convert wavelength, λ, to frequency, f, and vice versa
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Time domain representation of Amplitude Modulated (AM) signal
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of angle CW modulated signal
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Conversions of power levels between W, dBW and dBm.
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Time domain representation of Angle Modulated (FM/PM) signal
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in heterodyne receivers
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k·T·Δf - available noise power
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Series-to-parallel conversions
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for good conductor
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transformerlike couplings, capacitive and inductive transformers
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Optimal generator and load admittances of a 2-port for maximum gain
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Leeson's noise model
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Antennas are an integral part of any wireless system. They are used to efficiently transform guided electric signals into freely propagating electromagnetic waves. Many different types of antennas exist, ranging from simple structures consisting of a single straight wire to complex phase controlled antenna arrays with many hundreds of carefully spaced radiating elements. A number of important characteristics are used to describe an antenna. Among these are:
When defining the different antenna parameters, usually only the far-field from the antenna is considered. This is done due to the fact that the far-field from any antenna is a TEM wave propagating in the r direction of the spherical coordinate system, where the antenna is located in origin, see figure below.
The spherical coordinate system with the antenna at origin.
The far-field is defined as the electromagnetic field in the region for which the distance r is larger than the far-field distance R_{ff} , given by^{*}
Where D is the maximum physical dimension of the antenna, and λ_{0} is the wavelength corresponding to the operating frequency.
A theoretical antenna, which radiates its energy uniformly in all directions in space is called an isotropic antenna. In practice it is impossible to construct such an antenna, but the concept is useful for defining other antenna parameters, such as the antenna directivity and gain.
Directivity of the antenna describes how the antenna radiates power in different directions. The directivity D(θ, φ) is the ratio of the radiation intensity U in the direction (θ, φ), to the radiation intensity averaged over all directions.
The term "directivity" is often used with no particular direction specified. When this is the case, the direction of maximum directivity is assumed. The directivity of an antenna is often expressed in decibels with respect to the directivity of a reference antenna. An isotropic antenna with D = 1 is often employed as the reference antenna and the term dBi is used.
The antenna bandwidth describes the range of frequencies over which the antenna is able to efficiently radiate or receive energy. The antenna bandwidth is typically specified in terms of VSWR or |S_{11}| over a frequency range. The antenna is typically assumed to operate efficiently when VSWR < 2 or |S_{11}| < −10 dB.
For some applications it is impossible to meet the gain or radiation pattern requirements with one antenna. By combining multiple antennas into a one- or two-dimensional antenna array an improved antenna performance can be achieved.
An antenna array can be used too achieve:
A simple way to understand the theory behind antenna arrays is by considering the following example: n isotropic antennas, spaced by λ_{0}/2 are positioned along the z-axis direction, as illustrated in figure below.
The geometry of the example array.
A plane wave is arriving at an angle θ to the z-axis. The E-field of the wave as a function of position can be expressed as:
Where r is the position of a receiving antenna element and
is the wave vector, which describes the spatial phase variation of a plane wave^{*}. The signal at the terminals of each of the antenna elements can thus be expressed as:
For the entire array, the received signal will be the sum of the signal from each of the antennas:
A plot of the magnitude of Y(θ) versus the angle and the number of antennas n is shown in figure below
Magnitude of an antenna array output vs. θ and n.
From the figure it can be seen that although the individual antennas are isotropic, when placed in an array, their combined radiation pattern will gain directivity. The directivity will depend on the number of elements in the array – the more elements, the more directive the antenna.
The total radiation pattern, F_{a }, of a non-steered array (in a phase controlled array a complex weight is applied to the
individual elements) is typically expressed through the array function, which takes the positions of the individual antennas into account:
Where F_{i}(θ,φ) is the radiation pattern of a single antenna element, or the individual antenna element, if different antenna types are used in the same array.
The antenna impedance relates the voltage and the current at the antenna input terminals V_{t} = I_{t}Z_{A}. The antenna impedance is generally a complex, frequency dependent quantity, which can be expressed as Z_{A} = R_{A} + jX_{A}, where R_{A} is the antenna resistance and X_{A} is the antenna reactance.
Attenuators are electronic devices, which are used to reduce the power of a given signal without distorting the signal waveform. Attenuators are typically passive and are made of resistor divider networks. The two commonly used attenuator networks are called the π-pad and the T-pad attenuators. The circuit diagrams for the two network types are shown in the following figures.
π-padattenuator
T-padattenuator
As it can be seen from the figure, both attenuator types are symmetrical. This is done in order to equate the impedance on the ports, thereby making them interchangeable. The input and output impedances of the ports are typically designed
to match the characteristic impedance of the system where the attenuator is going to be used, Z_{in} = Z_{out} = Z_{0}. The design equations for a π-pad attenuator are:
,
.
And for a T-pad attenuator:
.
Where K = 10^{LdB/20} is the ratio of current, voltage or power, corresponding to a desired attenuation L_{dB}.
In addition to the radiation pattern, antennas are also characterized by their beamwidths and sometimes sidelobe levels.
The main lobe (beam) of the antenna is the region around the direction of maximum radiation, while the antenna sidelobes are smaller beams, radiating in other directions.
The half power beamwidth, or sometimes just the beamwidth, of an antenna, is typically defined as
the angular separation over which the radiation pattern decreases by 3 dB from the peak of the main beam. These parameters
are illustrated in the example radiation pattern below.
An example radiation pattern with indications of the main lobe, the sidelobes and the beamwidth
Often, when dealing with sinusoidal signals, it is cumbersome to operate with the real valued signal notations.
In order to simplify the mathematical operations, a representation of the real signals using complex phasors is often used. The advantages of using complex signal notation in mathematical analysis are:
A phasor is a complex number that is a function of time. Let us consider the complex number e^{jφ}, where φ = 2πf_{0}t = ω_{0}t. This number, also called a complex exponential, is depicted as the tip of the red vector in the complex plane in the figure below.
e^{jφ} in the complex plane.
As time t increases,φ, the phase angle of the complex exponential increases, while the amplitude of the signal is kept constant. The spiral path created by the phasor motion is shown in the following figure, where time is added as the third dimension.
The continuous motion of the phasor’s tip as a function of time and the real and imaginary parts of e^{jφ}.
The real and imaginary parts of e^{jφ} are shown as projections onto the real/time and the imaginary/time planes respectively, demonstrating Euler’s identity:
e^{jφ }= cos (φ) + j sin (φ)
If a second phasor, e^{-jφ}, which is rotating in the opposite direction of e^{jφ} is introduced into the first figure (shown in blue), the sum of the two complex exponentials, due to the nature of complex conjugation, will always result in a real valued number:
e^{jφ} + e^{-jφ} = cos (φ) + j sin (φ) + cos (φ) − j sin (φ)
e^{jφ} + e^{-jφ} = 2 cos (φ)
cos (φ) = 1/2 (e^{jφ} + e^{-jφ})
A similar expression can be derived for a sine:
sin (φ) = j/2 (e^{-jφ} - e^{jφ}).
Using these equations, a real signal can be written in complex form
Coplanar Waveguide with Lower Ground Plane, aka CPWG, is a popular type of planar transmission line, and is a part of the coplanar waveguide transmission line family.
The geometry of a CPWG transmission line.
CPWG, similarly to the microstrip transmission line, consists of a conductive trace of width W, deposited on a grounded dielectric substrate of thickness d. The difference lies in the conductive trace being flanked by ground planes from both sides, with g being the gap distance.
The CPWG lines have a number of advantages over microstrip lines:
Similar to microstrip transmission lines, the design equations for the CPWG lines are based on approximations to the static or quasi-static solutions.
Directional couplers are passive devices, whose purpose is to couple a certain part of the energy flowing in a transmission line to another port. Directional couplers are often symmetrical 4-port devices. A block diagram symbol for a directional coupler is shown in the following figure.
A block diagram symbol for a directional coupler.
As it can be seen from the figure, the 4 ports are called the input port, the transmited port, the coupled port and the isolated port. An important property that defines a directional coupler is that it only couples energy flowing in the direction (from the input port to the transmitted port) to the coupled port. The energy flowing into the transmitted port is coupled to the isolated port, which is often terminated in a matched load.
A directional coupler is characterized by the following parameters:
.
.
.
In an ideal directional coupler the insertion loss will entirely consist of the coupling loss.
.
The coupler parameters can also be expressed in terms of a scattering matrix, which for an ideal directional coupler will look like^{**}:
.
Where κ and τ are complex frequency dependent quantities. Insertion loss and coupling factor can be expressed by:
.
One of the widely used methods of constructing directional couplers is by using coupled transmission lines, as shown in the following figure:
Layout of the directional coupler based on coupled lines.
The parameters of the coupler are determined by the geometry – the width of the transmission lines, the length of the parallel sections, and the separation distance between them. For TEM transmission lines, such as the stripline and the coaxial lines these parameters can be determined through techniques such as the even-odd mode analysis and conformal mapping. For quasi-TEM lines, such as microstrip lines, the results can be obtained by numerical simulations or quasi-static techniques^{*}.
Antenna gain is a term derived from the directivity, which takes the antenna radiation efficiency into account:
G(θ,φ) = D(θ,φ)η_{rad}
Similar to directivity, when no particular direction is specified, the direction of maximum gain is assumed. Gain is also often expressed in decibel with respect to the gain of a reference antenna, for which a lossless isotropic antenna is often
employed.
Gain is sometimes expressed through another parameter, denoted the antenna effective area or the antenna aperture A_{e},
G = 4πA_{e}/λ^{2}
which is a measure of how effective an antenna is at receiving. A_{e} is related to physical area of the antenna through A_{e} = Aρ_{e }, where ρ_{e} is the antenna aperture efficiency.
One of the most widely used planar transmission line types is a microstrip transmission line. This type of transmission line is particularly popular due to simple fabrication process, as it can be integrated on a printed circuit board (PCB). The geometry of a microstrip line is depicted in the figure below:
The geometry of a microstrip transmission line.
As it can be seen from the figure, the microstrip line consists of a thin conductor of width W on a grounded dielectric substrate of thickness d. The relative permittivity of the dielectric substrate is ε_{r}. The analysis of the microstrip line is complicated by the fact that a small part of the fields propagate through the air above the conductor, while the rest propagates through the dielectric, as shown in the figure below:
Electric and magnetic field lines around a microstrip line.
Due to the difference in the dielectric properties of the two media, the microstrip transmission line can not support a pure TEM wave, as the phase velocity of the wave will be different in the air (v_{p} = c) and inside the dielectric (v_{p} = c / √ε_{r}).
In practical applications the dielectric thickness is chosen to be electrically thin: d << λ. By doing this, the fields can be considered to be quasi-TEM, and good approximations for the transmission line parameters can be obtained by curvefitting the static or quasi-static solutions^{*}. The phase velocity and the propagation constant can be expressed as:
Where k_{0} is the wave number of a plane wave in free space k_{0 }= ω√μ_{0}ε_{0}, and ε_{e} is the effective dielectric constant, which satisfies 1 < ε_{e} < ε_{r} and can be interpreted as the dielectric constant of a homogeneous dielectric medium that equivalently replaces air and dielectric regions of the microstrip line. ε_{e} is dependent on such factors as the substrate thickness, conductor width and the frequency. An approximation for the effective dielectric constant for a microstrip line is given by:
The width of the microstrip line for a given characteristic impedance and substrate ε_{r} can be calculated from the W/d ratio found by applying the following design equation:
Where terms A and B are:
Linear two-ports, such as linear amplifiers for example, often have a fixed gain over the specified bandwidth. If the output power is plotted versus the input power, as shown in figure below, the relationship will be linear and the slope of the line will be equal to the gain. In real amplifiers, as the magnitude of the input signal grows larger, at some point the amplifier will start to saturate and the gain will start to decrease. When this happens, the amplifier is said to be operating in the compression region.
Illustration of the 1 dB compression point, which corresponds to the input power that causes the gain to decrease 1 dB.
The linearity of an amplifier is typically described by its 1 dB compression point (P1dB), which is the input power level that causes the gain to decrease 1 dB from the expected linear output (in some component datasheets P1dB is specified as the output level, at which a 1 dB drop occurs). When designing circuits that contain amplifiers, it is important to keep the input signal level below P1dB, as the amplifier will start producing harmonics of the input signal on its output, when it enters the non-linear region.
The second and above order harmonics, which are produced by an amplifier operating in the non-linear region, typically lie outside of the amplifier bandwidth and cause no problems. However, non-linearity also produces a mixing effect if two or more signals are present at the input. If the two signals are close together in frequency, some of the generated frequencies, called the intermodulation products, can occur within the amplifier bandwidth and will thus interfere with the main signals, causing intermodulation distortion (IMD). The problem is illustrated in the figure below.
Illustration of intermodulation products present in an amplifier output signal due to non-linearity.
The third-order intercept point describes the capability of an amplifier to suppress the 2f_{1} − f_{2} and 2f_{2} − f_{1} two-tone, 3^{rd}-order intermodulation distortion. In this approach the 3^{rd}-order intercept point (IP3) is defined as the theoretical location, where the two 3rd-order products and the theoretical output signal become equal in power, as the input power is increased. This point is illustrated in the first figure.
One of the most popular types of planar antennas, are patch antennas, sometimes referred to as microstrip antennas. Patch antennas are particularly attractive due to their simplicity, low cost and ease of fabrication, as they, similarly to microstrip transmission lines, can be manufactured during the standard PCB manufacturing process.
A simple rectangular patch antenna with a microstrip feedline.
A simple rectangular patch antenna, fed by a microstrip transmission line, is shown in figure above. As it can be seen from the figure, the antenna consists of an electrically thin (h « λ_{0}) conductive patch placed on a dielectric substrate with relative permittivity ε_{r}, above a ground plane. The operating frequency of the antenna is determined by the length of the patch, l, which is chosen to be close to λ_{0} / 2. The approximate center frequency is then given by:
f_{0} ≈ c / (2l √ε_{r})
The principle of operation of a rectangular patch antenna can be understood by considering the antenna as an open circuited transmission line. As the current at the end of the transmission line is zero, due to the open end, it takes its maximum value at the center of the patch. The voltage, on the other hand, is 90° out of phase, so it takes its maximum value at the open end of the transmission line, its minimum value at the center, and its maximum negative value at the feed point, as illustrated in figure below.
The voltage and current distributions over the length of a rectangular patch antenna.
The E-field lines underneath the patch antenna are also sketched in this figure. The fringing fields near the surface of the patch antenna appear to have a horizontal component in the same direction at both edges, adding up in phase, and thus giving rise to the radiation^{*}. The fringing fields are also responsible for a down shift in the actual resonance frequency of the patch, compared to the one calculated using formula above. This shift occurs due to an apparent extension of the patch length by the fringing fields, which can be modelled as radiating slots. This apparent extension can be approximated by^{**}:
The input impedance of a theoretical patch antenna is infinitely high, due to the 0 current at the feed point. In practice, the input impedance of a patch antenna with l = W is Z_{in} ≈ 400 Ω. As such high impedance is impractical, a number of ways exist to lower it. Some of the most popular ways are:
The two latter methods are illustrated below.
Patch antenna with inset feed.
Patch antenna with coaxial feed.
Rectangular patch antennas are linearly polarized, with the direction of polarization going along the length of the patch. These antennas are generally very narrowband, with a bandwidth as low as 3% of the center frequency. The bandwidth can be slightly improved by increasing the width of the patch. A number of methods have been proposed to make significant improvements of the patch bandwidth.
The polarization of an antenna describes the orientation of the electric field of the radio wave emitted by the antenna in relation to the surface of the Earth. The polarization is dependent on the type of the antenna, its construction and its spatial orientation. Generally, the polarization should be considered as a sum over time of projections of the electric field onto a plane perpendicular to the motion of the radio wave. In most cases, the polarization varies over time, which gives the projected shape an elliptical form. Special cases of polarization are circular polarization (where both axes of the ellipse are equal) and linear polarization (where the projection falls onto one of the axes).
Practical antennas are never polarized in a single mode. Hence, a parameter called cross polarization is used to describe the ratio of the opposite polarization component to the desired polarization component.
The power accepted by the antenna depends on the antenna resistance,R_{A}, and the current at the antenna input terminals,I_{t },:
P_{t} = 1/2 R_{A} |I_{t}|^{2} .
For a lossless antenna, all of the accepted power is converted into unguided electromagnetic waves and radiated. If the antenna is lossy, a part of the accepted power is dissipated by the antenna and converted into heat, while the remaining
part is radiated. The antenna radiation efficiency, η_{rad}, is defined as the ratio of the radiated power to the accepted power:
η_{rad} = P_{rad }/ P_{t} .
A graph, that shows the relative field strength versus the direction at a fixed distance from the antenna in the far-field is called the radiation pattern of the antenna. The radiation pattern is a plot of a three-dimensional function F(θ,φ), which varies with both θ and φ in a spherical coordinate system. An example of a radiation pattern is shown in the following figure.
An example radiation pattern of a patch antenna
Sometimes, in order to avoid complex three-dimensional plots, the plot is given as the magnitude of the normalized field strength versus θ for a constant φ (called the E-plane pattern) and the magnitude of the normalized field strength vs. φ for θ = π/2 (called the H-plane pattern)^{*}.
When dealing with high frequency components such as transmission lines, filters, amplifiers, attenuators and couplers, a component can typically be represented as an n-port network.
A 2-port network with incident and reflected waves at both ports
As it is difficult to measure the individual voltages and currents at high frequencies (microwave range and above), but easier to measure the ratios of the incident and the reflected waves, a scattering matrix approach is often more convenient for the description of n-port networks. A 2-port network, such as the one depicted above, can thus be characterized by using a scattering matrix:
Where the generally complex matrix elements, the scattering parameters or the S-parameters, represent the relations between the incident and the reflected waves of the network ports as follows:
V_{1}^{− }= S_{11}V_{1}^{+} + S_{12}V_{2}^{+}
V_{2}^{− }= S_{21}V_{1}^{+} + S_{22}V_{2}^{+}
S_{11} and S_{22} are the reflection coefficients on port 1 and 2, respectively, when the opposite port is terminated in a matched load. S_{21} represents the transmission from port 1 to port 2, when port 2 is terminated in a matched load and vice versa for S_{12}.
Knowledge of the reflection coefficients can be used to calculate the input impedances of the network ports:
Z_{n} = Z_{0}·(1 + Γ_{n}) / (1 − Γ_{n}) = Z_{0}·(1 + Snn) / (1 − Snn)
Here you find book chapters in .pdf format as well as references to the relevant quizzes and RF calculators.
Continuous wave
AM, FM, PM
Parallel and series resonators
R, L, C
Linear, Active Two-Port Networks. Optimal power gain, stability, reciprocity, ...
2-ports
Sources of noise, noise in semiconductors, distortion, ...
k·T·Δf
Power and nonlinear RF-amplifiers
Amp.
Basics: feedback and negative conductance criteria; circuits
(_{˜})
Mixer basics, differential stage, Gilbert-cell, detectors
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The maximum possible power which goes to two sidebands is approximately 33 %
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the relative amplitude of the fifth pair of sidebands of an FM signal with β_{eff }= 7 is 0.3479
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The 3rd pair of sidebands are spaced from the carrier by 3 x 3kHz = 9 kHz.
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The bandwidth by Carson's rule is 2x(9kHz + 3kHz) = 24 kHz.
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The impedance magnitude of a high-Q parallel resonator reaches maximum at resonance frequency.
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The impedance magnitude of a high-Q series resonator reaches maximum at resonance frequency.
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The impedance phase of resonators at resonance frequency is 0.
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The amplitude characteristic of a first-order low-pass circuit has an out-of-band roll-off of 20 dB / decade.
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A two-port that can deliver net-power to a surrounding network at a given frequency is called active.
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Reciprocity of a network implies that y_{12}=y_{21}
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The maximum frequency of oscillation, f_{max} usually is higher than the cut-off frequency, f_{T}.
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A unilateral (y_{12}=0) two-port network is absolutely stable if the real part conductances g_{11 }> 0; g_{22} >
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The mean squared noise current, i^{ 2}, from shot noise in a frequency banf Δf is given by 2q·I·Δf.
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The mean squired short-circuit noise current from thermal noise i^{ 2} = 4k·T·Δf / R
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All odd ordered Taylor expansion terms would contribute to the fundamental signal component.
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The 2nd order term in the Taylor expansion causes a DC contribution.
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Class A amplifiers provide maximum linearity.
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Class AB amplifiers provide maximum output power.
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Most output power from a given transistor is achieved when θ ≈ 245°.
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Increasing conduction angle will lead to decrease of efficiency and increase of output power.
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The frequency of oscillation is equal to the resonance frequency if the only ohmic component resides across the capacitors of the resonator.
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The amplifier compensates for losses in the feedback circuit.
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To ensure start of oscillation, the loop gain must start being grater than one.
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The oscillator noise characteristics will degrade if you increase bandwidth of the resonance circuit.
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The primary function of a mixer is to multiply two signals.
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Conversion gain of a mixer is the ratio between IF signal and RF signal.
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With a large signal input to LO port, the upper differential pairs in Gilbert-Cell mixer operate like switches.
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The oscillator noise characteristics will degrade if you increase bandwidth of the resonance circuit.
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