# User manual | Square Law and Linear Detection Application Note 986 ```Square Law
and Linear Detection
Application Note 986
100
Introduction
Above about -10 dBm the slope is closer to linear but may vary about
30% for different values of frequency, diode capacitance, and load
resistance. The slope may be controlled by tuning at the proper power
level. Linear detection is used in power monitors. In some applications
the linearity is important because the detected voltage is a measure of
power input.
RL = 100K
DETECTED VOLTAGE (mV)
Schottky diode detectors are used to detect small signals close to the
noise level and to monitor large signals well above the noise. From the
noise level up to about -20 dBm (Figure 1) the slope of the response
curves is constant. This is the square law region. Video receivers
usually operate in this range. The diode receives the signal directly
from the antenna in most systems, although a preamplifier may be used
to improve sensitivity. This type of receiver is used in short range radar
or in counter-measure equipment where the sensitivity of the more
complicated superheterodyne receiver is not needed.
10
100K
1K
1
5082-2800
0.1
FREQUENCY 2 GHz
0.01
-40
-30
INPUT POWER (dBm)
-20
Figure 1. Square Law Response
2.4
Detection Law
2.2
Over a wide range of input power level, P, the output voltage, V,
follows the formula
P)
∝
At low levels, below -20 dBm, ∝ is two. This is the square law region.
When DC bias current is used (usually microamperes), the diode
impedance is independent of power level so the tuning can be done at
any level. Usually the diode is tuned at -30 dBm. The detected voltage
at this level is called the voltage sensitivity.
1.8
1.6
1.4
1.2
1.0
0.8
At higher power levels the diode impedance changes with power. At
these levels, the value of ∝ can be as low as 0.8 (Figure 2). The slope is
related to diode capacitance, frequency, and load resistance. When the
circuit is retuned at each power level, the output and the slope
increase.
5082-2824 DIODE 2 GHz FREQUENCY
100 K OHM LOAD RESISTANCE
TUNED AT -30 dBm
2.0
DETECTION LAW
V=K(
1K
5082-2824
-40
-30
-20 -10
0
10
POWER INPUT (dBm)
20
Figure 2. Variation of Detection Law
with Input Power
30
2
Frequency and Diode Capacitance –
Effect on Voltage Sensitivity
The diode junction capacitance shunts the junction resistance.
Detected voltage is reduced because some of the input current flows
through the capacitance and never reaches the resistance where
detection takes place. The effect is more serious at higher frequencies
because capacitance susceptance is proportional to frequency.
Similarly, the effect is more serious for higher capacitance diodes.
To illustrate these effects, measurements of voltage sensitivity at 20
microamperes bias were made (Figure 1) with a 5082-2800 diode with a
zero bias junction capacitance of 1.40 pF and a 5082-2824 diode with
0.88 pF.
100
Ratio of Voltage Sensitivity at 2 GHz to Voltage Sensitivity
at 1 GHz
1K
Measured Ratio
0.30
Computed Ratio
-2824
100 K
1K
0.29
0.46
0.30
100 K
0.52
0.48
Two values of load resistance were used. The detected voltage is about
70% less at the higher frequency for the 5082-2800 diode, about 50% less
for the lower capacitance 5082-2824 diode. Load resistance has little
effect on the ratio. The results are summarized in the table. These
measurements agree with the computed ratios. See Appendix.
Frequency and Diode Capacitance –
Effect on Slope in High Level Detection
Figure 3 shows how the slope of the detection characteristic becomes
steeper at higher frequencies. The output voltage at 2 GHz is nearly
equal to the 1 GHz voltage above 22 dBm. The main reason for this
behavior is the lower value of junction resistance at higher power. This
is explained in greater detail in the Appendix.
Since frequency and diode capacitance appear in the degradation
factor as the product fC, the increasing slope at higher frequency also
happens at higher capacitance. This is shown in Figure 4 where the
detection characteristic for the higher capacitance 5082-2800 diode is
steeper than the characteristic for the -2824 diode.
OUTPUT VOLTAGE (VOLTS)
-2800
Load Resistance
10
1 GHz
1
2 GHz
0.1
-10
-2
30
100
2 GHz FREQUENCY
100 K OHM LOAD RESISTANCE
TUNED AT -30 dBm
10
5082-2824
VBR 22 VOLTS
Cjo 0.95 pF
1
5082-2800
VBR 86 VOLTS
Cjo 1.38 pF
Effect of Breakdown Voltage
In Figure 4 the curves cross so that above 22 dBm the voltage detected
by the -2800 is higher than the voltage detected by the lower
capacitance -2824 diode. The degradation factor analysis does not
explain this crossover. It is related to the effect of breakdown voltage.
6
14
22
POWER INPUT (dBm)
Figure 3. Higher Frequency Increases
Slope
VOLTAGE OUTPUT (VOLTS)
Diode
5082-2800 DIODE 100 K OHM LOAD
RESISTANCE TUNED AT -30 dBm
0.1
-10
-2
6
14
22
POWER INPUT (dBm)
Figure 4. Higher Capacitance
Increases Slope
30
3
The -2800 diode has a high voltage breakdown and so does not exhibit
this reverse conduction effect.
Effect of Load Resistance on Detection Slope
Figure 5 shows that the slope of the detection curve increases when
the load resistance is decreased from 100 K ohms to 1 K ohms. The
detection law increases from 0.9 to 1.0. The Appendix analysis shows
that the rectified current at high power causes a decrease in the
degradation factor due to low C and Rj. When the load resistance is
decreased the rectified current increases, as seen in Figure 6. The
degradation factor decreases so the slope is steeper.
100
5082-2824 DIODE 2 GHz FREQUENCY
VOLTAGE OUTPUT (VOLTS)
At high power levels the negative portion of the input signal is large
enough to cause reverse conduction. This negative detected voltage
reduces the output level and the curve levels off. At higher levels the
negative detected voltage will predominate and the curve will reverse
direction and have a negative slope.
Summary
RL = 100 K
RL = 1 K
1
0.1
-10
-2
6
14
22
POWER INPUT (dBm)
30
Figure 5. Load Resistance Effects
Slope
10
Tuning for Linear Response
5082-2824 DIODE 2 GHz FREQUENCY
DIODE CURRENT (mA)
In applications where the detector is used as a power meter linear
detector response is needed. Microprocessors are often available to
correct the diode response but this involves added expense. Reducing
the load resistance can often produce the desired response but the
sensitivity is reduced. Figure 7 shows how the sensitivity can be
improved while the response is corrected. When the slope is too
shallow (the usual case) the correction can be made by tuning at a
higher level instead of tuning at -30 dBm. In this case the tuning level
was +20 dBm. If the circuit is tuned for maximum output at 20 dBm the
response will be too steep. A few iterations are necessary to get the
response shown.
10
1
RL = 1 K
0.1
RL = 100 K
0.01
-30
-20
-10
0
10
20
POWER INPUT (dBm)
30
Figure 6. Diode Current
Appendix
Detection Degradation
Parasitic series resistance and junction capacitance degrade the
performance of Schottky detectors. Some of the voltage applied across
the diode appears across the series resistance Rs and is not available to
be detected by the junction resistance Rj. A more serious effect is the
division of current between the junction resistance and the junction
capacitance Cj. The degradation factor is:
1+
RS
+ 4 π 2 f 2 Cj2 RS Rj
Rj
100
5082-2824 DIODE
2 GHz FREQUENCY
100 K OHM LOAD RESISTANCE
VOLTAGE OUTPUT (VOLTS)
At low power levels (receiver applications) detected voltage is
proportional to input power. This is the square law detection region. At
higher levels (power monitor applications) the slope of the transfer
characteristic depends on the frequency, diode capacitance, and load
resistance. Linear detection can be obtained by tuning the diode at a
high power input level.
10
TUNED AT +20 dBm
1
TUNED AT -30 dBm
0.1
-10
-2
6
14
22
POWER INPUT (dBm)
Figure 7. Tuning for Linear Response
30
The resistance values are 1400 ohms for junction resistance for both
diodes at 20 microamperes forward bias, 20 ohms for -2800 series
resistance, and 8 ohms for -2824 series resistance. Junction capacitance
at zero bias is 1.4 pF for the -2800 diode and 0.95 pF for the -2824 diode.
At forward bias the capacitance increases by the factor
VB
VB - VF
where VB is the barrier voltage, 0.64 voltage for these diodes, and VF is
the forward voltage, 240 mV for the -2800, 260 mV for the -2824. The
degradation factor is
1.01 + 3.42 f 2
for the -2800, and
1.01 + 0.57 f 2
for the -2824.
The ratio of this factor at 1 GHz to that at 2 GHz is 0.30 for the -2800
and 0.48 for the -2824.
This degradation factor also explains the increasing slope of the
detection curve at higher values of frequency and diode capacitance.
Consider the -2800 at 1 and 2 GHz. We have seen that the ratio of the
degradation factor for these frequencies at low power is 0.3.
The power supply voltage is 2.24 volts, 2 volts for 20 microamperes
through the 100 K resistor and 0.24 volts for the diode forward voltage.
At 1 watt input level the rectified current increases the diode current to
93 microamperes. This changes the diode voltage to 2.24 - 9.3 = -7.06
volts. This back bias lowers the junction capacitance to 0.5 pF. In
addition, the junction resistance decreases to 2800/93 = 300 ohms. The
degradation factor is now
1.07+ 0.059 f 2
and the 1 to 2 GHz ratio is 0.86. At 2 GHz the slope increases to bring
the curves closer together at the higher levels.
Notice that the -2800 diode junction capacitance of 1.4 pF is reduced to
0.5 pF at higher input power level. Diodes that are rejected for high
capacitance at lower levels may be quite acceptable for high level
detection.
www.semiconductor.agilent.com
Data subject to change.
Copyright © 1999 Agilent Technologies, Inc.
5953-4444 (11/99)
```
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