richards

Convert lumped element circuit to distributed element circuit using Richards' transformation

Since R2021b

Syntax

cktOut = richards(cktIn,opFreq)

txOut = richards(LorCobj,opFreq)

[txOut,nodes] = richards(LorCobj,opFreq)

[txOut,nodes] = richards(___,stubmode=stubType)

Description

cktOut = richards(cktIn,opFreq) applies Richards' transformation on the circuit cktIn and returns the circuit object cktOut at the given reference frequency opFreq. In the cktOut all capacitors and inductors are replaced by electrical-length-based transmission line objects txlineElectricalLength.

Note

You can apply Richard's transformation only to circuits where all negative terminals of the ports share the same node.

txOut = richards(LorCobj,opFreq) convert a capacitor or inductor LorCobj into an electrical-length-based transmission line object txOut at opFreq.

[txOut,nodes] = richards(LorCobj,opFreq) also returns a vector of suggested nodes nodes to connect txOut if LorCobj is connected to a circuit.

[txOut,nodes] = richards(___,stubmode=stubType) specify the stub type of the output transmission line by using a name-value argument.

Examples

Apply Richards' Transformation to RF Filter

Create lowpass LC-Pi Chebyshev filter with the passband frequency of 1 GHz, passband attenuation of 0.5 dB, and filter order of 5.

Fp = 1e9;          
Ap = 0.5;          
Ord = 5;           
cktIn = rffilter("FilterType","Chebyshev","ResponseType","Lowpass","Implementation","LC Pi","PassbandFrequency", ...
    Fp,"PassbandAttenuation",Ap,"FilterOrder",Ord);
opFreq = 1e9;

Convert the lumped elements of the RF filter to a distributed element using Richards' transformation.

cktOut = richards(cktIn,opFreq)

cktOut = 
  circuit: Circuit element

    ElementNames: {'C_tx'  'L_tx'  'C_1_tx'  'L_1_tx'  'C_2_tx'}
        Elements: [1×5 txlineElectricalLength]
           Nodes: [0 1 2 3 4 5 6]
            Name: 'unnamed'

Apply Richards' Transformation to Inductor

Create an inductor with the inductance of 5 nH.

LorCobj= inductor(5e-9);

Create a circuit.

ckt = circuit('new_circuit1');

Add a resistor and the inductor you created earlier to the circuit.

add(ckt,[1 2],LorCobj);
add(ckt,[2 3],resistor(100));

Set the ports and display the results.

setports(ckt,[1 0],[3 0])
disp(ckt)

  circuit: Circuit element

    ElementNames: {'L'  'R'}
        Elements: [1×2 rf.internal.circuit.RLC]
           Nodes: [0 1 2 3]
            Name: 'new_circuit1'

Apply Richards' transformation to the inductor at 1 GHz and display nodes to connect the transmission line.

[txOut,nodes] = richards(LorCobj,1e9)

txOut = 
  txlineElectricalLength: ElectricalLength element

                  Name: 'L_tx'
                    Z0: 31.4159
            LineLength: 0.7854
    ReferenceFrequency: 1.0000e+09
           Termination: 'Short'
              StubMode: 'Series'

nodes = 1×4

     1     2     0     0

The nodes in this example represent the nodes at which a 2-port transmission line representing the series L. The returned node value is set to -1 when a ground node cannot be determined from the circuit.

Input Arguments

cktIn — Input RF circuit

circuit object | lcladder object | rffilter object | matchingnetwork object

Input RF circuit, specified as a circuit, lcladder, rffilter, or matchingnetwork object.

LorCobj — Inductor or capacitor

inductor object | capacitor object

Inductor or capacitor, specified as an inductor or capacitor object.

opFreq — Operating frequency

positive scalar

Operating frequency at which the Richards' transformation is applied, specified as a positive scalar.

stubType — Stub type

'Series' (default) | 'Shunt'

Stub type of txOut, specified as a 'Series' or 'Shunt'.

Output Arguments

cktOut — Output circuit

circuit object

Output circuit, returned as a circuit object.

txOut — Electrical-length-based transmission line

txlineElectricalLength object

Electrical-length-based transmission line, returned as a txlineElectricalLength object.

nodes — Nodes to connect txOut

vector

Nodes to connect txOut, returned as a vector.

Algorithms

Richards' Transformation

This figure shows how Richards' transformation converts a circuit with capacitors and inductors into an abstract transmission line model [1].

References

[1] Pozar, David M. Microwave Engineering. 4th ed. Hoboken, NJ: Wiley, 2012.

Version History

Introduced in R2021b