Isomorphic Circuits of Independent Amplitude Tunable Voltage-Mode Bandpass Filters and Quadrature Sinusoidal Oscillators

: This paper presents isomorphic circuits of voltage-mode (VM) non-inverting bandpass ﬁlters (NBPFs) and VM quadrature sinusoidal oscillators (QSOs) with independent amplitude control functionality. The proposed VM NBPFs and VM QSOs exhibit low-output impedance and independent amplitude control, which are important for easily cascading the VM operation and independent control of the amplitude gain. The proposed isomorphic circuits employ three LT1228 commercial integrated circuits (ICs), two grounded capacitors, two grounded resistors and one ﬂoating resistor. The use of grounded capacitors is beneﬁcial for the implementation of the IC. Both NBPFs have a high-input impedance and have a wide range of independent amplitude tunable passband gain without affecting the quality factor (Q) and center frequency (f o ). The Q and f o parameters of the proposed NBPFs are orthogonal tunability. By feeding back each input signal to the output response of the NBPF, two VM fully uncoupled QSOs are also proposed. The proposed VM fully uncoupled QSOs have two quadrature sinusoidal waveforms with two low-output impedances and one independent amplitude tunable sinusoidal waveform. The frequency of oscillation (FO) and the condition of oscillation (CO) are fully uncoupled and controlled electronically. The performances of the proposed isomorphic circuits have been tested with a ± 5 volt power supply and are demonstrated by experimental measurements which conﬁrm the theoretical assumptions.


Introduction
The LT1228 is a commercial integrated circuit (IC) using bipolar or complementary metal oxide semiconductor (CMOS) technology. It combines a transconductance amplifier (OTA) and a current feedback amplifier (CFA). LT1228 is one of the attractive components used for realizing voltage-mode (VM) analog circuits. It has wider electronic tunability, higher accuracy, higher frequency applicability and simplicity of implementation, so many topologies based on LT1228 application circuits have been published in the literature [1][2][3][4][5][6][7][8].
An active VM non-inverting bandpass filter (NBPF) is one of the important filters for analog signal processing, and its transfer function can be expressed as V NBPF V in = ds as 2 + bs + c (1) where the coefficients a, b, c and d are real numbers, V in is the NBPF input signal and V NBPF is the output response. According to the feedback theory, the VM quadrature sinusoidal oscillator (QSO) can be synthesized by connecting the input voltage of the NBPF to the output response. Therefore, Equation (1) can be rewritten as Equation (2): Based on Equations (1) and (2), the VM QSO can be realized by zeroing the input voltage of the NBPF, and the characteristic equation (CE) of the VM QSO can be expressed as Equation (3): The two poles of Equation (3) are obtained as follows: In Equation (3), the oscillator will oscillate if the condition of oscillation (CO) and the frequency of oscillation (FO) are as given by Equations (5) and (6), respectively: For a positive feedback system, if the two poles of Equation (4) are located in the right half of the complex plane, the circuit can be built, and the sinusoidal oscillation can be maintained. Therefore, the CO of Equation (5) is modified to become Equation (7): Recently, two interesting VM circuits were proposed in the literature [21,22]. The circuit proposed in [21] employs five OTAs combined with two grounded capacitors. The circuit proposed in [22] also employs five OTAs combined with two grounded capacitors. However, the passband gain of these two circuits cannot be adjusted independently without affecting the quality factor (Q) and resonance angular frequency (ω o ), and it cannot be transformed into an electronically controllable QSO with an independent amplitude control function [21,22]. An interesting VM QSO with gain controllability of the quadrature voltage output was recently proposed in [23]. The circuit in Figure 5 of [23] employs three LT1228s, two grounded capacitors, two grounded resistors and three floating resistors, but the resistances of the circuit in Figure 5 of [23] are not the smallest. According to the feedback theory, this study proposes isomorphic circuits of the VM NBPFs and VM QSOs with independent amplitude control functionality. Each of the isomorphic circuits contains three LT1228 commercial ICs, two grounded capacitors, two grounded resistors and one floating resistor. Table 1 lists the available circuits and compares them according to the relevant standards. The additional performance comparisons with previous circuits are summarized in Tables 2 and 3. As shown in Tables 1-3, the proposed VM NBPFs and VM QSOs exhibit high-input and low-output impedance and independent amplitude control, which are important for easily cascading the VM operation and independent control of the amplitude gain. The range of the measured QSO amplitude can be adjusted from 1.51 to 24.05, and the measured oscillation frequency can be varied from 82.15 kHz to 1629 kHz. This means that the proposed circuits are sufficient for the typical impedance sensor applications [23]. Because the proposed isomorphic circuits have independent amplitude control functions and a wide tunable frequency range, the applications will be beneficial in the literature [23,24].

The Proposed Isomorphic Circuits
Each of the isomorphic circuits' symbols of an LT1228 commercial IC from Linear Technology [25] is shown in Figure 1a, and its packed IC is shown Figure 1b. The LT1228 IC contains an OTA with a specially devised CFA. Figure 1c shows the equivalent circuit of the LT1228 IC, and the ideal port relations of LT1228 can be characterized by the following hybrid matrix [23]: where g m is the transconductance gain of the LT1228 IC and R T is the transresistance gain of the CFA. Ideally, R T approaches infinity, and the g m is controlled by the direct current (DC) bias current (I B ) of LT1228. The g m value is equal to 10 times the value of I B and can be expressed as [23,25] where V DD is the positive voltage supply and V EE is the negative voltage supply. V BE is a bias voltage of the bipolar junction transistor and is approximately equal to 0.65 volts. R B is a bias control resistor used to obtain the DC bias current I B . The proposed two isomorphic circuits are shown in Figures 2 and 3. Each of the circuits contains three LT1228 commercial ICs, two grounded capacitors, two grounded resistors and one floating resistor. The use of grounded capacitors is beneficial for the implementation of the IC. Each input voltage of the VM NBPF (Figures 2a and 3a) is applied to the positive voltage terminal of the LT1228, so it has a high-input impedance and can be easily cascaded without any voltage buffers. The output impedances of terminals V BP2 and V BP5 (Figures 2a and 3a, respectively) are very small, so they can also be directly connected to the next stage without any voltage buffers. Moreover, the VM NBPF responses of V BP3 and V BP6 shown in Figures 2a and 3a, respectively, can achieve a wide range of independent amplitude adjustable passband gains without affecting the parameters of f o and Q.   V BP1 V in = sC 1 g m2 R 1 s 2 C 1 C 2 R 1 + sC 1 + g m1 g m3 R 1 (10) Based on the three NBPF responses from Equations (10)- (12), the NBPF parameters of f o and Q are given by From Equations (13) and (14), the NBPF parameters of f o and Q have orthogonal tunability by tuning the g m1 for f o first and then the grounded resistor R 1 for Q without affecting parameter f o .
The passband gains of the three NBPF responses are given by From Equations (15) and (16), it is important to note that the passband gains of the NBPF responses are independently adjustable by g m2 without affecting the NBPF parameters of f o and Q. Moreover, the resistors R 2 or R 3 in Equation (16) can achieve a wide range for the independent amplitude adjustable passband gain of the NBPF without affecting parameters f o and Q. It is interesting to note that, as shown in Figure 2a, by feeding back the input signal to the output response of the NBPF, a VM fully uncoupled QSO can be realized as shown in Figure 2b. This means that the input signal V in is connected to V BP1 (Figure 2a), and Equation (10) becomes Therefore, the CE of Figure 2b is expressed as Based on Equation (18), the CO and FO of the proposed VM QSO (Figure 2b) are given by Equations (19) and (20), respectively: FO : f o = 1 2π g m1 g m3 C 1 C 2 (20) According to Equations (19) and (20), the CO can be fully controlled independently without affecting the FO by adjusting g m2 or R 1 , and the FO can be also fully controlled independently without affecting the CO by adjusting g m1 or g m3 . Hence, the VM QSO ( Figure 2b) provides a fully uncoupled tuning law for the CO and FO. Routine analysis of the QSO structure, shown in Figure 2b, yielded the following two quadrature voltage outputs and a sinusoidal output waveform with independent amplitude control: According to Equation (21), the magnitude ratio and phasor of output voltages V o1 and V o2 are given as Equations (23) and (24) show that the voltage magnitude ratio of V o1 and V o2 is in unity, and the voltage phase shift of V o1 and V o2 is 90 • . Thus, the proposed first VM QSO has a unity voltage amplitude ratio and a quadrature voltage phase shift.
Similarly, routine analysis of the VM NBPF circuit (Figure 3a) yielded the following three NBPF responses: Based on three NBPF responses of Equations (25)- (27), the f o , Q, H BP4 , H BP5 and H BP6 are given by From Equations (28)-(31), the NBPF parameters of f o and Q have orthogonal tunability, and the passband gains are independently adjustable by g m2 without affecting the NBPF parameters f o and Q. Note that the resistors R 5 and R 6 in Equation (31) can independently achieve a wide range of NBPF adjustable passband gain without affecting the NBPF parameters f o and Q.
By feeding back the input signal to the output response of the VM NBPF as in Figure 3a, a VM fully uncoupled QSO can be realized as shown in Figure 3b. This means that the input signal V in is connected to V BP4 (Figure 3a), and Equation (25) becomes Therefore, the CE from Figure 3b is expressed as Based on Equation (33), the CO and FO of the proposed VM QSO (Figure 3b) are given by Equations (34) and (35), respectively: According to Equations (34) and (35), the CO can be fully controlled independently without affecting the FO by adjusting g m2 or R 4 , and the FO can also be fully controlled independently without affecting the CO by adjusting g m1 or g m3 . Hence, the VM QSO shown in Figure 3b provides a fully uncoupled tuning law for the CO and FO. Routine analysis of the VM QSO structure (Figure 3b) yielded the following two quadrature voltage outputs and a sinusoidal output waveform with independent amplitude control: According to Equation (36), the magnitude ratio and phasor of output voltages V o4 and V o5 are given as Equations (38) and (39) show that the voltage magnitude ratio of V o4 and V o5 is in unity, and the voltage phase shift of V o4 and V o5 is −90 • . Thus, the proposed second VM QSO has a unity voltage amplitude ratio and a quadrature voltage phase shift.

Simulation and Experimental Measurements
The proposed isomorphic circuits of the VM NBPFs and VM QSOs were simulated with Cadence OrCAD PSpice simulation software version 16.6 to confirm the theory and implemented by three off-the-shelf LT1228 ICs, two grounded capacitors and three resistors to verify the performance of the isomorphic circuits. The isomorphic circuits adopted a DC bias of ±5 V for the voltage supplies, and the voltage output was measured in the time domain with a Tektronix DPO 2048B oscilloscope, the voltage output was measured in the frequency domain with a Keysight E5061B-3L5 network analyzer, and the voltage output in the frequency spectrum was measured with a Keysight-Agilent N9000A CXA signal analyzer. The measurement photos of the isomorphic circuits are shown in Figures 4 and 5. analysis of the VM QSO structure (Figure 3b) yielded the following two quadrature voltage outputs and a sinusoidal output waveform with independent amplitude control: According to Equation (36), the magnitude ratio and phasor of output voltages Vo4 and Vo5 are given as Equations (38) and (39) show that the voltage magnitude ratio of Vo4 and Vo5 is in unity, and the voltage phase shift of Vo4 and Vo5 is −90°. Thus, the proposed second VM QSO has a unity voltage amplitude ratio and a quadrature voltage phase shift.

Simulation and Experimental Measurements
The proposed isomorphic circuits of the VM NBPFs and VM QSOs were simulated with Cadence OrCAD PSpice simulation software version 16.6 to confirm the theory and implemented by three off-the-shelf LT1228 ICs, two grounded capacitors and three resistors to verify the performance of the isomorphic circuits. The isomorphic circuits adopted a DC bias of ±5 V for the voltage supplies, and the voltage output was measured in the time domain with a Tektronix DPO 2048B oscilloscope, the voltage output was measured in the frequency domain with a Keysight E5061B-3L5 network analyzer, and the voltage output in the frequency spectrum was measured with a Keysight-Agilent N9000A CXA signal analyzer. The measurement photos of the isomorphic circuits are shown in

First Proposed Isomorphic Circuit Simulation and Experimental Results
The theoretical, simulated and measured NBPF responses of VBP2 and VBP3 (Figure 2a) are shown in Figures 6 and 7, respectively, as depicted in Equations (11) and (12). In Figure  6, the NBPF voltage gain at the voltage output of VBP2 could be adjusted independently by gm2 without affecting parameters fo and Q, as depicted in Equations (13)- (15). In Figure 7, the wide range of independent amplitude adjustable NBPF voltage gain at the voltage output of VBP3 could also be independently adjusted by R2 without affecting the NBPF parameters fo and Q, as depicted in Equations (13), (14) and (16). The component values used in Figure 6 were C1 = C2 = 0.96 nF, gm1 = gm3 = 1 mS and R1 = R2 = R3 = 1 kΩ for the theoretical value of fo = 165.8 kHz, and only gm2 changed, with different values of 1, 1.5, 2 and 3 mS. This resulted in an NBPF at a voltage output of VBP2 with voltage gains of 0, 3.52, 6.02 and 9.54 dB. Figure 8 shows the measurement results of the VBP2 gain response.

First Proposed Isomorphic Circuit Simulation and Experimental Results
The theoretical, simulated and measured NBPF responses of V BP2 and V BP3 (Figure 2a) are shown in Figures 6 and 7, respectively, as depicted in Equations (11) and (12). In Figure 6, the NBPF voltage gain at the voltage output of V BP2 could be adjusted independently by g m2 without affecting parameters f o and Q, as depicted in Equations (13)- (15). In Figure 7, the wide range of independent amplitude adjustable NBPF voltage gain at the voltage output of V BP3 could also be independently adjusted by R 2 without affecting the NBPF parameters f o and Q, as depicted in Equations (13), (14) and (16). The component values used in Figure 6 were C 1 = C 2 = 0.96 nF, g m1 = g m3 = 1 mS and R 1 = R 2 = R 3 = 1 kΩ for the theoretical value of f o = 165.8 kHz, and only g m2 changed, with different values of 1, 1.5, 2 and 3 mS. This resulted in an NBPF at a voltage output of V BP2 with voltage gains of 0, 3.52, 6.02 and 9.54 dB. Figure 8 shows the measurement results of the V BP2 gain response.
The component values used in Figure 7 were C 1 = C 2 = 0.96 nF, g m1 = g m2 = g m3 = 1 mS and R 1 = R 3 = 1 kΩ for a theoretical value of f o = 165.8 kHz, and only R 2 changed, with different values of 1, 4, 11 and 19 kΩ. This resulted in an NBPF at a voltage output of V BP3 with voltage gains of 6.02, 13.98, 21.58 and 26 dB. Figure 9 shows the measurement results of the V BP3 gain response. To display the independent electronically adjusted passband gains from Figure 2a, the value of g m2 was changed from 1 mS to 4 mS, and the NBPF voltage gain at the voltage output of V BP2 could be changed from 0 dB to 12.04 dB. As shown in Figure 10, the proposed NBPF in Figure 2a exhibited independent electronic adjustment of passband gains at the voltage output of V BP2 . To display the maximum operating passband gains of Figure 2a, the value of R 2 was changed from 1 kΩ to 25 kΩ, and the NBPF voltage gain at the voltage output of V BP3 could be changed from 6.02 dB to 28.3 dB. As is shown in Figure 11, the proposed NBPF from Figure 2a exhibited a wide range of independent amplitude adjustable passbands at a voltage output of V BP3 . Figure 12 shows the theoretical, simulated and measured NBPF responses on the V BP2 output terminal with different values of R 1 when C 1 = C 2 = 0.96 nF, R 2 = R 3 = 1 kΩ and g m1 = g m2 = g m3 = 1 mS. Figure 13 shows the measurement results of the V BP2 gain response with different Q values. In this case, the four values of R 1 were changed to 1, 1.5, 2 and 3 kΩ, and the measured values of Q were 1.17, 1.53, 2.08 and 3. As shown in Figures 12 and 13, the Q parameter of the NBPF could be independently controlled by adjusting the different values of R 1 without affecting the parameters of f o , as depicted in Equations (13) and (14). For an experimental test, Figure 14 shows the input and output voltage waveforms of the NBPF response at the V BP2 output terminal, which could be extended to an amplitude of 120 mVpp without signification distortion. As shown in Figure 14, the input voltage waveform was a 165.8-kHz sinusoidal waveform, and the output voltage waveform was measured to be a 161.7-kHz sinusoidal waveform with a phase difference of 0.23 • .
To evaluate the performance of the proposed isomorphic circuit, the total harmonic distortion (THD) and the intermodulation distortion (IMD) of the isomorphic circuit were measured. The THD is defined as the ratio of the sum of the powers of all harmonic components (except the fundamental frequency) to the power of fundamental frequency. The IMD is defined as the two-tone test signals of different frequencies mixed together to form an additional signal at the excitation frequency. The THD and IMD can be expressed as follows [26,27]: where V 1 is the fundamental frequency voltage content and V n (n = 2, 3, . . . , N) is the nth harmonic voltage content: where p(f 2 ± (n − 1))f 1 is the root mean square value of the nth-order intermodulation component at the sum and difference of the two tones and p(f 2 ) is the root mean square value of the fundamental component at the excitation frequency f 2 . Figure 15 shows the frequency spectrum of the NBPF response at the V BP2 output terminal. The NBPF response center frequency measured at the output of V BP2 was about 165.8 kHz. The THD, including the fundamental harmonic through the sixth harmonic components of Figure 15, was about 1.08%. Figure 16 shows the THD result of the NBPF response at the V BP2 output terminal with different input voltages by keeping a constant center frequency of 165.8 kHz. Figure 17 shows the spectrum of the NBPF at the output of V BP2 , which was obtained by applying two-tone signals f 1 and f 2 around the theoretical value of f o = 165.8 kHz for IMD characterization. In Figure 17, the low-frequency tones of f 1 = 164.8 kHz and the high-frequency tones of f 2 = 166.8 kHz were used with an input amplitude of 63 mVpp. The measured value of the third-order IMD (IMD3) was about −40.1 dBc, and the thirdorder intercept (TOI) was about −7.95 dBm. Figure 18 shows the 1-dB compression point (P1dB) of the NBPF measured at the output of V BP2 between the output power and the input power when the center frequency was 165.8 kHz. The measured output power P1dB was about −11.4 dBm. Figure 19 shows the phase noise performance of the NBPF measured at the output of V BP2 . The measured value of the phase noise of V BP2 was less than −83.86 dBc/Hz at a 30-Hz offset.              According the NBPF frequency spectrum measured in Figure 15, the dynamic range between the fundamental tone and the largest spurious of the spurious-free dynamic range was approximately 40.79 dB. Thus, the FoM of the proposed VM NBPF at the voltage output of V BP2 was approximately computed to be 3.07 × 10 6 .
To investigate the VM fully uncoupled QSO in Figure 2b, Figure 20 shows the measured waveforms of the quadrature voltage outputs V o1 and V o2 in the time domain. The component values used in Figure 20 were C 1 = C 2 = 0.96 nF, g m1 = g m2 = g m3 = 1 mS and R 1 = R 2 = R 3 = 1 kΩ for the theoretical value of f o = 165.8 kHz. The measured FO in Figure 20 was 162.3 kHz, which was close to the theoretical value of 165.8 kHz. Figure 21 shows the spectrum of the VM fully uncoupled QSO at the V o2 output terminal. The measured FO was 160.95 kHz, which was close to the theoretical value of 165.8 kHz. The difference between the amplitudes of the fundamental and second harmonics was 41.87 dB, and the calculated THD was about 0.81%. As shown in Figure 21, the third harmonic and subsequent harmonics were not visible in the spectrum because these harmonics were lower than the noise floor. This means that the THD of the first proposed VM fully uncoupled QSO was small. Figure 22 shows the phase noise performance of the QSO measured at the output of V o2 . The measured value of the phase noise of V o2 was less than −40.18 dBc/Hz at a 30-Hz offset. Figure 23 shows the FO tuning range of the measured output voltage V o2 , where C 1 = C 2 = 0.96 nF, g m2 = 1 mS and R 1 = R 2 = R 3 = 1 kΩ, and only g m1 = g m3 were varied from the values of 0.5-10 mS. As is shown in Figure 23, the measured oscillation frequency was electronically and linearly varied from 82.15 kHz to 1629 kHz. In Figures 20-23, the g m2 value should have been slightly adjusted to satisfy the CO as depicted in Equation (19). Figure 24 shows the relationship between the measured amplitude ratio of the quadrature output voltages V o1 and V o2 and the FO as depicted in Equation (23). Figure 25 shows the relationship between the measured phase difference of the quadrature output voltages V o1 and V o2 and the FO as depicted in Equation (24). Figure 26 shows the experimental results of the voltage gain of V o3 when only R 2 was changed while maintaining C 1 = C 2 = 0.96 nF, g m1 = g m2 = g m3 = 1 mS, g m2 = 1.05 mS and R 1 = R 3 = 1 kΩ. As is shown in Figure 26, the proposed VM fully uncoupled QSO (Figure 2b) exhibited a wide range of independent amplitude adjustable V o3 voltage gains. The measured operating amplitude control range could be adjusted from 1.51 to 24.05. Table 4 summarizes the performance of the first proposed isomorphic circuit.

Second Proposed Isomorphic Circuit Simulation and Experimental Results
The theoretical, simulated and measured NBPF responses of the V BP5 and V BP6 (Figure 3a) are shown in Figures 27 and 28, respectively, as depicted in Equations (26) and (27). In Figure 27, the NBPF voltage gain at the voltage output of V BP5 could be adjusted independently by g m2 without affecting the parameters f o and Q as depicted in Equations (28)-(30). In Figure 28, the wide range of the independent amplitude adjustable NBPF voltage gain at the voltage output of V BP6 could also be independently adjusted by R 5 without affecting the NBPF parameters f o and Q as depicted in Equations (28), (29) and (31). The component values used in Figure 27 were C 3 = C 4 = 0.96 nF, g m1 = g m3 = 1 mS and R 4 = R 5 = R 6 = 1 kΩ for the theoretical value of f o = 165.8 kHz, and only g m2 changed, with different values of 1, 1.5, 2 and 3 mS. This resulted in an NBPF at a voltage output of V BP5 with voltage gains of 0, 3.52, 6.02 and 9.54 dB. Figure 29 shows the measurement results of the V BP5 gain response. The component values used in Figure 28 were C 3 = C 4 = 0.96 nF, g m1 = g m2 = g m3 = 1 mS and R 4 = R 6 = 1 kΩ for a theoretical value of f o = 165.8 kHz, and only R 5 changed, with different values of 1, 4, 11 and 19 kΩ. This resulted in an NBPF at a voltage output of V BP6 with voltage gains of 6.12, 13.9, 21.3 and 25.52 dB. Figure 30 shows the measurement results of the V BP6 gain response. To display the independent electronically adjusted passband gains from Figure 3a, the value of g m2 was changed from 1 mS to 4 mS, and the NBPF voltage gain at the voltage output of V BP5 could be changed from 0 dB to 12.04 dB. As is shown in Figure 31, the proposed NBPF from Figure 3a exhibited independent electronic adjustment of the passband gains at a voltage output of V BP5 . To display the maximum operating passband gains from Figure 3a, the value of R 5 was changed from 1 kΩ to 25 kΩ, and the NBPF voltage gain at the voltage output of V BP6 could be changed from 6.02 dB to 28.3 dB. As is shown in Figure 32, the proposed NBPF from Figure 3a exhibited a wide range of independent amplitude adjustable passbands at the voltage output of V BP6 . Figure 33 shows the theoretical, simulated and measured NBPF responses on the V BP5 output terminal with different values of R 4 when C 3 = C 4 = 0.96 nF, R 5 = R 6 = 1 kΩ and g m1 = g m2 = g m3 = 1 mS. Figure 34 shows the measurement results of the V BP5 gain response with different Q values. In this case, the four values of R 4 were changed to 1, 1.5, 2 and 3 kΩ, and the measured values of Q were 1.14, 1.59, 2.06 and 2.98, respectively. As is shown in Figures 33 and 34, the Q parameter of the NBPF could be independently controlled by adjusting the different values of R 4 without affecting the parameters of f o , as depicted in Equations (28) and (29). For the experimental test, Figure 35 shows the input and output voltage waveforms of the NBPF response at the V BP5 output terminal, which could be extended to an amplitude of 120 mVpp without signification distortion. As shown in Figure 35, the input voltage waveform was a 165.8-kHz sinusoidal waveform, and the output voltage waveform was measured as a 165.8-kHz sinusoidal waveform with a phase difference of 0.57 • . Figure 36 shows the frequency spectrum of the NBPF response at the V BP5 output terminal. The NBPF response center frequency measured at the output of V BP5 was 165.8 kHz. The THD, including the fundamental harmonic through the sixth harmonic components of Figure 36, was about 1.12%. Figure 37 shows the THD result of the NBPF response at the V BP5 output terminal with different input voltages while keeping a constant center frequency of 165.8 kHz. Figure 38 shows the spectrum of the NBPF at the output of V BP5 , which was obtained by applying two-tone signals f 1 and f 2 around the theoretical value of f o = 165.8 kHz for IMD characterization. In Figure 38, the low-frequency tones of f 1 = 164.8 kHz and the high-frequency tones of f 2 = 166.8 kHz were used with input amplitudes of 63 mVpp. The measured value of the IMD3 was about −39.66 dBc, and the TOI was about −5.753 dBm. Figure 39 shows the P1dB of the NBPF measured at the output of V BP5 between the output power and the input power when the center frequency was 165.8 kHz. The measured output power P1dB was about −9.77 dBm. Figure 40 shows the phase noise performance of the NBPF measured at the output of V BP5 . The measured value of the phase noise of V BP5 was less than −73.87 dBc/Hz at a 30-Hz offset. According to the NBPF frequency spectrum measured in Figure 36, the dynamic range between the fundamental tone and the largest spurious of the spurious-free dynamic range was approximately 40.86 dB. Thus, the FoM of the proposed VM NBPF at the voltage output of V BP5 was approximately computed to be 3.08 × 10 6 .                  To investigate the VM fully uncoupled QSO in Figure 3b, Figure 41 shows the measured waveforms of the quadrature voltage outputs V o4 and V o5 in the time domain. The component values used in Figure 41 were C 3 = C 4 = 0.96 nF, g m1 = g m2 = g m3 = 1 mS and R 4 = R 5 = R 6 = 1 kΩ for the theoretical value of f o = 165.8 kHz. The measured FO in Figure 41 was 161.3 kHz, which was close to the theoretical value of 165.8 kHz. Figure 42 shows the spectrum of the VM fully uncoupled QSO at the V o5 output terminal. The measured FO was 164 kHz, which was close to the theoretical value of 165.8 kHz. The difference between the amplitudes of the fundamental and second harmonics was 49.67 dB, and the calculated THD was about 0.33%. As is shown in Figure 42, the third harmonic and subsequent harmonics were not visible in the spectrum, because these harmonics were lower than the noise floor. This means that the THD of the second proposed VM fully uncoupled QSO was small. Figure 43 shows the phase noise performance of the QSO measured at the output of V o5 . The measured value of the phase noise of V o5 was less than −34.7 dBc/Hz at a 30-Hz offset. Figure 44 shows the FO tuning range of the measured output voltage V o5 , where C 3 = C 4 = 0.96 nF, g m2 = 1 mS and R 4 = R 5 = R 6 = 1 kΩ, and only g m1 = g m3 were varied at values from 0.5 mS to 10 mS. As is shown in Figure 44, the measured oscillation frequency was electronically and linearly varied from 80.83 kHz to 1626 kHz. In Figures 41-44, the g m2 value should have been slightly adjusted to satisfy the CO, as depicted in Equation (34). Figure 45 shows the relationship between the measured amplitude ratio of the quadrature output voltages V o4 and V o5 and the FO as depicted in Equation (38). Figure 46 shows the relationship between the measured phase difference of the quadrature output voltages V o4 and V o5 and the FO as depicted in Equation (39). Figure 47 shows the experimental results of the voltage gain of V o6 when only R 5 was changed while maintaining C 3 = C 4 = 0.96 nF, g m1 = g m3 = 1 mS, g m2 = 1.05 mS and R 4 = R 6 = 1 kΩ. As is shown in Figure 47, the proposed VM fully uncoupled QSO (Figure 3b) exhibited a wide range of independent amplitude adjustable V o6 voltage gains. The measured operating amplitude control range could be adjusted from 1.51 to 24.03. Table 5 summarizes the performance of the second proposed isomorphic circuit.

Concluding Remarks
This paper proposes isomorphic circuits with independent amplitude control function VM NBPFs and VM fully uncoupled QSOs. The proposed VM NBPFs and VM fully uncoupled QSOs employ three LT1228 commercial ICs, two grounded capacitors and three resistors. The proposed isomorphic circuits exhibit low-output impedance and independent amplitude control, which are important for easily cascading the VM operation and independent control of the amplitude gain. The Q and f o parameters of all the proposed NBPF responses are orthogonally adjustable. The passband gain of all the proposed NBPFs can be adjusted independently without affecting Q and f o . The measured passband gains of the first proposed NBPF response can be independently tuned to 27.86 dB, and the measured phase noise is less than −83.86 dBc/Hz at a 30-Hz offset. The measured spurious-free dynamic range of the first proposed NBPF response was approximately 40.79 dB, and the calculated FoM was 3.07 × 10 6 . The measured passband gains of the second proposed NBPF response can be independently tuned to 27.58 dB, and the measured phase noise is less than −73.87 dBc/Hz at a 30-Hz offset. The measured spurious-free dynamic range of the second proposed NBPF response was approximately 40.86 dB, and the calculated FoM was 3.08 × 10 6 . By feeding back each input signal to the NBPF response, the VM fully uncoupled QSO could be obtained. During FO tuning, the voltage amplitude ratio of the quadrature output was equal. The FO could be electronically and linearly controlled by the bias current of the LT1228 IC, and one sinusoidal waveform could be independently controlled without any additional active devices. The experimental measurements confirmed the workability of the isomorphic circuits. Since the proposed isomorphic circuits have independent amplitude control functions for the NBPFs and QSOs, these isomorphic circuits are expected to have a wide variety of applications in the future, such as in phase-sensitive detection, signal processing, instrumentation, telecommunication and power systems.