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, ZLoad, and a particular source impedance, Zs.
 * The impedance seen looking into the input port will be ZS if the output port is terminated by ZLoad.
 * The impedance seen looking into the output port will be ZLoad if the input port is terminated by ZS.

Reference figures for attenuator component calculation


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.


 * ZS 	= the output impedance of the source.
 * ZLoad      = the input impedance of the load.
 * Zin 	= the impedance seen looking into the input port when ZLoad is connected to the output port. Zin is a function of the load impedance.
 * Zout 	= the impedance seen looking into the output port when Zs is connected to the input port. Zout is a function of the source impedance.
 * Vs 	= source open circuit or unloaded voltage.
 * Vin 	= voltage applied to the input port by the source.
 * Vout 	= voltage applied to the load by the output port.
 * Iin 	= current entering the input port from the source.
 * Iout 	= current entering the load from the output port.
 * Pin	= Vin Iin = power entering the input port from the source.
 * Pout	= Vout Iout = power absorbed by the load from the output port.
 * Pdirect	= the power that would be absorbed by the load if the load were connected directly to the source.
 * Lpad	= 10 log10 (Pin / Pout ) always.  And if Zs = ZLoad then Lpad = 20 log10 (Vin / Vout ) also.      Note, as defined, Loss ≥ 0 dB
 * Linsertion = 10 log10 (Pdirect / Pout ).  And if Zs = ZLoad then Linsertion = Lpad.
 * Loss ≡ Lpad.  Loss is defined to be Lpad.

Symmetric T pad resister calculation

 * $$A = 10^{-Loss/20} \qquad R_a = R_b = Z_S \frac {1 - A} {1 + A} \qquad R_c =   \frac {Z_s^2 - R_b^2   } {2 R_b } \qquad \, $$ 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.}

\, $$

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.