Understanding L-Pad attenuator

Question

I'm not understanding the concept of L-Pad attenuator other than seeing it as another ordinary voltage divider to attenuate a voltage. Electronics-Tutorials: L-pad Attenuator gives the following description:

L-pad attenuators are commonly used in audio applications to reduce a larger or more powerful signal while matching the impedance between the source and load in provide maximum power transfer. However, if the impedance of the source is different to the impedance of the load, the L-pad attenuator can be made to match either impedance but not both.

I understand maximum power transfer occurs when input impedance matches output impedance (or \$ Z_{in} = Z_{out} \$), and that \$ \text{dB} = 20log({V_{out} \over V_{in}}) \$. In terms of impedance matching, looking into the attenuator from the left, \$ Z_{in} = (Z_L || R_2)+R_1 \$ and from the right, \$ Z_{out} = (Z_s + R_1) || R_2 \$ in series. Hence for the two impedances to match and if I'm given a voltage attenuation ratio, I'll have exactly the following two equations to solve for two unknown values of \$ R_1 \$ and \$ R_2 \$:

1. Matching input and output impedance: \$ (Z_L || R_2)+R_1 = (Z_s + R_1) || R_2 \$.
2. Voltage attenuation ratio(): \$ V_{out} = V_{in} \big({{R_2 || Z_L} \over {({R_2 || Z_L})+Z_S+R1}}\big) \$

Aren't these two equations all I need to match input and output impedance while achieving a specific attenuated voltage? Why are there these logarithmic equations with a 'K' value solving for \$ R_1 \$ and \$ R_2 \$? And what does it mean when they say

L-pad attenuator can be made to match either impedance but not both

Answer 1

L-pad attenuator can be made to match either impedance but not both

What it means is that a simple L-pad made from two resistors can do one of two things: -

• Provide an attenuation
• Match two impedances

But you cannot simultaneously match two impedances AND have independent control of the desired attenuation level. The reason is because there are only two resistors and there are not enough variables (R1 and R2) to get both a defined attenuation AND a defined impedance matching circuit.

To get both, you need at least three resistors. This webpage gives an example and shows the two formulas that define R1 and R2: -

As you should be able to see, if you define \$R_{IN}\$ and \$R_L\$ from the outset, you get unique values for R1 and R2. Those unique values are fixed by \$R_{IN}\$ and \$R_L\$ and produce an attenuation that is fixed. In other words, it is the value of \$R_{IN}\$ and \$R_L\$ that produce R1 and R2 and that, in turn, means no control over what the attenuation is.

Answer 2

Andy Aka's answer is definitely correct and provides good insight into the problem. Here I will just try to provide an answer to

Why are there these logarithmic equations with a 'K' value solving for \$R_1\$ and \$R_2\$?

Such equations provide a direct link between the attenuation you want to obtain and the resistors in your L-pad attenuator when source and load impedances are the same (\$Z_S=Z_L\$, as you can see in the tutorial you mention) and you want to match your attenuator to the source impedance (not an optimal choice, as the tutorial itself points out).

\$K\$ is defined as an attenuation, i.e. the inverse of the "gain" \$V_{OUT}/V_{IN}\$ (if we choose to consider voltages): \$K\triangleq V_{IN}/V_{OUT}\$. In this way, what the tutorial calls \$dB\$ is of course just \$K\$ expressed in decibel: \$dB=20\log(K)\$.

Then, \$V_{OUT}=V_{IN}\frac{R_2||Z_L}{R_1+R_2||Z_L}=V_{IN}\frac{Z_S-R1}{Z_S}\$ since \$Z_S=R_1+R_2||Z_L\$ is the condition for source matching. After some algebra we obtain \$R_1=Z_S\frac{K-1}{K}=Z_S\frac{10^{dB/20}-1}{10^{dB/20}}\$ as reported in the tutorial.

The formula for \$R_2\$ can be obtained by substituting the above result for \$R_1\$ into \$Z_S\$ and expanding the parallel resistor \$R_2||Z_L\$: \$Z_S=Z_S\frac{K-1}{K}+\frac{R_2Z_L}{R_2+Z_L}\$. Then recall that \$Z_S=Z_L\$ and solve in \$R_2\$.

The above allows you to set the desired attenuation \$K\$ through a circuit that is matched to your source only, as you can easily see by taking the values obtained in the tutorial: \$Z_S=Z_L=R_2=8\,\Omega\$ and \$R_1=4\,\Omega\$.