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Amplifiers

In this article, we will examine the basics of propagation and motivate the use of amplifiers for RF transmission. We will also discuss some important specifications of amplifiers that are typically approached by practicing engineers.

Transmission

To understand power in RF transmission, we need to go back to high school physics and do a little thought experiment.

Imagine a Dyson Sphere, an artificial hollow sphere of matter around a star. The energy (in Joules) that the encompassed star provides spreads out uniformly in all directions. The Luminosity or Power the star puts out is based on Joules/second, or Watts. The Brightness or power received at the surface of the Dyson-sphere received is based on the distance from the star and the surface of the inner surface of the sphere (and is measured in Watts/m^2). Suppose the Dyson-sphere has a radius of r, then the surface area of that sphere is 4{pi}r^2.

If the total output power (or Luminosity) of the star is L Watts, and the surface area of that sphere is 4{pi}r^2, So the energy flowing through each square meter of the sphere every second is:

B = L/{4{pi}r^2}

We call that the star’s brightness, B, and is measured in Watts/meter^2. This is known as a form of the “Inverse Square Law”, as the relationship is 1/{r^2} not 1/{r}

If the distance, r, is doubled then the brightness, B, (or received brightness) decreases by a factor of 1/{2^2} or a quarter the brightness.

If the distance, r, is tripled then the brightness, B, (or received brightness) decreases by a factor of 1/{2^3} or a eighth the brightness.

You can also look at this from the opposite way - if you need to read a book on the surface of the Dyson-sphere, what kind of output power do you need from the star, to get the appropriate amount of brightness? To double the brightness, you need to increase the power of the star by 4 times.

If you still are not sure lets go way back to the middle school butter gun analogy, by watching the excellent video, put together by Roger Linsell of http://www.fizzics.org/.

Free Space Path Loss

As energy is transmitted by any transceiver it becomes dispersed across space and absorbed by the surrounding environment. This is typically referred to as path loss or loss or attenuation of energy over a distance. Although generally not accurate in most situations, the most common model for path loss is free-space path loss (FSPL). FSPL is the energy loss in an obstacle-free, line-of-sight path through free space. FSPL is proportional to the square of the distance:

L=({4 pi d}/lambda)^2 = ({4 pi d f}/c)^2,

where d is the distance (meters) from the transmitter, f is the signal or carrier frequency (hertz), lambda is the wavelength (meters), and c is the speed of light (meters per second). The equation above also assumes the transmitter is isotropic and is measured from the far field.

This equation is typically provided in dB which can be calculated as:

L_{dB}=10log_10(({4 pi d f}/c)^2) = 20log_10(({4 pi d f}/c)) = 20log_10(d) + 20log_10(f) + 20log_10({4 pi}/c).

Let us get some perspective on this equation from looking at the idea of receiver sensitivity (RS), which is the minimum power a signal can be received at and still recovered. For WiFi 802.11ac, this requirement is -93 dBm[FIX ME and ref]. Operating at 2.4 GHz and examining distance as a function of transmit power we can observe the limited range of Pluto alone in orange in the figure below. However, by adding a 20 dBm amplifier to Pluto we can substantially increase its range.

Practical things

Theory of Operation

In a basic sense, amplifiers can be thought of as a tradeoff between signal gain and signal corruption. Gain is simply the ratio of the input to output signal amplitude. However, signal corrupt comes fin many forms but usually categorized as noise and distortion.

Power Supply Rejection Ratio (PSSR)

If the supply of an amplifier changes, its output should not, but it typically does. If a change of X volts in the supply produces an output voltage change of Y volts, then the PSRR on that supply (referred to the output, RTO) is X/Y. The dimensionless ratio is generally called the power supply rejection ratio (PSRR), and Power Supply Rejection (PSR) if it is expressed in dB. However, PSRR and PSR are almost always used interchangeably, and there is little standardization within the semiconductor industry.

PSSR can be expressed as a positive or negative value in dB, depending on whether the PSRR is defined as the power supply change divided by the output voltage change, or vice-versa. There is no accepted standard for this in the industry, and both conventions are in use.

Log and dB

The power PdBm in dBm is equal to 10 times the base 10 logarithm of the power PmW in milliwatts (mW) divided by 1 milliwatt (mW):

P_{dBm} = 10 log_{10}({P_{mW}/{1mW}})

Power in mW Power in dBm
0.1 mW -10 dBm
0.3 mW -5 dBm
1 mW 0 dBm
3.2 mW 5 dBm
10 mW 10 dBm
32 mW 15 dBm
100 mW 20 dBm
316 mW 25 dBm

A doubling of output power (from 1mW to 2mW) is only +3dBm. A gain of +20dBm, is output power increasing by a factor of 100 times in mW.

Peak to Average

The peak-to-average power ratio (PAPR) is the peak amplitude squared (giving the peak power) divided by the RMS value squared (giving the average power).

PAPR_{%} = {(t_{peak})^2}/{(t_{rms})^2}  ; PAPR_{dB} = 10 log_{10}{(t_{peak})^2}/{(t_{rms})^2}

whether expressed in percent in dB, PAPR is dimensionless quantity.

P1dB

Normally, there is a direct relationship between input and output of an amplifier. As the input increases, the output increases the same amount. However, once the input reaches a certain level, the output signal begins to soft-limit, or compress. A parameter of interest here is the 1 dB compression point. This is the point where the output signal is compressed 1dB from an ideal input/output transfer function. This is shown in the figure within the region where the ideal slope = 1 line becomes dotted, and the actual response exhibits compression (solid).

Typically, the 1 dB compression point is a function of frequency, and as one would expect, the distortion is worse at higher frequencies.

Noise

Two techniques are typically used for analyzing the noise, but each can be cumbersome. Noise spectral density (NSD) defines the noise power per unit bandwidth. It is represented in mean-square dBm/Hz or dBFS/Hz for ADCs and rms nV/√Hz for amplifiers. This incompatibility in units provides an obstacle to calculating system noise when an amplifier is driving an ADC, or a DAC is driving an amplifier.

Noise figure (NF) is the log ratio of input SNR to the output SNR expressed in decibels. This specification, commonly used by RF engineers, makes sense in a purely RF world, but attempting to use NF calculations in a signal chain with an ADC can lead to misleading results.

Distortion

When a spectrally pure sinewave passes through an amplifier (or other active device), various harmonic distortion products are produced depending upon the nature and the severity of the non-linearity. However, simply measuring harmonic distortion produced by single tone sinewaves of various frequencies does not give all the information required to evaluate the amplifier's potential performance in a communications application. In most communications systems there are a number of channels which are “stacked” in frequency. It is often required that an amplifier be rated in terms of the intermodulation distortion (IMD) produced with two or more specified tones applied.

Intermodulation distortion products are of special interest in the IF and RF area, and a major concern in the design of radio receivers. Rather than simply examining the harmonic distortion or total harmonic distortion (THD) produced by a single tone sinewave input, it is often required to look at the distortion products produced by two tones.

As shown in the figure, two tones f1 and f2 will produce second and third order intermodulation products.

The example shows the second and third order products produced by applying two frequencies, f1 and f2, to a nonlinear device.

Order of Mixing Location of mixing products/Harmomics
first order 2f_{1}, 2f_{2}, 3f_{1}, 3f_{2}, 4f_{1}, 4f_{2}
second order f_{1} - f_{2}, f_{1} + f_{2}
third order 2f_{1} - f_{2}, 2f_{2} - f_{1}, 2f_{1} + f_{2}, 2f_{2} + f_{1}

The second order products located at f2 + f1 and f2 – f1 are located far away from the two tones, and may be removed by filtering. The third order products located at 2f1 + f2 and 2f2 + f1 may likewise be filtered. The third order products located at 2f1 – f2 and 2f2– f1, however, are close to the original tones, and filtering them is difficult.

Third order IMD products are especially troublesome in multi-channel communications systems where the channel separation is constant across the frequency band. Third-order IMD products from large signals (blockers) can mask out smaller signals.

Power Supply Limits

The relationship between dBm (power with respect to 1mW) and Volts peak-peak for a sinusoidal signal in a 50-Ohm systems is:

dBm mW Voltsrms Voltspp
-10 0.1 70.711 mV 199.970 mV
0 1 223.607 mV 632.360 mV
5 3.16 0.398 V 1.125 V
10 10 0.707 V 2.000 V
15 31.6 1.257 V 3.556 V
20 100 2.236 V 6.324 V
25 316 3.976 v 11.246 V

The question is, how do we get +20dBm (6.324V peak-peak) out of a system, when the power supply is limited to 5V? The trick is in how we connect the output stage. The output stage (RFOUT) is connected to Vcc through the inductor L1. From a DC perspective, inductors become short circuits, and RFOUT is setting at 5.0V, allowing a 10Vpeak-peak swing from the amplifier. This is also why it is AC-coupled by the output capacitor before it attaches to the antenna.

Power Dissipation

Supply Current in the ADL5606 datasheet is specified for a max on 390mA at 5.0V (or 1.95W). Although the ADL5606 is packaged in a thermally efficient 4 mm × 4 mm, 16-lead LFCSP (thermal resistance from junction to air (θJA) is 52.9°C/W. This 1.95W dissipation times the θJA is a temperature increase of 103.2 °C. Given a maximum junction temperature of 150°C, we either only use the part in sub 46.845°C environments, or work harder on spreading the heat out of the package, onto the PCB.

S Parameters

Scattering or S-parameters are commonly used to characterize 2-port networks. An S-parameter indicates the amount of power leaving one port of the network, given power entering another (or the same) port of the network. For a two port network (one input, one output), we number the ports 1 and 2, and measure 4 different ratios:

Measurement Leaving Port Entering Port
S11 1 1
S12 1 2
S21 2 1
S22 2 2

s21.jpg

In the case of S21, the suffix “21” denotes the power leaving port 2, with power delivered to port 1. Note that in the RF world, S-parameters are measured using a 50Ω system. The source impedance driving port 1 must be 50Ω, and the load impedance presented to port 2 must also be 50Ω.

Depending on what you are measuring - each measurement can be more important than another. For example, for an amplifier, S21 measures the gain of the amplifier, with port 1 being the input and port 2 being the output. If an amplifier's S21=0 dB, the amplifier has a gain of 1. For an antenna, S11 represents how much power is reflected from the antenna back, and therefore is known as the reflection coefficient (or return loss). If an antenna's S11=0 dB, then all the power being sent to the antenna is reflected from the antenna and nothing is radiated.

S Parameters are normally measured over frequency, as a good amplifier will have a flat gain over frequency. When the gain varies over frequency, it is often called ripple. Ripple is evident once S21 is measured over frequency.

Measurements

Signal To Noise Ratio

When we consider the performance of a communication link, in the most basic sense, we are interested in the bandwidth and power of the transmitted signal. Bandwidth is measured from the power spectral density of signal and is also proportional to the bit rate. We define average energy per bit as:

overline{E}_b = {overline{E}_s}/{log_2(M)}

where M is the order of the modulation scheme and overline{E}_s is the average symbol energy. Note that overline{E}_b is of units Joules/bit. When considering an Additive White Gaussian Noise (AWGN) channel, we can write the power efficiency as {overline{E}_b}/{N_0}, where N_0 is the noise power spectral density (PSD). Since AWGN is flat in frequency, or equal across all frequencies, channel noise can typically be measured over 1 Hz of spectrum. Therefore, N_0 is in units Watts/Hz right Joules. {overline{E}_b}/{N_0} can be alternatively understood as normalized signal-to-noise ratio (SNR) and is related by the bitrate R_b and channel bandwidth B

SNR = {overline{E}_b}/{N_0}*{R_s}/{B}.

SNR is commonly expressed in SNR in decibels (dB) and the equation above can be rewritten as:

SNR_{dB} = 10log_{10}({overline{E}_b}/{N_0}) + 10log_{10}({R_s}/{B})

When determining SNR of a signal it is important to understand that signals are band limited unlike noise. Therefore, as the observation bandwidth of the signal is increased the SNR becomes worse.

Noise Generation and Power

Calculation of power is a non-trivial exercise in practice and in many situations loosely defined. This is especially true when determining SNR. Let us first consider a unique example where we have scaled exponentials (s(t) = alpha exp(j pi f_c t)) in AWGN. A resulting FFT provided in the figure below, generated from some simple code for two different source signals:

  r = signal+noise;
  % View averaged spectrum
  freq = linspace(-bandwidth/2,bandwidth/2,fftLen);
  R = reshape(r,fftLen,frames);
  R = fftshift(fft(R));
  R_mean = mean(abs(R),2)/fftLen;
  plot(freq,10*log10(R_mean));

An obvious question to ask would be “What is the SNR of these signals?”. However, we cannot directly derive the SNR from this plot alone. If we are using a spectrum or vector analyzer we need knowledge of the Resolution Bandwidth (RBW), which is a function of the FFT bin count and observation bandwidth. Since we already know the observation bandwidth (1 MHz), if we used a FFT of size 1024, then the true SNR is ~10 dB. An important aspect to note from the figure above is the shape of the signals r_1 and r_2. In generation these two signals only differ in f_c, although they appear quite different in the figure. r_2 was strictly chosen to be within a FFT bin and r_2 strattles bins. This difference is due to the window effect of the FFT applied to the signals. This is commonly known as scalloping and will be dependent on the window chosen. fred harris (yes that capitalization is correct) provides a detailed overview of different windows and how to compensate for their effects in [harris1978]. When performing digital signal processing it is alway important to understand effects of discrete computations over the infinite resolution of our written equations.

Now let us connect this to the theoretical concepts. We know that the power spectral density (PSD) is obtained through a Fourier Transform of a signal's autocorrelation function. Formally written as:

S_{XX}(f) = int{-infty}{infty}{R_{XX}(tau)e^{-j2 pi f tau} d\tau}

where the autocorrelation of our process or signal X(t) is:

R_{XX}(t_1,t_2) = E[X(t_1)X^{ast}(t_2)].

In the case of AWGN this autocorrelation is simply:

R_{XX}(t_1,t_2) = sigma^2(if t_1=t_2, otherwise)

where sigma^2 is the variance of the noise. Therefore, S_{XX}=sigma^2 for all frequencies within the observation bandwidth. Therefore, the power of the AWGN signal is simply P_N = sigma^2*B. This result provides a mechanism for determining power from the PSD or frequency domain of a signal. However, it can be useful to calculate signal power from the time domain, and from Parseval's theorem we know power is identical between domains. For an AWGN signal calculating the variance directly provides P_N, but this is not true for non-zero mean processes. In that case, power can be simply calculated based on the squared RMS or squared mean of the process samples. However, a significant amount of data should be collected to get a representative power value of the signal. In MATLAB for a complex input signal the power is simply:

signalpower = sqrt(mean(signal.*conj(signal)))^2;

However, the power unit is a bit tricker to determine. An instrument will provide the I/Q data in voltage based on some input impedance. Therefore, we can calculate the power directly in dBm easily:

P_{dBm} = 10log_{10} ({P_{RMS}}/{1mW}) = 10log_{10} ({{I^2+Q^2}/2}/{R*1mW})

where R is the input resistance of the device. For an SDR I/Q values will be based on some uncalibrated ADC values and we typically define them of unit dBFS. Resulting in a power calculation that is not relatable to Watts. If it was desired to use an SDR an a instrument we would need to relate each output of the ADC to a specific input voltage at the antenna. This mapping would allow us to accurate measure quantities in SI units instead of relative units.

S Parameters

This data was taken on a Keysight ENA E5080A:

First we calibrate things with a cable, and connector, to make sure we see what is happening. We expect this to be a flat line, with 0dB of gain. (it is a cable after all).

E5080A Calibration

Then we can look at the S12 of the amplifier board. Here we can see gain between 2 and 3 GHz, with the flat part being between 2.4 and 2.5 GHz, just like we hope.

If we vary the amplitude at a constant frequency, we can see the P1dB point at +5dBm. In order to keep things operating in the linear region, we should make sure not to drive the amplifer board with more than +5dBm.

Results

The yellow line is an antenna, the red line is with the same antenna and the amplifier. You can see the +20dB of transmission at 2.4GHz.

university/tools/pluto/users/amp.1548076282.txt.gz · Last modified: 21 Jan 2019 14:11 by Robin Getz