User:Constant314/Component values for resistive pads and attenuators

Component values for resistive pads and attenuators new article content ...

This section concerns pi-pads, T-pads and L-pads made entirely from resisters and terminated on each port with a purely real resistance.

• All impedances, currents, voltages and two-port parameters will be assumed to be purely real. For practical applications, this assumption is often close enough.

• The pad is designed for a particular load impedance, Z_Load, and a particular source impedance, Z_s.

• The impedance seen looking into the input port will be Z_S if the output port is terminated by Z_Load.
• The impedance seen looking into the output port will be Z_Load if the input port is terminated by Z_S.

Reference figures for attenuator component calculation

This circuit is used for the general case, all T-pads, all pi-pads and L-pads when the source impedance is greater than or equal to the load impedance.

The L-pad computation assumes that port 1 has the highest impedance. If the highest impedance happens to be the output port, then use this figure.

Unique resister designations for Tee, Pi and L pads.

The attenuator two-port is generally bidirectional. However in this section it will be treated as though it were one way. In general, either of the two figures above applies, but the figure on the left (which depicts the source on the left) will be tacitly assumed most of the time. In the case of the L-pad, the right figure will be used if the load impedance is greater than the source impedance.

Each resister in each type of pad discussed is given a unique designation to decrease confusion.

The L-pad component value calculation assumes that the design impedance for port 1 (on the left) is equal or higher than the design impedance for port 2 (on the right).

Terms used

• Pad will include pi-pad, T-pad, L-pad, attenuator, and two-port.
• Two-port will include pi-pad, T-pad, L-pad, attenuator, and two-port.
• Input port will mean the input port of the two-port.
• Output port will mean the output port of the two-port.
• Symmetric means a case where the source and load have equal impedance.
• Loss means the ratio of power entering the input port of the pad divided by the power absorbed by the load.
• Insertion Loss means the ratio of power that would be delivered to the load if the load were directly connected to the source divided by the power absorbed by the load when connected through the pad.

Symbols used

Passive, resistive pads and attenuators are bidirectional two-ports, but in this section they will be treated as unidirectional.

• Z_S = the output impedance of the source.
• Z_Load = the input impedance of the load.
• Z_in = the impedance seen looking into the input port when Z_Load is connected to the output port. Z_in is a function of the load impedance.
• Z_out = the impedance seen looking into the output port when Z_s is connected to the input port. Z_out is a function of the source impedance.
• V_s = source open circuit or unloaded voltage.
• V_in = voltage applied to the input port by the source.
• V_out = voltage applied to the load by the output port.
• I_in = current entering the input port from the source.
• I_out = current entering the load from the output port.
• P_in = V_in I_in = power entering the input port from the source.
• P_out = V_out I_out = power absorbed by the load from the output port.
• P_direct = the power that would be absorbed by the load if the load were connected directly to the source.
• L_pad = 10 log₁₀ (P_in / P_out ) always. And if Z_s = Z_Load then L_pad = 20 log₁₀ (V_in / V_out ) also. Note, as defined, Loss ≥ 0 dB
• L_insertion = 10 log₁₀ (P_direct / P_out ). And if Z_s = Z_Load then L_insertion = L_pad.
• Loss ≡ L_pad. Loss is defined to be L_pad.

Symmetric T pad resister calculation

$A=10^{-Loss/20}\qquad R_{a}=R_{b}=Z_{S}{\frac {1-A}{1+A}}\qquad R_{c}={\frac {Z_{s}^{2}-R_{b}^{2}}{2R_{b}}}\qquad \,$ see Valkenburg p 11-3

Symmetric pi pad resister calculation

$A=10^{-Loss/20}\qquad R_{x}=R_{y}=Z_{S}{\frac {1+A}{1-A}}\qquad R_{z}={\frac {2R_{x}}{\left({\frac {R_{x}}{Z_{0}}}\right)^{2}-1}}]\qquad \,$ see Valkenburg p 11-3

L-Pad for impedance matching resister calculation

If a source and load are both resistive (i.e. Z₁ and Z₂ have zero or very small imaginary part) then a resistive L-pad can be used to match them to each other. As shown, either side of the L-pad can be the source or load, but the Z₁ side must be the side with the higher impedance.

$R_{q}={\frac {Z_{m}}{\sqrt {\rho -1}}}\qquad R_{p}=Z_{m}{\sqrt {\rho -1}}\qquad Loss=20\log _{10}\left({\sqrt {\rho -1}}+{\sqrt {\rho }}\quad \right)\quad {\text{where}}\quad \rho ={\frac {Z_{1}}{Z_{2}}}\quad Z_{m}={\sqrt {Z_{1}Z_{2}}}{\text{ }}\,$ see Valkenburg p 11-3

Large positive numbers means loss is large. The loss is a monotonic function of the impedance ratio. Higher ratios require higher loss.

Converting T-pad to pi-pad

$R_{z}={\frac {R_{a}R_{b}+R_{a}R_{c}+R_{b}R_{c}}{R_{c}}}\qquad R_{x}={\frac {R_{a}R_{b}+R_{a}R_{c}+R_{b}R_{c}}{R_{b}}}\qquad R_{y}={\frac {R_{a}R_{b}+R_{a}R_{c}+R_{b}R_{c}}{R_{a}}}.\qquad {\text{See Hayt, page 494, problem 8. }}\,$

Converting pi-pad to T-pad

$R_{c}={\frac {R_{x}R_{y}}{R_{x}+R_{y}+R_{z}}}\qquad R_{a}={\frac {R_{z}R_{x}}{R_{x}+R_{y}+R_{z}}}\qquad R_{b}={\frac {R_{z}R_{y}}{R_{x}+R_{y}+R_{z}}}\qquad {\text{see Hayt, page 494, problem, 8.}}\,$

Conversion between two-ports and pads

T-pad to impedance parameters

The impedance parameters for a passive two-port are

$V_{1}=Z_{11}I_{1}+Z_{12}I_{2}\qquad V_{2}=Z_{21}I_{1}+Z_{22}I_{2}\qquad with\qquad Z_{12}=Z_{21}\,$

It is always possible to represent a resistive t-pad as a two-port. The representation is particularly simple using impedance parameters as follows:

$Z_{21}=R_{c}\qquad Z_{11}=R_{c}+R_{a}\qquad Z_{22}=R_{c}+R_{b}\,$

Impedance parameters to T-pad

The preceding equations are trivially invertible, but if the loss is not enough, some of the t-pad components will have negative resistances.

$R_{c}=Z_{21}\qquad R_{a}=Z_{11}-Z_{21}\qquad R_{b}=Z_{22}-Z_{21}\,$

Impedance parameters to pi-pad

These preceding T-pad parameters can be algebraically converted to pi-pad parameters.

$R_{z}={\frac {Z_{11}Z_{22}-Z_{21}^{2}}{Z_{21}}}\qquad R_{x}={\frac {Z_{11}Z_{22}-Z_{21}^{2}}{Z_{22}-Z_{21}}}\qquad R_{y}={\frac {Z_{11}Z_{22}-Z_{21}^{2}}{Z_{11}-Z_{21}}}\qquad$

Pi-pad to admittance parameters

The admittance parameters for a passive twp port are

$I_{1}=Y_{11}V_{1}+Y_{12}V_{2}\qquad I_{2}=Y_{21}V_{1}+Y_{22}V_{2}\qquad with\qquad Y_{12}=Y_{21}\,$

It is always possible to represent a resistive pi pad as a two-port. The representation is particularly simple using admittance parameters as follows:

$Y_{21}={\frac {1}{R_{z}}}\qquad Y_{11}={\frac {1}{R_{x}}}+{\frac {1}{R_{z}}}\qquad Y_{22}={\frac {1}{R_{y}}}+{\frac {1}{R_{z}}}\,$

Admittance parameters to pi-pad

The preceding equations are trivially invertible, but if the loss is not enough, some of the pi-pad components will have negative resistances.

$R_{z}={\frac {1}{Y_{21}}}\qquad R_{x}={\frac {1}{Y_{11}-Y_{21}}}\qquad R_{y}={\frac {1}{Y_{22}-Y_{21}}}\,$

General case, determining impedance parameters from requirements

Because the pad is entirely made from resisters, it must have a certain minimum loss to match source and load if they are not equal.

The minimum loss is given by

$Loss_{min}=20\ log_{10}\left({\sqrt {\rho -1}}+{\sqrt {\rho }}\quad \right)\,\quad {\text{where}}\quad \rho ={\frac {\max[Z_{S},Z_{Load}]}{\min[Z_{S},Z_{Load}]}}\,$

Although a passive matching two-port can have less loss, if it does it will not be convertable too a resistive attenuator pad.

$A=10^{-Loss/20}\qquad Z_{11}=Z_{S}{\frac {1+A^{2}}{1-A^{2}}}\qquad Z_{22}=Z_{Load}{\frac {1+A^{2}}{1-A^{2}}}\qquad Z_{21}=2{\frac {A{\sqrt {Z_{S}Z_{Load}}}}{1-A^{2}}}\,$

Once these parameters have been determined, they can be implemented as a T or pi pad as discussed above.

References

• Hayt, William; Kemmerly, Jack E. (1971), Engineering Circuit Analysis (2nd ed.), McGraw-Hill, ISBN 0070273820
• Valkenburg, Mac E. van (1998), Reference Data for Engineers: Radio, Electronics, Computer and Commmuication (eight ed.), Newnes, ISBN 0750670649

External links

• A page that will perform these calculations.