Network parameters

Convert a network's scattering parameters to and from its impedance, admittance, chain (ABCD) and transfer parameters, at the reference impedances of your choice. A conversion takes one matrix, or an array of matrices with the frequency along the dimensions after the first two; an in-place form, ending in !, writes into an array of the same shape.

See the API overview for other topics, closed-form networks for networks to convert, and connections to join them.

Convert and renormalize

A line of 40 Ω, 1 cm long, between 50 Ω ports, the measurement system. Its chain matrix gives its scattering parameters at 50 Ω, and its impedance matrix carries them to 40 Ω ports, where the line is matched. At 50 Ω the reflection is that of a line section between two equal media, Γ(1 - exp(-2iθ))/(1 - Γ² exp(-2iθ)) with Γ = (40 - 50)/(40 + 50) and θ its electrical length; at 6 GHz the line is half a wavelength long and reflects nothing.

julia

using JosephsonCircuits
w = 2pi .* [4e9, 5e9, 6e9]
len, vp = 0.01, 1.2e8
θ = w .* len ./ vp
# one chain matrix per frequency, 2×2×3
abcd = JosephsonCircuits.ABCD_tline(40.0, θ)
S = JosephsonCircuits.ABCDtoS(abcd; portimpedances = 50.0)
Z = JosephsonCircuits.StoZ(S; portimpedances = 50.0)
S40 = JosephsonCircuits.ZtoS(Z; portimpedances = 40.0)
Γ = (40 - 50)/(40 + 50)
closed = Γ .* (1 .- exp.(-2im .* θ)) ./ (1 .- Γ^2 .* exp.(-2im .* θ))
@assert isapprox(S[1, 1, :], closed; atol = 1e-12)
@assert maximum(abs, S40[1, 1, :]) < 1e-12
(S11 = S[1, 1, :], closed = closed, S11at40 = S40[1, 1, :])

(S11 = ComplexF64[-0.16664157760048176 + 0.09386396066909475im, -0.056935658077457256 + 0.09621019761221972im, -3.45882099876391e-33 + 2.7554552980815448e-17im], closed = ComplexF64[-0.16664157760048165 + 0.09386396066909468im, -0.05693565807745726 + 0.09621019761221974im, -8.43614877747295e-35 + 2.7554552980815445e-17im], S11at40 = ComplexF64[3.81140974547136e-16 - 1.0902954179146984e-16im, -3.290813860454552e-16 - 3.181927392442096e-16im, 3.276858658674052e-16 + 9.487917790209044e-17im])

A complex reference impedance defines pseudo-waves; see ZtoS.

JosephsonCircuits.ABCDtoS Function

julia

ABCDtoS(ABCD;portimpedances=50.0)

Convert the 2 port chain (ABCD) matrix ABCD to the scattering parameter matrix S and return the result. Assumes a port impedance of 50 Ohms unless specified with the portimpedances keyword argument: a number for both ports, a vector of one or two impedances, the first for the input port and the last for the output port, or a matrix of such columns, one per frequency. The waves are the pseudo-waves of ZtoS, with the square root of each port impedance.

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.ABCDtoS! Method

julia

ABCDtoS!(ABCD::AbstractMatrix, portimpedances)

See ABCDtoS for description.

JosephsonCircuits.AtoB Function

julia

AtoB(A)

Convert the chain (ABCD) matrix A to the inverse chain matrix B and return the result. Note that despite the name, the inverse of the chain matrix is not equal to the inverse chain matrix, inv(A) ≠ B.

Examples

julia

julia> A = Complex{Float64}[1.0 0.0;1/50 1.0];JosephsonCircuits.AtoB(A)
2×2 Matrix{ComplexF64}:
  1.0+0.0im  0.0+0.0im
 0.02+0.0im  1.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.AtoB! Method

julia

AtoB!(B::AbstractMatrix,A::AbstractMatrix,tmp::AbstractMatrix)

See AtoB for description.

JosephsonCircuits.AtoS Function

julia

AtoS(A; portimpedances = 50.0)

Convert the chain (ABCD) matrix A to the scattering parameter matrix S and return the result. The first half of the ports are the inputs of the chain matrix and the second half its outputs, each with its own port impedances: portimpedances is a scalar, a vector with one value per port, or a matrix with one row per port and one column per frequency, 50 Ohms unless specified. A is the chain matrix of StoA, [V1; I1] = A*[V2; -I2] with the currents into the ports.

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.AtoS! Method

julia

AtoS!(S::AbstractMatrix, A::AbstractMatrix, tmp::AbstractMatrix,
    sqrtportimpedances1, sqrtportimpedances2)

See AtoS for description.

JosephsonCircuits.AtoY Function

julia

AtoY(A)

Convert the chain (ABCD) matrix A to the admittance matrix Y and return the result.

Examples

julia

julia> A = Complex{Float64}[-1 -50.0;0 -1];JosephsonCircuits.AtoY(A)
2×2 Matrix{ComplexF64}:
 0.02+0.0im  0.02+0.0im
 0.02+0.0im  0.02+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.AtoY! Method

julia

AtoY!(Y::AbstractMatrix,A::AbstractMatrix,tmp::AbstractMatrix)

See AtoY for description.

JosephsonCircuits.AtoZ Function

julia

AtoZ(A)

Convert the ABCD matrix A to the impedance matrix Z and return the result.

Examples

julia

julia> A = Complex{Float64}[1.0 0.0;1/50 1.0];JosephsonCircuits.AtoZ(A)
2×2 Matrix{ComplexF64}:
 50.0+0.0im  50.0+0.0im
 50.0+0.0im  50.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.AtoZ! Method

julia

AtoZ!(Z::AbstractMatrix,A::AbstractMatrix,tmp::AbstractMatrix)

See AtoZ for description.

JosephsonCircuits.BtoA Function

julia

BtoA(B)

Convert the inverse chain matrix B to the chain (ABCD) matrix A and return the result. Note that despite the name, the inverse of the chain matrix is not equal to the inverse chain matrix, inv(A) ≠ B.

Examples

julia

julia> B = Complex{Float64}[1.0 0.0;1/50 1.0];JosephsonCircuits.BtoA(B)
2×2 Matrix{ComplexF64}:
  1.0+0.0im  0.0+0.0im
 0.02+0.0im  1.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.BtoA! Method

julia

BtoA!(A::AbstractMatrix,B::AbstractMatrix,tmp::AbstractMatrix)

See BtoA for description.

JosephsonCircuits.BtoS Function

julia

BtoS(B; portimpedances = 50.0)

Convert the inverse chain (ABCD) matrix B to the scattering parameter matrix S and return the result. The first half of the ports are the inputs of the chain matrix and the second half its outputs, each with its own port impedances: portimpedances is a scalar, a vector with one value per port, or a matrix with one row per port and one column per frequency, 50 Ohms unless specified.

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006 with change of overall sign (suspected typo).

JosephsonCircuits.BtoS! Method

julia

BtoS!(S::AbstractMatrix, B::AbstractMatrix, tmp::AbstractMatrix,
    sqrtportimpedances1, sqrtportimpedances2)

See BtoS for description.

JosephsonCircuits.BtoY Function

julia

BtoY(B)

Convert the inverse chain matrix B to the admittance matrix Y and return the result.

Examples

julia

julia> B = Complex{Float64}[-1 -50.0;0 -1];JosephsonCircuits.BtoY(B)
2×2 Matrix{ComplexF64}:
 0.02+0.0im  0.02+0.0im
 0.02+0.0im  0.02+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.BtoY! Method

julia

BtoY!(Y::AbstractMatrix,B::AbstractMatrix,tmp::AbstractMatrix)

See BtoY for description.

JosephsonCircuits.BtoZ Function

julia

BtoZ(B)

Convert the inverse chain matrix B to the impedance matrix Z and return the result.

Examples

julia

julia> B = Complex{Float64}[1.0 0.0;1/50 1];JosephsonCircuits.BtoZ(B)
2×2 Matrix{ComplexF64}:
 50.0+0.0im  50.0+0.0im
 50.0+0.0im  50.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006 with change of sign on B21 and B22 terms (suspected typo).

JosephsonCircuits.BtoZ! Method

julia

BtoZ!(Z::AbstractMatrix,B::AbstractMatrix,tmp::AbstractMatrix)

See BtoZ for description.

JosephsonCircuits.StoA Function

julia

StoA(S; portimpedances = 50.0)

Convert the scattering parameter matrix S to the chain (ABCD) matrix A and return the result. The first half of the ports are the inputs of the chain matrix and the second half its outputs, each with its own port impedances: portimpedances is a scalar, a vector with one value per port, or a matrix with one row per port and one column per frequency, 50 Ohms unless specified. With the voltages V and the currents I into the ports, [V1; I1] = A*[V2; -I2]: the output currents are taken flowing out of the output ports, so that the chain matrix of networks in cascade is the product of theirs.

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.StoA! Method

julia

StoA!(A::AbstractMatrix, S::AbstractMatrix, tmp::AbstractMatrix,
    sqrtportimpedances1, sqrtportimpedances2)

See StoA for description.

JosephsonCircuits.StoABCD Function

julia

StoABCD(S;portimpedances=50.0)

Convert the scattering parameter matrix S to the 2 port chain (ABCD) matrix and return the result. Assumes a port impedance of 50 Ohms unless specified with the portimpedances keyword argument, one impedance for both ports or one for each, as for ABCDtoS. A two port which transmits nothing, S[2,1] == 0, has no chain matrix, and a SingularException is thrown.

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.StoABCD! Method

julia

StoABCD!(S::AbstractMatrix, portimpedances)

See StoABCD for description.

JosephsonCircuits.StoB Function

julia

StoB(S; portimpedances = 50.0)

Convert the scattering parameter matrix S to the inverse chain (ABCD) matrix B and return the result. Note that despite the name, the inverse of the chain matrix is not equal to the inverse chain matrix, inv(A) ≠ B: with the currents into the ports, [V2; I2] = B*[V1; -I1], the chain matrix of the network seen from its outputs. The first half of the ports are the inputs of the chain matrix and the second half its outputs, each with its own port impedances: portimpedances is a scalar, a vector with one value per port, or a matrix with one row per port and one column per frequency, 50 Ohms unless specified.

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.StoB! Method

julia

StoB!(B::AbstractMatrix, S::AbstractMatrix, tmp::AbstractMatrix,
    sqrtportimpedances1, sqrtportimpedances2)

See StoB for description.

JosephsonCircuits.StoT Function

julia

StoT(S)

Convert the scattering parameter matrix S to a transmission matrix T and return the result. The first half of the ports are the inputs and the second half the outputs, and T relates the waves at the inputs to those at the outputs, [b1; a1] = T*[a2; b2], so that the transmission matrix of networks in cascade, the outputs of each joined to the inputs of the next, is the product of theirs, T1*T2.

Examples

julia

julia> S = Complex{Float64}[0.0 1.0;1.0 0.0];JosephsonCircuits.StoT(S)
2×2 Matrix{ComplexF64}:
 1.0+0.0im  -0.0+0.0im
 0.0-0.0im   1.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.StoT! Method

julia

StoT!(T::AbstractMatrix,S::AbstractMatrix,tmp::AbstractMatrix)

See StoT for description.

JosephsonCircuits.StoY Function

julia

StoY(S;portimpedances=50.0)

Convert the scattering parameter matrix S to an admittance parameter matrix Y and return the result. Assumes a port impedance of 50 Ohms unless specified with the portimpedances keyword argument, a scalar, vector, or matrix of port impedances. A complex port impedance defines pseudo-waves, as described in ZtoS.

Examples

julia

julia> S = Complex{Float64}[0.0 0.999;0.999 0.0];JosephsonCircuits.StoY(S)
2×2 Matrix{ComplexF64}:
  19.99+0.0im  -19.99+0.0im
 -19.99+0.0im   19.99+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.StoY! Method

julia

StoY!(Y::AbstractMatrix,S::AbstractMatrix,tmp::AbstractMatrix,oneoversqrtportimpedances)

In place version of StoY, writing into Y with tmp as scratch and oneoversqrtportimpedances the reciprocal square roots of the port impedances.

JosephsonCircuits.StoZ Function

julia

StoZ(S;portimpedances=50.0)

Convert the scattering parameter matrix S to an impedance parameter matrix Z and return the result. Assumes a port impedance of 50 Ohms unless specified with the portimpedances keyword argument, a scalar, vector, or matrix of port impedances. A complex port impedance defines pseudo-waves, as described in ZtoS.

Examples

julia

julia> S = Complex{Float64}[0.0 0.0;0.0 0.0];JosephsonCircuits.StoZ(S)
2×2 Matrix{ComplexF64}:
 50.0+0.0im   0.0+0.0im
  0.0+0.0im  50.0+0.0im

julia> S = Complex{Float64}[0.0 0.999;0.999 0.0];JosephsonCircuits.StoZ(S)
2×2 Matrix{ComplexF64}:
 49975.0+0.0im  49975.0+0.0im
 49975.0+0.0im  49975.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.StoZ! Method

julia

StoZ!(Z::AbstractMatrix,S::AbstractMatrix,tmp::AbstractMatrix,sqrtportimpedances)

See StoZ for description.

JosephsonCircuits.TtoS Function

julia

TtoS(T)

Convert the transmission matrix T to a scattering parameter matrix S and return the result; the inverse of StoT, which gives the convention.

Examples

julia

julia> T = Complex{Float64}[1.0 0.0;0.0 1.0];JosephsonCircuits.TtoS(T)
2×2 Matrix{ComplexF64}:
 0.0-0.0im  1.0+0.0im
 1.0+0.0im  0.0-0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006 with change of sign on T11 and T21 terms (suspected typo).

JosephsonCircuits.TtoS! Method

julia

TtoS!(S::AbstractMatrix,T::AbstractMatrix,tmp::AbstractMatrix)

See TtoS for description.

JosephsonCircuits.YtoA Function

julia

YtoA(Y)

Convert the admittance matrix Y to the chain (ABCD) matrix A and return the result.

Examples

julia

julia> Y = Complex{Float64}[1/50 1/50;1/50 1/50];JosephsonCircuits.YtoA(Y)
2×2 Matrix{ComplexF64}:
 -1.0+0.0im  -50.0+0.0im
  0.0+0.0im   -1.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006, with the overall sign changed (a suspected typo there).

JosephsonCircuits.YtoA! Method

julia

YtoA!(A::AbstractMatrix,Y::AbstractMatrix,tmp::AbstractMatrix)

See YtoA for description.

JosephsonCircuits.YtoB Function

julia

YtoB(Y)

Convert the admittance matrix Y to the inverse chain matrix B and return the result.

Examples

julia

julia> Y = Complex{Float64}[1/50 1/50;1/50 1/50];JosephsonCircuits.YtoB(Y)
2×2 Matrix{ComplexF64}:
 -1.0+0.0im  -50.0+0.0im
  0.0+0.0im   -1.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.YtoB! Method

julia

YtoB!(B::AbstractMatrix,Y::AbstractMatrix,tmp::AbstractMatrix)

See YtoB for description.

JosephsonCircuits.YtoS Function

julia

YtoS(Y;portimpedances=50.0)

Convert the admittance parameter matrix Y to a scattering parameter matrix S and return the result. portimpedances is a scalar, vector, or matrix of port impedances. Assumes a port impedance of 50 Ohms unless specified with the portimpedances keyword argument.

Examples

julia

julia> Y = Complex{Float64}[1/50.0 0.0;0.0 1/50.0];JosephsonCircuits.YtoS(Y)
2×2 Matrix{ComplexF64}:
 0.0+0.0im  0.0-0.0im
 0.0-0.0im  0.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.YtoS! Method

julia

YtoS!(S::AbstractMatrix,Y::AbstractMatrix,tmp::AbstractMatrix,sqrtportimpedances)

See YtoS for description.

JosephsonCircuits.YtoZ Function

julia

YtoZ(Y)

Convert the admittance matrix Y to the impedance matrix Z, its inverse, and return the result.

JosephsonCircuits.YtoZ! Method

julia

YtoZ!(Z, Y, tmp)

See YtoZ for description.

JosephsonCircuits.ZtoA Function

julia

ZtoA(Z)

Convert the impedance matrix Z to the ABCD matrix A and return the result.

Examples

julia

julia> Z = Complex{Float64}[50.0 50.0;50.0 50.0];JosephsonCircuits.ZtoA(Z)
2×2 Matrix{ComplexF64}:
  1.0+0.0im  0.0-0.0im
 0.02+0.0im  1.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.ZtoA! Method

julia

ZtoA!(A::AbstractMatrix,Z::AbstractMatrix,tmp::AbstractMatrix)

See ZtoA for description.

JosephsonCircuits.ZtoB Function

julia

ZtoB(Z)

Convert the impedance matrix Z to the inverse chain matrix B and return the result.

Examples

julia

julia> Z = Complex{Float64}[50.0 50;50 50];JosephsonCircuits.ZtoB(Z)
2×2 Matrix{ComplexF64}:
  1.0+0.0im  0.0-0.0im
 0.02+0.0im  1.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.ZtoB! Method

julia

ZtoB!(B::AbstractMatrix,Z::AbstractMatrix,tmp::AbstractMatrix)

See ZtoB for description.

JosephsonCircuits.ZtoS Function

julia

ZtoS(Z;portimpedances=50.0)

Convert the impedance parameter matrix Z to a scattering parameter matrix S and return the result. portimpedances is a scalar, vector, or matrix of port impedances. Assumes a port impedance of 50 Ohms unless specified with the portimpedances keyword argument. A complex port impedance Z_k defines the pseudo-waves a_k = (V_k + Z_k I_k)/(2 sqrt(Z_k)) and b_k = (V_k - Z_k I_k)/(2 sqrt(Z_k)), which a load of impedance Z_k matches, rather than power waves.

Examples

julia

julia> Z = Complex{Float64}[0.0 0.0;0.0 0.0];JosephsonCircuits.ZtoS(Z)
2×2 Matrix{ComplexF64}:
 -1.0+0.0im   0.0+0.0im
  0.0+0.0im  -1.0+0.0im

References

Russer, Peter. Electromagnetics, Microwave Circuit, And Antenna Design for Communications Engineering, Second Edition. Artech House, 2006.

JosephsonCircuits.ZtoS! Method

julia

ZtoS!(S::AbstractMatrix,Z::AbstractMatrix,tmp::AbstractMatrix,oneoversqrtportimpedances)

In place version of ZtoS, writing into S with tmp as scratch and oneoversqrtportimpedances the reciprocal square roots of the port impedances.

JosephsonCircuits.ZtoY Function

julia

ZtoY(Z)

Convert the impedance matrix Z to the admittance matrix Y, its inverse, and return the result.

JosephsonCircuits.ZtoY! Method

julia

ZtoY!(Y, Z, tmp)

See ZtoY for description.