuit including: a capacitor, C.sub.1, connected at a first end to the current input Y; a voltage output, V.sub.o, connected to the first end of the capacitor; a first end of an alternating voltage source, V.sub.in, connected to a second end of the capacitor, C1; a second end of an alternating voltage source, V.sub.in, connected to ground.

14. The tunable impedance multiplier of claim 13, where the resistor R.sub.0 is provided by a second operational transconductance amplifier, OTA.sub.2, where the second operational transconductance amplifier includes an inverting voltage input, a non-inverting voltage input, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the inverting voltage input is connected to the voltage output P and the non-inverting voltage input is connected to ground.

15. The tunable impedance multiplier of claim 13, wherein the resistor R.sub.0 is provided by a second operational transconductance amplifier, OTA.sub.2, where the second operational transconductance amplifier includes an inverting voltage input, a non-inverting voltage input, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the non-inverting input is connected to the voltage output P and the inverting input is connected to ground.

16. A method for tunably multiplying an impedance, comprising: connecting an alternating current source, i.sub.x, to a first current input of a current feedback operational amplifier, CFOA; connecting a second current input of the CFOA to an impedance, Z; connecting a first output of the CFOA to an inverting input of a first operational transconductance amplifier, OTA.sub.1 having a specified gain, g.sub.m; connecting a second output of the CFOA to a grounded resistance, R.sub.o; connecting a non-inverting input of the OTA.sub.1 to ground; connecting a current output, I.sub.o, of the OTA.sub.1 to the first current input of the CFOA; sweeping the amplitude of a first bias current source, i.sub.B1, connected to a bias input of the OTA.sub.1 over a range of frequencies of alternating current i.sub.x; determining the −3dB points of the current output corresponding to the range of frequencies; and calculating the equivalent input impedance based on the equation $\mathrm{Zin}=\frac{Z}{\mathrm{gm}XR0}.$

17. The method for tunably multiplying an impedance of claim 16, further comprising: providing the grounded resistance, R.sub.o, by connecting an inverting input of a second operational transconductance amplifier, OTA.sub.2 to a second voltage output of the CFOA, connecting the non-inverting input of the OTA.sub.2 to ground and connecting the output of the OTA.sub.2 to the inverting input of the OTA.sub.2; connecting a second bias current source, i.sub.B2, to a bias input of the OTA.sub.2; sweeping the amplitudes of the first bias current source, i.sub.B1, of the OTA.sub.1 and second bias current source, i.sub.B2, of the OTA.sub.2 over a range of frequencies of alternating current i.sub.x, wherein the value of the impedance, Z, is positive and multiplied by the factor $\frac{\mathrm{IB}2}{\mathrm{IB}1}.$

18. The method for tunably multiplying an impedance of claim 16, further comprising: providing the grounded resistance, R.sub.o, by connecting a non-inverting input of a second operational transconductance amplifier, OTA.sub.2 to a second voltage output of the CFOA, connecting the inverting input of the OTA.sub.2 to ground and connecting the output of the OTA.sub.2 to the inverting input of the OTA.sub.2; connecting a second bias current source, i.sub.B2, to a bias input of the OTA.sub.2; sweeping the amplitudes of the first bias current source, i.sub.B1, of the OTA.sub.1 and second bias current source, i.sub.B2, of the OTA.sub.2 over a range of frequencies of alternating current i.sub.x, wherein the value of the impedance, Z, is negative and multiplied by the factor $\frac{\mathrm{IB}2}{\mathrm{IB}1}.$

19. A method for tunably multiplying an impedance, comprising: connecting an alternating current source, i.sub.x, to a first current input of a current feedback operational amplifier, CFOA; connecting a second current input of the CFOA to an impedance, Z; connecting a first output of the CFOA to an inverting input of a first operational transconductance amplifier, OTA.sub.1 having a specified gain, g.sub.m; connecting a non-inverting input of the OTA.sub.1 to ground; connecting a current output of the OTA.sub.1 to the first current input of the CFOA; connecting an inverting input of a second operational transconductance amplifier OTA.sub.2 to a second voltage output of the CFOA, connecting the non-inverting input of the OTA.sub.2 to ground and connecting the output of the OTA.sub.2 to the inverting input of the OTA.sub.2; connecting a second bias current source, i.sub.B2, to a bias input of the OTA.sub.2; sweeping the amplitudes of the first bias current source, i.sub.B1, of the OTA.sub.1 and second bias current source, i.sub.B2, of the OTA.sub.2 over a range of frequencies of alternating current i.sub.x, determining the −3dB points of the current output corresponding to the range of frequencies; calculating the equivalent input impedance based on the equation $\mathrm{Zin}=\frac{Z}{\mathrm{gm}XR0};$ and calculating the multiplication of the impedance, Z, based on the equation multiplication= $Z\frac{\mathrm{IB}2}{\mathrm{IB}1}.$

20. The method for tunably multiplying an impedance of claim 19, further comprising providing the alternating current source, i.sub.x, by connecting a capacitor, C.sub.1, to the first current input of the CFOA; connecting a voltage output, V.sub.o, to the first end of the capacitor; connecting a first end of an alternating voltage source, V.sub.in, to a second end of the capacitor, C1; connecting a second end of an alternating voltage source, V.sub.in, to ground; sweeping the alternating voltage source over a range of frequencies; sweeping the amplitudes of the first bias current source, i.sub.B1, of the OTA.sub.1 and second bias current source, i.sub.B2, of the OTA.sub.2 at each of the frequencies; wherein the voltage output, V.sub.o, is high pass filtered to a subset of the range of frequencies and the amplitude of the voltage output, V.sub.o, is tuned by the multiplication of the impedance, Z.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0016] A more complete appreciation of this disclosure and many of the attendant advantages thereof will be readily obtained as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings, wherein:

[0017] FIG. 1A illustrates a circuit diagram of the tunable grounded impedance multiplier;

[0018] FIG. 1B illustrates the circuit connections of an OTA;

[0019] FIG. 2A illustrates a low pass filter using capacitive impedance;

[0020] FIG. 2B illustrates a low pass filter circuit using the equivalent capacitance circuit;

[0021] FIG. 2C illustrates a high pass filter using resistive impedance;

[0022] FIG. 2D illustrates a high pass filter circuit using the equivalent resistance circuit;

[0023] FIG. 3 is a plot of the simulated and calculated capacitances versus the bias current I.sub.B;

[0024] FIG. 4 is a graph illustrating the frequency response for the low pass filter with tunable cut-off frequency;

[0025] FIG. 5 is a graph illustrating the simulation of the temperature effect when I.sub.B=1000 μA;

[0026] FIG. 6 is a graph illustrating the frequency response for the high pass filter using ideal and simulated resistors;

[0027] FIG. 7 is a graph illustrating the frequency response for the high pass filter using a simulated resistor;

[0028] FIG. 8 is a graph illustrating the experimental results obtained for a high pass filter (FIG. 2b) using the resistance multiplier;

[0029] FIG. 9 illustrates the resistor-less positive impedance multiplier;

[0030] FIG. 10 is a graph illustrating the simulation results for a high pass filter using the positive impedance multiplier of FIG. 9;

[0031] FIG. 11 illustrates a block diagram for a negative impedance multiplier;

[0032] FIG. 12 illustrates the circuit used to simulate a negative impedance multiplier;

[0033] FIG. 13 is a graph illustrating the gain and phase shift for the high pass filter using the negative impedance multiplier; and

[0034] FIG. 14 is a graph illustrating the frequency response of the impedance multiplier.

DETAILED DESCRIPTION

[0035] In the drawings, like reference numerals designate identical or corresponding parts throughout the several views. Further, as used herein, the words “a,” “an” and the like generally carry a meaning of “one or more,” unless stated otherwise. The drawings are generally drawn to scale unless specified otherwise or illustrating schematic structures or flowcharts.

[0036] Furthermore, the terms “approximately,” “approximate,” “about,” and similar terms generally refer to ranges that include the identified value within a margin of 20%, 10%, or preferably 5%, and any values therebetween.

[0037] Aspects of this disclosure are directed to a tunable impedance multiplier and methods for tunably multiplying an impedance. In an aspect, the tunable impedance multiplier may include a low pass filter circuit. In another aspect, the tunable impedance multiplier may include a high pass filter circuit. In an aspect, the tunable impedance multiplier is modified to provide positive tunable impedance multiplication. In a further aspect, the tunable impedance multiplier is modified to provide negative tunable impedance multiplication

[0038] A schematic of the tunable grounded positive and negative impedance multiplier 100 is shown in FIG. 1A. It includes a current feedback operational amplifier (CFOA) 102, an operational transconductance amplifier (OTA.sub.1) 104, a grounded resistor R.sub.o and the impedance Z to be scaled.

[0039] A schematic circuit showing a conventional CFOA is shown in FIG. 1B. For a CFOA, I.sub.y=0, V.sub.x=V.sub.y, I.sub.z=I.sub.x, and V.sub.w=V.sub.z. (See Yuce, E, Minaei, S, Ibrahim, M. “A novel full-wave rectifier/sinusoidal frequency doubler topology based on CFOAs”, Analog Integr Circ Sig Process, 2017) 93, pp. 351-362, incorporated herein by reference in its entirety). Note that in FIG. 1A, the symbol P has been used to represent the voltage due to the output resistance, rather than Z, to avoid confusion with the symbol Z which represents the impedance to be multiplied. The current through the resistor R0 has similarly been named I.sub.p for consistency. Supply voltages are shown as set at ±5 v for explanatory purposes, but may be different depending on the amplifier specifications.

[0040] The operational transconductance amplifier (OTA) is an amplifier whose differential input voltage produces an output current. Thus, it is a voltage controlled current source (VCCS). In the ideal OTA, the output current is a linear function of the differential input voltage, calculated as follows: I.sub.out=(V.sub.in+−V.sub.in−). g.sub.m. The gain, g.sub.m, is directly proportional to the bias current, I.sub.B, which provides the tuning capability of the impedance multiplier. The gain is indirectly proportional to the thermal voltage, V.sub.t, which caused instability in previous multipliers.

[0041] Therefore, with reference to FIG. 1A,

V.sub.y=V.sub.X=I.sub.x×Z  (1)

V.sub.w=V.sub.p=I.sub.p×R.sub.o  (2)

I.sub.0=g.sub.m(V.sup.+−V.sup.−)  (3)

where

[00004]$g_m=\frac{I_B}{2V_t}$

for a bipolar junction transistor (BJT) OTA, V.sub.y is the voltage at input Y of the CFOA 102, V.sub.x, is the voltage at input X of the CFOA 102, V.sub.t is the thermal voltage=25 mV at room temperature. Combining equations (1)-(3) and noting that I.sub.x=−I.sub.0, since I.sub.y=0 and that V.sup.+=0, the input impedance Z.sub.in can be expressed as:

[00005]$Z_{\mathrm{in}}=\frac{V_y}{I_x}=-\frac{V_y}{I_o}=\frac{V_y}{g_m\left(V^{-}\right)}=\frac{V_y}{g_m\times I_z{R}_o}=\frac{V_x}{g_m\times I_x{R}_o}=\frac{Z}{g_m\times R_o}$

or simply:

[00006]$\begin{array}{cc}\mathrm{Zin}=\frac{Z}{\mathrm{gm}XR0}.& \left(4\right)\end{array}$

[0042] With the transconductance of OTA.sub.1 given by g.sub.m=20 ×I.sub.B1 (at room temperature, V.sub.t=25 mV, therefore g.sub.m=I.sub.B1/2V.sub.t=20I.sub.B1), equation (4) can be rewritten as:

[00007]$\begin{array}{cc}\mathrm{Zin}=\frac{Z}{20X\mathrm{IB}1XR0}.& \left(5\right)\end{array}$

[0043] If the impedance Z, is replaced by a capacitor C, then equation (5) implements a capacitance multiplier whose value is tunable by the OTA bias current and is given by:

C.sub.eq=(20×I.sub.B1×R.sub.o)C.  (6)

[0044] If the impedance Z is replaced by a resistor R, then equation (5) implements a resistance multiplier whose value is tunable by the OTA.sub.1 bias current and is given by:

[00008]$\begin{array}{cc}\mathrm{Req}=\frac{R}{20X\mathrm{IB}1XR0}.& \left(7\right)\end{array}$

[0045] A. Simulation Results

[0046] To verify the functionality of the impedance multiplier of the present disclosure, the capacitance and resistance multipliers obtained from FIG. 1B were used to design a low pass filter (LPF) and a high pass filter (HPF) as shown in FIG. 2A, 2B, 2C, 2D. The functionality of the circuits of FIG. 2B, 2D was confirmed using a PSPICE simulation program.

[0047] The capacitance circuit of FIG. 2A was used in a low pass filter circuit as shown in more detail in FIG. 2B. In a non-limiting example, the circuit was designed with R.sub.o=R.sub.1=1 kΩ, C=100 nF, and commercially available integrated circuits AD844 (CFOA), LM13700N (OTA) were used. The amplifier bias supplies were connected to a ±5V power supply. The bias current I.sub.B1 was swept from 50 μA to 2000 μA (2000 μA is the maximum bias current of the OTA). The −3dB frequency was measured, from which the value of the equivalent capacitance was calculated, as the −3dB frequency equals the cutoff frequency, 1/RC. Plots of the simulated and calculated capacitances are shown in FIG. 3. It is evident from FIG. 3 that the calculated and simulated results are in close agreement for I.sub.B≤2000 μA.

[0048] Frequency response simulation for the low pass filter was also carried out. The cut-off frequency was tuned using the bias current I.sub.B. The results obtained are shown in FIG. 4 as a function of four values of the bias current I.sub.B, which are I.sub.B=50 micro amps, 0.5 mA, 1 mA, and 2 mA.

[0049] Since the operation of the circuit of FIG. 1A is dependent on g.sub.m, which is a function of temperature, it is important to examine the effect of temperature variation on the performance of the circuits of FIG. 1A, 2A-2B. The simulation results obtained from FIG. 2A and shown in FIG. 5 indicate that the cut-off frequency only slightly changes with temperature.

[0050] The high pass filter circuit of FIG. 2C is shown in more detail in FIG. 2D. In a non-limiting example, the circuit was simulated with C=100 nF, Z=10 kΩ and R.sub.o=10Ω. The amplifier bias supplies were connected to a ±5V power supply. The bias current was set to 5 μA. Referring to the frequency response shown in FIG. 6, it is clear that the −3dB point is at 0.6 Hz. This frequency at the −3 dB point =1/RC. Since the capacitor value is known, the resistor was determined to be a resistance of 2.5 MΩ. This means that the resistor Z was scaled (multiplied) by 250 times.

[0051] In a non-limiting example which illustrates the tunability of the resistor, the capacitor was set to 10 nF and the bias current, I.sub.B, was varied from 50 μA to 200 μA. The response of the high pass filter is shown in FIG. 7. It is clear from FIG. 7 that the cut-off frequency varies with the bias current.

[0052] B. Experimental results

[0053] Experimental verification was carried out using the high pass filter circuit with C=10 nF, and Z=10 kΩ and the bias current was varied from 50 μA to 200 μA. A plot of the experimental results is shown in FIG. 8.

[0054] It is clear from FIG. 8 that the experimental results are in excellent agreement with the simulation shown in FIG. 7.

[0055] In order to minimize the temperature dependence, the circuit in FIG. 1A was modified as shown in FIG. 9 by replacing the resistance R.sub.o with an operational transconductance amplifier OTA.sub.2 106.

[0056] With reference to FIG. 9, OTA.sub.2 106 is configured as a resistor equal to 1/g.sub.m2. Thus, equation (4) can be rewritten as:

[00009]$\begin{array}{cc}\mathrm{Zin}=Z\frac{\mathrm{IB}2}{\mathrm{IB}1}.& \left(8\right)\end{array}$

where I.sub.B1 and I.sub.B2 are the bias currents for OTA1 and OTA2 respectively.

[0057] It is clear from equation (8), that the impedance can be scaled up or down using the bias currents I.sub.B1 and I.sub.B2. Moreover, it is free of passive components and it is temperature insensitive.

[0058] In a non-limiting example, the modified design was simulated as a resistance multiplier in a high pass filter. The bias current I.sub.B1 was fixed to 10μA and I.sub.B2 was varied from 50μA to 2000μA, Z=10Ωand C=100 nF. The simulation results shown in FIG. 10 confirm the functionality of the design.

[0059] The circuit of the present disclosure provides tunable positive (as shown above) and negative impedance multiplication. A tunable negative impedance multiplier can be achieved if OTA2 is configured as a negative resistor as shown in FIG. 11. The equivalent impedance Z.sub.in is given by:

[00010]$\begin{array}{cc}{Z}_{\mathrm{in}}=-Z\frac{I_{B1}}{I_{B2}}& \left(9\right)\end{array}$

[0060] The first embodiment is described with respect to FIG. 1A, 2B, 2D, 9 and 11. The first embodiment is a tunable impedance multiplier, comprising a current feedback operational amplifier, CFOA 102, having a current input Y, a current input X, a voltage output W and a voltage output P; a first operational transconductance amplifier, OTA.sub.1 104, having a gain g.sub.m, a first positive voltage input V+, a first negative voltage input V−, a first bias current input I.sub.B1, a positive supply voltage, a negative supply voltage, and a current output I.sub.o. A resistor R.sub.o is connected to the voltage output P at a first end and is connected to ground at a second end. An impedance, Z, is connected to the input X at a first end and is connected to ground at a second end. The voltage output, W, is connected to the voltage input, V, of OTA.sub.1 and the current output, I.sub.o, is connected as feedback to the current input Y of the CFOA.

[0061] The impedance Z may be either a resistor or a capacitor. The equivalent input impedance, Z.sub.in, of the impedance multiplier is given by the equation

[00011]$\mathrm{Zin}=\frac{Z}{\mathrm{gm}XR0}.$

[0062] If g.sub.m=20×I.sub.B.sub.1, the equivalent input impedance, Z.sub.in, of the impedance multiplier is given by the equation

[00012]$\mathrm{Zin}=\frac{Z}{20X\mathrm{IB}1XR0}.$

[0063] If the impedance Z is a capacitor, C, and g.sub.m=20×I.sub.B.sub.1 the equivalent capacitance of the impedance multiplier is given by C.sub.eq=(20×I.sub.B1×R.sub.o) C.

[0064] If the impedance Z is a resistor, g.sub.m=20×I.sub.B1 and the equivalent resistance of the impedance multiplier is given by

[00013]$\mathrm{Req}=\frac{R}{20X\mathrm{IB}1XR0}.$

[0065] FIG. 9 illustrates a positive impedance multiplier which is temperature insensitive in which the resistor R.sub.o is provided by a second operational transconductance amplifier, OTA.sub.2 906, where the second operational transconductance amplifier includes an inverting voltage input, V.sub.2.sup.−, a non-inverting voltage input, V2+, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the inverting voltage input is connected to the voltage output P and the non-inverting voltage input is connected to ground.

[0066] In FIG. 9, the impedance Z may be a resistor or a capacitor and the equivalent input impedance, Z.sub.in, of the impedance multiplier is given by the equation

[00014]$\mathrm{Zin}=Z\frac{\mathrm{IB}2}{\mathrm{IB}1}.$

[0067] FIG. 11 illustrates a negative impedance multiplier which is temperature insensitive in which the resistor R.sub.o is provided by a second operational transconductance amplifier, OTA.sub.2, where the second operational transconductance amplifier includes an inverting voltage input, a non-inverting voltage input, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the non-inverting input is connected to the voltage output P and the inverting input is connected to ground. In FIG. 11, the impedance Z is a resistor or a capacitor and the equivalent input impedance, Z.sub.in, of the impedance multiplier is given by the equation

[00015]$\mathrm{Zin}=-Z\frac{\mathrm{IB}2}{\mathrm{IB}1}.$

[0068] Referring back to FIG. 2A, the impedance multiplier can include a low pass filter circuit having a resistor, R.sub.1, connected at a first end to the current input Y; a voltage output, V.sub.o, connected to the first end; wherein a first end of an alternating voltage source, V.sub.in, is connected to a second end of the resistor, R1; and a second end of the alternating voltage source, V.sub.in, is connected to ground.

[0069] Additionally, the resistance R.sub.o may be provided by a second operational transconductance amplifier, OTA.sub.2, as shown with respect to FIG. 9, where the second operational transconductance amplifier includes an inverting voltage input, a non-inverting voltage input, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the inverting voltage input is connected to the voltage output P and the non-inverting voltage input is connected to ground.

[0070] Alternatively, the resistance R.sub.o may be provided by a second operational transconductance amplifier, OTA.sub.2, as shown with respect to FIG. 11, where the second operational transconductance amplifier includes an inverting voltage input, a non-inverting voltage input, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the non-inverting input is connected to the voltage output P and the inverting input is connected to ground. The circuit of FIG. 11 provides temperature insensitivity, thus greater stability. The equivalent impedance for this circuit is positive.

[0071] The tunable impedance multiplier may further comprise a high pass filter circuit shown in FIG. 2D, including a capacitor, C.sub.1, connected at a first end to the current input Y; a voltage output, V.sub.o, connected to the first end of the capacitor; a first end of an alternating voltage source, V.sub.in, connected to a second end of the capacitor, C1 and a second end of an alternating voltage source, V.sub.in, connected to ground.

[0072] The tunable impedance multiplier having a high pass filter circuit as shown in FIG. 2D may have the resistor R.sub.o provided by a second operational transconductance amplifier, OTA.sub.2 as shown in FIG. 11, where the second operational transconductance amplifier includes an inverting voltage input, a non-inverting voltage input, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the inverting voltage input is connected to the voltage output P and the non-inverting voltage input is connected to ground. This circuit configuration provides high pass filtering and positive impedance multiplication.

[0073] Alternately, R.sub.0 may provided by a second operational transconductance amplifier, OTA.sub.2, as shown in FIG. 11, where the second operational transconductance amplifier includes an inverting voltage input, a non-inverting voltage input, a bias input, I.sub.B2, a second positive and a second negative supply voltage, and a second current output which is connected to the inverting voltage input, wherein the non-inverting input is connected to the voltage output P and the inverting input is connected to ground. This circuit configuration provides high pass filtering and negative impedance multiplication.

[0074] The second embodiment is illustrated with respect to FIG. 1A, 2B, 2D, 9 and 11.

[0075] Referring to FIG. 1A, the second embodiment describes a method for tunably multiplying an impedance, comprising connecting an alternating current source, i.sub.x, to a first current input, Y, of a current feedback operational amplifier, CFOA 102; connecting a second current input X of the CFOA to an impedance, Z; connecting a first output, W, of the CFOA to an inverting input, V.sup.−, of a first operational transconductance amplifier, OTA.sub.1 104 having a specified gain, g.sub.m; connecting a second output, P, of the CFOA to a grounded resistance, R.sub.o; connecting a non-inverting input, V.sup.+, of the OTA.sub.1 to ground; connecting a current output, I.sub.o, of the OTA.sub.1 to the first current input of the CFOA. Tuning and determining the output comprises sweeping the amplitude of a first bias current source, i.sub.B1, connected to a bias input of the OTA.sub.1 over a range of frequencies of alternating current i.sub.x; determining the −3dB points of the current output corresponding to the range of frequencies and calculating the equivalent input impedance based on the equation

[00016]$\mathrm{Zin}=\frac{Z}{\mathrm{gm}XR0}.$

[0076] Referring to FIG. 9, the method includes providing temperature insensitivity and positive impedance multiplication by providing the grounded resistance, R.sub.o, by connecting an inverting input of a second operational transconductance amplifier, OTA.sub.2 106 to a second voltage output, P, of the CFOA, connecting the non-inverting input of the OTA.sub.2 to ground and connecting the output of the OTA.sub.2 to the inverting input of the OTA.sub.2. Tuning and determining the output comprises connecting a second bias current source, i.sub.B2, to a bias input of the OTA.sub.2; sweeping the amplitudes of the first bias current source, i.sub.B1, of the OTA.sub.1 and second bias current source, i.sub.B2, of the OTA.sub.2 over a range of frequencies of alternating current i.sub.x, wherein the value of the impedance, Z, is positive and multiplied by the factor

[00017]$\frac{\mathrm{IB}2}{\mathrm{IB}1}.$

[0077] Alternatively, referring to FIG. 11, the method includes providing temperature insensitivity and negative impedance multiplication by providing the grounded resistance, R.sub.o, by connecting a non-inverting input of a second operational transconductance amplifier, OTA.sub.2 to a second voltage output of the CFOA, connecting the inverting input of the OTA.sub.2 to ground and connecting the output of the OTA.sub.2 to the inverting input of the OTA.sub.2; connecting a second bias current source, i.sub.B2, to a bias input of the OTA.sub.2; sweeping the amplitudes of the first bias current source, i.sub.B1, of the OTA.sub.1 and second bias current source, i.sub.B2, of the OTA.sub.2 over a range of frequencies of alternating current i.sub.x, wherein the value of the impedance, Z, is negative and multiplied by the factor

[00018]$\frac{\mathrm{IB}2}{\mathrm{IB}1}.$

[0078] The third embodiment is described with respect to FIG. 1A, 2B, 2D, 9 and 11. The third embodiment describes a method for tunably multiplying an impedance, comprising connecting an alternating current source, i.sub.x, to a first current input, Y, of a current feedback operational amplifier, CFOA 102; connecting a second current input, X, of the CFOA to an impedance, Z; connecting a first output, W, of the CFOA to an inverting input, V.sup.−, of a first operational transconductance amplifier, OTA.sub.1 having a specified gain, g.sub.m; connecting a non-inverting input, V.sup.+, of the OTA.sub.1 to ground; connecting a current output of the OTA.sub.1 to the first current input of the CFOA; connecting an inverting input, V.sub.2.sup.−, of a second operational transconductance amplifier OTA.sub.2 106 to a second voltage output, P, of the CFOA, connecting the non-inverting input, V.sub.2.sup.+of the OTA.sub.2 to ground and connecting the output of the OTA.sub.2 to the inverting input of the OTA.sub.2; connecting a second bias current source, i.sub.B2, to a bias input of the OTA.sub.2. Tuning and determining the output comprises sweeping the amplitudes of the first bias current source, i.sub.B1, of the OTA.sub.1 and second bias current source, i.sub.B2, of the OTA.sub.2 over a range of frequencies of alternating current i.sub.x, determining the −3dB points of the current output corresponding to the range of frequencies; calculating the equivalent input impedance based on the equation

[00019]$\mathrm{Zin}=\frac{Z}{\mathrm{gm}XR0}$

and calculating the multiplication of the impedance, Z, based on the equation=

[00020]$Z\frac{\mathrm{IB}2}{\mathrm{IB}1}.$

[0079] Additionally, the circuit of the third embodiment may be operated as a high pass filter by providing the alternating current source, i.sub.x, by connecting a capacitor, C.sub.1, to the first current input of the CFOA; connecting a voltage output, V.sub.o, to the first end of the capacitor; connecting a first end of an alternating voltage source, V.sub.in, to a second end of the capacitor, C1; connecting a second end of an alternating voltage source, V.sub.in, to ground; sweeping the alternating voltage source over a range of frequencies; sweeping the amplitudes of the first bias current source, i.sub.B1, of the OTA.sub.1 and second bias current source, i.sub.B2, of the OTA.sub.2 at each of the frequencies; wherein the voltage output, V.sub.0, is high pass filtered to a subset of the range of frequencies and the amplitude of the voltage output, V.sub.o, is tuned by the multiplication of the impedance, Z.

[0080] To confirm the functionality of the design of the negative impedance multiplier, the multiplier was used to replace Z.sub.eq in a voltage divider circuit as shown in FIG. 12. The values used in simulation are as follows: R.sub.1=10kΩ, I.sub.B1100 μA, I.sub.B2=20 μA, and z=5kΩ(Z.sub.in=−5kΩ). The output voltage was calculated to be −Vi. It is evident from the simulation results shown in FIG. 13 that the output voltage is negative and is in close agreement with the calculation. The offset shown comes from CFOA 102 (AD844).

[0081] The tunable impedance multiplier functions well in the frequency range 2 Hz to −7 MHz as can be shown in FIG. 14.

[0082] The tunable positive and negative impedance multiplier of the present disclosure was compared to previous multipliers and a summary of comparison is shown in Table I. It is clear from Table I that the tunable impedance multiplier outperforms known multipliers as it is tunable and can multiply both resistance and capacitance. Further, the multiplier can invert the sign of the impedance modified as in the circuit of FIG. 11.

TABLE-US-00001 TABLE I SUMMARY OF PERFORMANCE COMPARISON [1] [8] [9] [11] This work Multiplication factor 1660 525 28 140 400 # of passive elements 7 0 0 0 0 Tunability NO Yes No Yes Yes R & C multiplier NO NO NO NO Yes

[0083] A grounded impedance scaling circuit that can tunably multiply a capacitor or a resistor was presented. The design may use commercially available integrated circuits. The grounded tunable multiplier was shown to perform well as either a low pass or high pass filter with a tunable cut-off frequency. The multipler of the present disclosure further is capable of being a positive or a negative impedance multiplier. FIG. 9 and FIG. 11 may be combined (not shown) by providing a double pole double pole switch, equivalent switch box, or switching transistor circuit at the inputs of OTA.sub.2.

[0084] Obviously, numerous modifications and variations of the present invention are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the invention may be practiced otherwise than as specifically described herein.