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RF and microwave
zToGamma gammaToZ vswr returnLoss mismatchLoss lineInput abcdLine abcdSeries abcdShunt sToZ zToS sToY yToS zToY yToZ sToAbcd abcdToS zToAbcd abcdToZ yToAbcd abcdToY
zToGamma(Z, [Z0]) MathMath’s own
The reflection coefficient Γ = (Z − Z0)/(Z + Z0) of an impedance Z on a line or port of impedance Z0.
Z0 is the reference impedance: real and positive, the same at every port, and 50 Ω when left out.
Element by element, so a vector of impedances gives a vector of Γ, ready for a Smith chart (Format → Smith chart on a plot). gammaToZ goes back.
Units
Z in ohms or a plain number of ohms; Γ has no units.
Refused
- Z = −Z0, whose reflection is infinite.
Γ≔zToGamma((100+50i)⋅Ω)
Γ=0.4+0.2i
gammaToZ(Γ, [Z0]) MathMath’s own
The impedance Z0·(1 + Γ)/(1 − Γ) whose reflection coefficient on Z0 is Γ.
Z0 is the reference impedance: real and positive, the same at every port, and 50 Ω when left out.
Units
In ohms when Z0 carries units, as the default does.
Refused
- Γ = 1, an open circuit, whose impedance is infinite.
gammaToZ(0.4+0.2i)=100+50i Ω
gammaToZ(−0.2,75)=50
vswr(Γ) MathMath’s own
The voltage standing wave ratio (1 + |Γ|)/(1 − |Γ|) of a reflection coefficient Γ.
Element by element. For an impedance, take zToGamma first.
Refused
- |Γ| of 1 or more, whose standing wave ratio is infinite.
- An argument with units: an impedance rather than Γ.
vswr(0.5)=3
vswr(zToGamma(100⋅Ω))=2
returnLoss(Γ) MathMath’s own
Return loss in decibels, −20·log|Γ|: positive, and larger for a better match.
Refused
- Γ = 0, a perfect match, whose return loss is infinite.
returnLoss(0.5)=6.021
returnLoss(zToGamma(60⋅Ω))=20.828
mismatchLoss(Γ) MathMath’s own
Mismatch loss in decibels, −10·log(1 − |Γ|²): the power lost to reflection.
Refused
- |Γ| of 1 or more, when no power gets through.
lineInput(ZL, Z0, θ) MathMath’s own
The impedance looking into a transmission line of impedance Z0 and electrical length θ, ended in a load ZL.
θ = β·ℓ, in radians or degrees: a quarter wave is π/2, or 90·deg. For a lossy line, θ is complex, β·ℓ − j·α·ℓ, with α in nepers per unit length.
An open end is ZL = ∞·Ω and a short is 0·Ω. Any argument can be a vector, as θ over a frequency sweep, and the result is a vector.
A lossy line's Z0 can be complex.
Units
In ohms when ZL or Z0 carries units.
Refused
- θ with units other than radians or degrees.
- ZL = −Z0, which has no steady state.
- An input that is exactly an open circuit.
lineInput(100⋅Ω,50⋅Ω,45⋅deg)=40−30i Ω
lineInput(0⋅Ω,50⋅Ω,8π)=20.711i Ω
abcdLine(Z0, θ) MathMath’s own
The ABCD matrix of a transmission line of impedance Z0 and electrical length θ.
(cos θ, j·Z0·sin θ; j·sin θ/Z0, cos θ). θ is as for lineInput. Multiply ABCD matrices, in order, to cascade parts; convert with abcdToS, abcdToZ or abcdToY.
Units
B is in ohms and C in siemens when Z0 carries units.
abcdToS(abcdLine(50⋅Ω,30⋅deg))=(00.866−0.5i0.866−0.5i0)
abcdSeries(Z) MathMath’s own
The ABCD matrix (1, Z; 0, 1) of an impedance Z in series with the line.
Multiply ABCD matrices, in order from port 1 to port 2, to cascade parts.
Units
B is in ohms when Z carries units.
T≔abcdSeries(10⋅Ω)⋅abcdShunt(100⋅Ω)⋅abcdSeries(20⋅Ω)
abcdToZ(T)=(110100100120) Ω
abcdShunt(Z) MathMath’s own
The ABCD matrix (1, 0; 1/Z, 1) of an impedance Z across the line, to ground.
Z is the part's impedance, not its admittance.
Units
C is in siemens when Z carries units.
Refused
- Z = 0, a short across the line, which has no ABCD matrix.
abcdToS(abcdShunt(25⋅Ω))=(−0.50.50.5−0.5)
sToZ(S, [Z0]) MathMath’s own
The Z (impedance) matrix of a network from its S matrix: Z0·(I + S)·(I − S)⁻¹.
Any number of ports: S is square, a row and column per port. A number is a one-port's S, its reflection coefficient.
Z0 is the reference impedance: real and positive, the same at every port, and 50 Ω when left out.
Units
Impedances and admittances can carry units or be plain numbers in ohms and siemens. A result is in ohms or siemens when an argument carried units; the default 50 Ω counts.
Refused
- A network with no Z matrix, as a part in series between ports on its own.
- A matrix that isn't square, or has units.
S≔(00.50.50)
sToZ(S)=(83.33366.66766.66783.333) Ω
zToS(Z, [Z0]) MathMath’s own
The S matrix of a network from its Z matrix: (Z − Z0·I)·(Z + Z0·I)⁻¹.
Any number of ports.
Z0 is the reference impedance: real and positive, the same at every port, and 50 Ω when left out.
Units
Impedances and admittances can carry units or be plain numbers in ohms and siemens. A result is in ohms or siemens when an argument carried units; the default 50 Ω counts.
zToS((110100100120)⋅Ω)=(0.0120.5810.5810.07)
sToY(S, [Z0]) MathMath’s own
The Y (admittance) matrix of a network from its S matrix: (I − S)·(I + S)⁻¹/Z0.
Any number of ports.
Z0 is the reference impedance: real and positive, the same at every port, and 50 Ω when left out.
Units
Impedances and admittances can carry units or be plain numbers in ohms and siemens. A result is in ohms or siemens when an argument carried units; the default 50 Ω counts.
Refused
- A network with no Y matrix, as a part across a port on its own.
sToY((00.50.50))=(0.033−0.027−0.0270.033) Ω1
yToS(Y, [Z0]) MathMath’s own
The S matrix of a network from its Y matrix: (I − Z0·Y)·(I + Z0·Y)⁻¹.
Any number of ports.
Z0 is the reference impedance: real and positive, the same at every port, and 50 Ω when left out.
Units
Impedances and admittances can carry units or be plain numbers in ohms and siemens. A result is in ohms or siemens when an argument carried units; the default 50 Ω counts.
yToS((0.02−0.01−0.010.02)⋅S)=(0.0670.2670.2670.067)
zToY(Z) MathMath’s own
The Y matrix of a network from its Z matrix: its inverse, which works for complex matrices too.
Units
In siemens when Z is in ohms.
Refused
- A network with no Y matrix: Z is singular.
zToY((110100100120)⋅Ω)=(0.037−0.031−0.0310.034) Ω1
yToZ(Y) MathMath’s own
The Z matrix of a network from its Y matrix: its inverse, which works for complex matrices too.
Units
In ohms when Y is in siemens.
Refused
- A network with no Z matrix: Y is singular.
yToZ((0.02−0.01i−0.01i0.02)⋅S)=(4020i20i40) Ω
sToAbcd(S, [Z0]) MathMath’s own
The ABCD (chain) matrix of a two-port from its 2 × 2 S matrix.
ABCD matrices multiply, in order, to cascade two-ports.
Z0 is the reference impedance: real and positive, the same at every port, and 50 Ω when left out.
Units
B is in ohms and C in siemens when Z0 carries units, as the default does.
Refused
- A two-port whose port 2 doesn't respond to port 1 (S21 = 0), which has no ABCD matrix.
- A matrix that isn't 2 × 2.
sToAbcd((00.50.50))=(1.250.01537.51.25)
abcdToS(M, [Z0]) MathMath’s own
The S matrix of a two-port from its ABCD matrix M.
Z0 is the reference impedance: real and positive, the same at every port, and 50 Ω when left out.
Units
M's B is in ohms and C in siemens, or plain numbers of them; A and D have no units.
A≔abcdShunt(150.48⋅Ω)⋅abcdSeries(37.35⋅Ω)⋅abcdShunt(150.48⋅Ω)
abcdToS(A)=(−2.876×10−60.5010.501−2.876×10−6)
zToAbcd(Z) MathMath’s own
The ABCD matrix of a two-port from its 2 × 2 Z matrix.
Refused
- Z21 = 0, when port 2 doesn't respond to port 1.
zToAbcd((110100100120)⋅Ω)=(1.10.01321.2)
abcdToZ(M) MathMath’s own
The Z matrix of a two-port from its ABCD matrix M.
Refused
- C = 0, as for a part in series alone, which has no Z matrix.
abcdToZ(abcdShunt(100⋅Ω))=(100100100100) Ω
yToAbcd(Y) MathMath’s own
The ABCD matrix of a two-port from its 2 × 2 Y matrix.
Refused
- Y21 = 0, when port 2 doesn't respond to port 1.
yToAbcd((0.1−0.1−0.10.1)⋅S)=(10101)
abcdToY(M) MathMath’s own
The Y matrix of a two-port from its ABCD matrix M.
Refused
- B = 0, as for a part to ground alone, which has no Y matrix.
abcdToY(abcdSeries(10⋅Ω))=(0.1−0.1−0.10.1) Ω1