Design of Attenuators
Updated to include also Tattenuators on 20120212.
If you are just looking for the excel sheet that helps with the
design of attenuators the link is provided here:
An excel sheet for attenuator design
Background
In RF systems it is often necessary to attenuate a signal. This has
to be done in an impedancematched environment where there shall be a
minimum of impedance mismatches and reflections. Two popular
attenuator topologies are the pi attenuator (so called because the
configuration of resistors resembles the Greek letter pi) and the
Tattenuator (for its topological similarity with the letter T). The
schematic of a pi attenuator is shown in figure 1 below:
Figure 1. The topology of a pi attenuator.
The schematic of a T attenuator is shown in figure 2 below:
Figure 2. The topology of a T attenuator.
You might wonder why two resistors have the same designator in the
figures. This is because we want the attenuators to be symmetrical so
that they provides the same function no matter which end of it connects
to the load and which connects to the source. There are two
requirements on the networks, namely to match the impedance and to
provide a specific value of attenuation, so it should be enough with
two degrees of freedom. We do therefore not need to keep all three
resistor values free and making this observation here makes the
derivations easier.
Derivation of pi attenuator formulas
So how does one select R1 and R2 for the pi attenuator? The two
requirements mentioned above yields one equation each.
First we want the impedance looking in from the left to be equal to
the system impedance Z_{0}:
R_{2}//(R_{1} +
R_{2}//Z_{0}) = Z_{0}
Where // denotes parallel connection of resistors.
Simplifying this yields:
R_{1} =
2*R_{2}*Z_{0}^{2}/(R_{2}^{2}
 Z_{0}^{2}) (I)
The other requirement is that the network shall provide a certain
attenuation. The formula for voltage division yields:
V_{out} = V_{in} *
(R_{2}//Z_{0}) / (R_{1} +
R_{2}//Z_{0})
The attenuation (gain) A is defined as:
A =
V_{out}/V_{in}
Combining the latter two formulas yields:
R_{2}*Z_{0}*(1A) =
A*R_{1}*( R_{2} + Z_{0} )
Inserting (I) in this formula and simplifying yields:
R_{2} =
Z_{0}*(1+A)/(1A) (II)
Inserting (II) in (I) and simplifying yields:
R_{1} =
Z_{0}*(1A^{2})/2A (III)
So these are the formulas to obtain the resistor values of a pi
attenuator given a certain system impedance and a desired attenuation.
Derivation of T attenuator formulas
The resistor values of T attenuators can be derived in a similar
manner as for the pi attenuator above.
First, let's find out the attenuation (gain) of the network when
loaded with the impedance matched load Z_{0}. Using the
formula for voltage division twice yields the expression:
V_{out} =
V_{in} * (R_{2}//(R_{1} + Z_{0})) /
(R_{1} + R_{2}//(R_{1} + Z_{0})) *
Z_{0}/(R_{1} + Z_{0})
Setting the gain A = V_{out}/V_{in} yields:
A = V_{out}/V_{in} =
(R_{2}//(R_{1} + Z_{0})) /
(R_{1} + R_{2}//(R_{1} + Z_{0})) *
Z_{0}/(R_{1} + Z_{0}) (IV)
Second, we want the impedance looking in from the left to be equal to
the system impedance Z_{0}:
R_{1} + R_{2}//(R_{1} +
Z_{0}) = Z_{0}
Here we can se that the somewhat complicated term
R_{2}//(R_{1} + Z_{0}) appears again just
like it did in (IV) and we can use this fact to solve for it to get
a simpler expression we can put into (IV) to simplify that equation:
R_{2}//(R_{1} +
Z_{0}) = Z_{0} 
R_{1} (IV)
(V) in (IV) now yields:
A =
(Z_{0}  R_{1}) / Z_{0} *
Z_{0}/(R_{1} + Z_{0}) =
(Z_{0}  R_{1}) / (R_{1} + Z_{0})
(VI)
Solving (VI) for R_{1} yields:
R_{1} =
Z_{0} * (1A)/(1+A) (VII)
We can put this expression for R_{1} into e.g. (IV) and
simplify to get:
R_{2} =
Z_{0} *
2A/(1A^{2}) (VIII)
So these are the formulas to obtain the resistor values of a T
attenuator given a certain system impedance and a desired attenuation.
Should I use a T attenuator or a pi attenuator?
It can be shown that the two attenuator topologies given here have
identical port impedances and transfer functions. If one is given a
black box (a box that one cannot look into) attenuator and an
ohmmeter, it is impossible to determine which topology is in the box
as all resistances will be the same.
From this one might conclude that it does not matter which of the
topologies one decides to implement, but in reality there may be other
considerations depending on what resistors are conveniently available
in the technology used (thin film vs. thick film) and the attenuation
one is after. Parasitics may also play a role at high frequencies when
determining which topology is better.
In general I would guess the pi attenuator has better frequency
response since there is less series inductance in the signal path and
also less series inductance in the shunt path. For normal SMD
resistors the series inductance is likely to dominate over the
parallel capacitance. So the pi version is generally to be preferred
if getting the maximum bandwidth is important.
Identifying unmarked attenuators
Occasionally one may encounter an unmarked attenuator and it can be
useful to have a quick and easy way of determining what attenuation it
has using a simple ohmmeter. The straightforwad method is to measure
the resistance from input to output or from input to GND. It is of
course trivial to calculate what these resistances will be for a given
attenuation when the individual resistor values are known and I
included formulas for this in the spread sheet below as well as rows
in the table with example values.
As mentioned above, there is no difference between the port
impedances of ideal pi or T attenuators with the same attenutation.
For attenuators with lots of attenuation (say more than 3040 dB) the
ohmmeter method is however not very precise as the measured
resistances change very little with changing attenuation. One may then
have to resort to actually measuring the attenuation using a network
analyzer (if conveniently available) or an ad hoc "DC network
analyzer" built from a DC supply that feeds one end of the attenuator
and measuring the output voltage at the other end. Applying 1.000V at
the input makes it easy to read out the unterminated gain using a
voltmeter. Be sure not to overload the attenuator though. Some small
attenuators might not be able to withstand even 1W (7V into 50 ohms),
but 1V is probably safe for all attenuators.
I included the unterminated voltage gain in the spread sheet and
table for this purpose.
Using standard resistor values
In real life one has to stick with standard resistor values and this
(as well as component tolerances) will result in nonideal behavior of
the attenuator. To get closer to the ideal values, one can use
parallel combinations of resistors like in the figure below:
Figure 3. A pi attenuator with parallel resistors to
get closer to the ideal resistor values using standard values.
There are various ways in which to measure the nonideality of a
given attenuator. One can easily calculate the input impedance and the
attenuation using the selected resistor values and compare those to
the desired ones. Based on the calculated input impedance, one can
also proceed to calculate other figures of merit, like the reflection
coefficient (Γ), the voltage standing wave ratio (VSWR) and the
return loss (RL). These numbers can be found as:
Γ =
(Z_{in}  Z_{0})/(Z_{in} +
Z_{0})
RL = 20*log_{10}(Γ)
VSWR = (1 + Γ)/(1  Γ)
If Z_{in} > Z_{0} then this simplifies to:
VSWR =
Z_{in}/Z_{0} (when
Z_{in} > Z_{0})
If instead Z_{in} < Z_{0} this formula applies:
VSWR =
Z_{0}/Z_{in} (when
Z_{in} < Z_{0})
The above formulas are used in the following excel sheet to aid
in the selection of resistor values for specific values of
Z_{0} and specific values of attenuation:
An excel sheet for attenuator design
Ideal and suggested values for some specific 50Ω pi attenuators
are also provided in the table below.
Attenuation
 [dB]
 1
 2
 3
 6
 10
 20
 30
 40
 Voltage gain
 [V/V]
 0.89
 0.79
 0.71
 0.501
 0.316
 0.1000
 0.0316
 0.0100
 R1 ideal
 [ohm]
 5.77
 11.61
 17.61
 37.35
 71.15
 247.5
 790
 2500
 R2 ideal
 [ohm]
 870
 436.2
 292.4
 150.5
 96.25
 61.11
 53.27
 51.01
 Endtoend resistance
 [ohm]
 5.75
 11.46
 17.10
 33.23
 51.95
 81.82
 93.87
 98.02
 EndtoGND resistance
 [ohm]
 436.21
 220.97
 150.48
 83.54
 61.11
 51.01
 50.10
 50.01
 Unterminated voltage gain
 [V/V]
 0.993
 0.974
 0.943
 0.801
 0.575
 0.198
 0.0632
 0.0200
 Suggested E24 values









 R1a
 [ohm]
 11
 24
 39
 75
 160
 680
 1000
 6200
 R1b
 [ohm]
 12
 22
 33
 75
 130
 390
 3600
 4300
 R2a
 [ohm]
 910
 470
 300
 160
 100
 62
 56
 51
 R2b
 [ohm]
 20000
 6200
 12000
 2700
 2700
 4700
 1100
 
 Zin
 [ohm]
 49.98
 49.91
 50.19
 50.15
 50.19
 50.07
 49.99
 50.01
 Zin error
 [%]
 0.04%
 0.19%
 0.38%
 0.29%
 0.37%
 0.13%
 0.02%
 0.01%
 Voltage gain

 0.89
 0.80
 0.70
 0.500
 0.315
 0.0999
 0.0319
 0.0098
 Attenuation
 [dB]
 1.0
 2.0
 3.0
 6.0
 10.0
 20.0
 29.9
 40.1
 Attenuation error
 [dB]
 0.00
 0.02
 0.04
 0.01
 0.04
 0.01
 0.08
 0.13
 Reflection coefficient

 0.0002
 0.0009
 0.0019
 0.0014
 0.0019
 0.0007
 0.0001
 0.0001
 VSWR

 1.000
 1.002
 1.004
 1.003
 1.004
 1.001
 1.000
 1.000
 Return loss
 [dB]
 73
 61
 54
 57
 55
 64
 82
 85

References
Here is a link to a page with some additional information about
attenutators, including information about other topologies:
Microwaves
101 attenuator page
