RF and Microwave Network Theory and Analysis

Why S-parameters are used at microwave frequencies

At microwave frequencies the Z, Y, h and ABCD parameters become hard to use because:

• They need ideal open or short circuits at the ports, which are hard to make at high frequency (stray capacitance and inductance, radiation from an open end).
• Active devices (transistors) often oscillate or are damaged when terminated in an open or short.
• Voltage and current are not uniquely defined in a waveguide, but incident and reflected power waves are.

S-parameters are measured with all ports terminated in a matched load (Z0), they relate incident and reflected waves directly, and they are measured easily with a vector network analyzer. Their magnitudes give power flow (|Sij|² = fraction of power), so return loss, insertion loss and isolation are read off directly.

S-matrix of a 3-port network

For a 3-port network, let aᵢ be the normalized incident wave and bᵢ the reflected (outgoing) wave at port i. Then

 | b1 |   | S11  S12  S13 | | a1 |
 | b2 | = | S21  S22  S23 | | a2 |
 | b3 |   | S31  S32  S33 | | a3 |

• Sᵢᵢ = bᵢ/aᵢ (all other ports matched): reflection coefficient at port i.
• Sᵢⱼ = bᵢ/aⱼ (all ports except j matched): transmission coefficient from port j to port i.

            port 3
              |
          +---+---+
 port 1 --|  [S]  |-- port 2
          +-------+

Properties of the S-matrix of a 3-port network

1. Reciprocity (symmetry): for a passive network without ferrites or plasma, Sᵢⱼ = Sⱼᵢ, i.e. [S] = [S]ᵀ.
2. Lossless (unitary) property: [S]ᵀ[S] = [I], so for each column Σₖ|Sₖᵢ|² = 1, and for two different columns Σₖ SₖᵢSₖⱼ = 0.
3. Matched ports: if all ports are matched, S11 = S22 = S33 = 0.
4. Phase shift property: if the reference plane at port n is moved by a length lₙ, Sᵢⱼ is multiplied by e^(−jβ(lᵢ + lⱼ)).
5. Key theorem: a 3-port network cannot be lossless, reciprocal and matched at all ports at the same time. If it is lossless and matched, it must be non-reciprocal (a circulator); if it is reciprocal and lossless, at least one port is mismatched (e.g. a tee junction).

Example 1: H-plane tee (reciprocal, lossless, port 3 matched)

       | 1/2   -1/2   1/√2 |
 [S] = | -1/2   1/2   1/√2 |
       | 1/√2   1/√2   0   |

Power fed at port 3 divides equally and in phase between ports 1 and 2; ports 1 and 2 are not matched (S11 = 1/2).

Example 2: ideal circulator (lossless, matched, non-reciprocal)

       | 0  0  1 |
 [S] = | 1  0  0 |      (1 -> 2 -> 3 -> 1)
       | 0  1  0 |

All ports are matched and the matrix is unitary but not symmetric, as the theorem requires.

Importance of S-parameters in microwave network analysis

• At microwave frequencies, voltage and current are difficult to measure (and not uniquely defined in waveguides), but power waves travelling in and out of a port are easy to measure.
• Z, Y, h parameters need open or short terminations; these are hard to realise at high frequencies and can make active devices oscillate. S-parameters are measured with matched (Z0) loads.
• S-parameters directly give reflection coefficient, VSWR, return loss, insertion loss, gain and isolation.
• They can be measured accurately with a vector network analyzer and cascaded (via T-parameters) for system design.

Derivation of S-parameters for a two-port network

        a1 ->           -> b2
  o-----------+-------+-----------o
  port 1      |  [S]  |      port 2
  o-----------+-------+-----------o
        <- b1           <- a2

At each port n the total voltage and current are the sum of incident (+) and reflected (−) waves on a line of characteristic impedance Z0:

 Vn = Vn+ + Vn-
 In = (Vn+ - Vn-)/Z0

Define the normalized incident and reflected waves (their squares are power):

 an = Vn+/√Z0 = (Vn + Z0·In)/(2√Z0)
 bn = Vn-/√Z0 = (Vn - Z0·In)/(2√Z0)
 Incident power   = |an|²/2
 Reflected power  = |bn|²/2

For a linear network the outgoing waves are linear combinations of the incoming waves:

 b1 = S11·a1 + S12·a2
 b2 = S21·a1 + S22·a2

 | b1 |   | S11  S12 | | a1 |
 | b2 | = | S21  S22 | | a2 |

Each parameter is found by making one incident wave zero, i.e. terminating that port in a matched load Z0 (a matched load reflects nothing, so aₖ = 0):

• S11 = b1/a1 |a2=0: input reflection coefficient with output matched.
• S21 = b2/a1 |a2=0: forward transmission coefficient (gain or loss).
• S22 = b2/a2 |a1=0: output reflection coefficient with input matched.
• S12 = b1/a2 |a1=0: reverse transmission (isolation).

Relation with impedance parameters

Using V = Z·I with normalized quantities (z = Z/Z0), substituting V = √Z0(a + b) and I = (a − b)/√Z0 gives

 √Z0 (a + b) = Z (a - b)/√Z0
 (Z - Z0·U) a = (Z + Z0·U) b
 [S] = (Z - Z0·U)(Z + Z0·U)^-1      (U = unit matrix)

For a one-port this reduces to S11 = (ZL − Z0)/(ZL + Z0) = Γ, the familiar reflection coefficient.

Useful results

• Return loss = −20 log|S11| dB; insertion loss = −20 log|S21| dB.
• Input VSWR = (1 + |S11|)/(1 − |S11|).
• Reciprocal network: S12 = S21; lossless network: |S11|² + |S21|² = 1 and S11·S12* + S21·S22* = 0.
• Example: an ideal matched attenuator of 3 dB has S11 = S22 = 0, S12 = S21 = 1/√2 ≈ 0.707.

Scattering (S) parameters describe a linear microwave network in terms of the incident waves (aₙ) and reflected waves (bₙ) at its ports, measured when all other ports are terminated in a matched load Z0. For a two-port:

 b1 = S11·a1 + S12·a2
 b2 = S21·a1 + S22·a2

• S11, S22: input and output reflection coefficients.
• S21: forward transmission (gain or insertion loss); S12: reverse transmission (isolation).
• aₙ = Vₙ⁺/√Z0, bₙ = Vₙ⁻/√Z0, so |aₙ|² and |bₙ|² are proportional to power.

Merits of S-parameters at microwave frequencies:

1. Easy measurement: they need matched terminations, not open or short circuits, which are hard to realise at microwave frequencies.
2. Safe for active devices: transistors stay stable when terminated in 50 Ω, whereas open/short terminations may cause oscillation.
3. Defined for waveguides: voltage and current are not unique in a waveguide, but incident and reflected waves are.
4. Direct physical meaning: |S11| gives return loss and VSWR; |S21|² gives power gain or insertion loss.
5. Measured by VNA with high accuracy, and the reference plane can be shifted mathematically (multiply by e^(−jβl)).
6. Network properties are visible: reciprocal → [S] symmetric; lossless → [S] unitary; matched → Sᵢᵢ = 0.
7. Used in amplifier design: stability factor, gain circles and noise circles are all computed from S-parameters.

Example: a matched 10 dB attenuator has S11 = S22 = 0 and |S21| = |S12| = 10^(−10/20) = 0.316.

Comparison of h-parameters and S-parameters

h-parameters (hybrid parameters) relate a mix of voltages and currents (V1, I2 in terms of I1, V2) and are used at low frequencies. S-parameters relate incident and reflected waves and are used at RF and microwave frequencies.

S-parameters of a two-port network

        a1 ->           -> b2
  o-----------+-------+-----------o
  port 1      |  [S]  |      port 2
  o-----------+-------+-----------o
        <- b1           <- a2

At port n, with line impedance Z0:

 an = Vn+/√Z0 = (Vn + Z0·In)/(2√Z0)   (incident wave)
 bn = Vn-/√Z0 = (Vn - Z0·In)/(2√Z0)   (reflected wave)

The network relates them linearly:

 | b1 |   | S11  S12 | | a1 |
 | b2 | = | S21  S22 | | a2 |

• S11 = b1/a1 (a2 = 0): input reflection coefficient, output terminated in Z0.
• S21 = b2/a1 (a2 = 0): forward transmission coefficient (gain or loss).
• S22 = b2/a2 (a1 = 0): output reflection coefficient, input terminated in Z0.
• S12 = b1/a2 (a1 = 0): reverse transmission coefficient (isolation).

Setting aₖ = 0 means port k is terminated in a matched load, so no wave is reflected back into the network.

Properties: reciprocal → S12 = S21; lossless → |S11|² + |S21|² = 1. Return loss = −20 log|S11| dB, insertion loss = −20 log|S21| dB, VSWR = (1 + |S11|)/(1 − |S11|).

Example: a transistor with |S21| = 3 has forward power gain |S21|² = 9 (9.54 dB) when both ports are matched to 50 Ω.

Importance of S-parameters at microwave frequencies

• Voltage and current cannot be measured directly at microwave frequencies (and are not unique in waveguides); incident and reflected power waves can.
• Z, Y and h parameters need open or short terminations, which are hard to build at high frequency and may make active devices oscillate. S-parameters are measured with matched loads.
• S-parameters give reflection (S11, S22), gain/loss (S21) and isolation (S12) directly, and are measured with a vector network analyzer.
• Network properties are simple to test: reciprocal → [S] = [S]ᵀ, lossless → [S] unitary.

S-matrix of an amplitude modulator

A microwave amplitude modulator is a two-port device whose transmission coefficient changes with a low-frequency modulating signal. A common form is a PIN diode mounted in shunt across a matched line (or waveguide). The diode's RF resistance R is controlled by its bias current, so the bias carries the modulating signal.

  a1 ->                       -> b2
  o-------------+-------------o
  port 1 (Z0)   |        port 2 (Z0)
               [R]  PIN diode, R(t)
                |   (bias = modulating signal)
  o-------------+-------------o

Assumption: the diode acts as a pure resistance with normalized shunt admittance y = Z0/R; ports are matched lines of Z0.

Step 1 – S11. With port 2 matched, port 1 sees Z0 in parallel with R, i.e. normalized admittance 1 + y:

 S11 = (1 - (1 + y))/(1 + (1 + y)) = -y/(2 + y)

Step 2 – S21. For a shunt element, the voltage across both ports is the same, so V2 = V1 = V1+(1 + S11):

 S21 = 1 + S11 = 2/(2 + y)

Step 3 – symmetry. The element is symmetric and reciprocal, so S22 = S11 and S12 = S21:

          1     | -y   2 |
 [S] = ------- · |       |
        2 + y   |  2  -y |

Step 4 – modulation. If the bias makes y vary with the modulating signal, the output is b2 = S21(t)·a1 = [2/(2 + y(t))]·a1. Choosing the bias so that |S21(t)| = k[1 + m·cos ωₘt] gives an AM wave:

 a1 = A cos ωc t
 b2 = kA [1 + m cos ωm t] cos ωc t

with modulation index m.

Limiting cases:

• y → 0 (R very high, diode reverse biased): S11 = 0, S21 = 1, full transmission (carrier peak).
• y → ∞ (R very low, diode forward biased): S11 = −1, S21 = 0, full reflection (carrier off).

Properties of the matrix:

• Reciprocal: S12 = S21.
• Not lossless: |S11|² + |S21|² = (y² + 4)/(2 + y)² < 1 for 0 < y < ∞; the missing power is absorbed in the diode resistance.
• Not matched: S11 ≠ 0 except at y = 0, so an isolator is often placed before the modulator to protect the source from the reflected power.

An ideal (matched) modulator is therefore often modeled simply as [S] = [[0, S21(t)], [S21(t), 0]] with S21(t) = k[1 + m cos ωₘt]·e^(−jθ).

Use of S-parameters for three-port analysis

A three-port network (tee junction, power divider, circulator) is described by a 3 × 3 S-matrix that relates outgoing waves b to incoming waves a, with each port terminated in a matched load Z0:

 | b1 |   | S11  S12  S13 | | a1 |
 | b2 | = | S21  S22  S23 | | a2 |
 | b3 |   | S31  S32  S33 | | a3 |

• Sᵢᵢ is the reflection coefficient at port i; Sᵢⱼ is the transmission from port j to port i.
• |Sᵢⱼ|² gives the fraction of power going from port j to port i, so power division, isolation and matching are read directly.
• Reciprocity (Sᵢⱼ = Sⱼᵢ), losslessness ([S]*ᵀ[S] = I) and matching (Sᵢᵢ = 0) can be checked from the matrix.
• Using these conditions one proves that a 3-port cannot be lossless, reciprocal and matched at all ports at once. Hence a lossless matched 3-port must be a non-reciprocal circulator, and a reciprocal lossless tee junction must have at least one mismatched port.

Example (ideal circulator): S21 = S32 = S13 = 1 and all other terms zero; power goes 1 → 2 → 3 → 1.

Return loss and insertion loss

• Return loss (RL): the ratio, in dB, of incident power to reflected power at a port. It shows how well the port is matched.

 RL = 10 log(Pin/Pref) = -20 log|S11|  dB

A large RL means a good match (e.g. |S11| = 0.1 gives RL = 20 dB); a short or open gives RL = 0 dB.

• Insertion loss (IL): the ratio, in dB, of power delivered to the load without the device to power delivered with the device inserted; for a matched network it is the loss from input to output port.

 IL = 10 log(Pin/Pout) = -20 log|S21|  dB

For example, |S21| = 0.9 gives IL = 0.92 dB.

A two-port network is any linear circuit with an input port (1) and an output port (2), such as an amplifier, filter, attenuator or transmission line section. It is modeled as a "black box" described only by the relations between the port voltages, currents or waves.

      I1 ->                     <- I2
  o---------+-------------+---------o
  +         |             |         +
  V1        |  Two-port   |        V2
  -         |  network    |         -
  o---------+-------------+---------o
   port 1                     port 2

Four variables (V1, I1, V2, I2) exist; any two can be written in terms of the other two, giving the parameter sets below.

1. Impedance (Z) parameters

 V1 = Z11·I1 + Z12·I2
 V2 = Z21·I1 + Z22·I2

Z11 = V1/I1 at I2 = 0 (output open); Z21 = V2/I1 at I2 = 0; Z12 = V1/I2 at I1 = 0; Z22 = V2/I2 at I1 = 0. Units: ohm. Reciprocal → Z12 = Z21.

2. Admittance (Y) parameters

 I1 = Y11·V1 + Y12·V2
 I2 = Y21·V1 + Y22·V2

Measured with the opposite port short-circuited (V = 0). [Y] = [Z]⁻¹. Units: siemens.

3. Hybrid (h) parameters

 V1 = h11·I1 + h12·V2
 I2 = h21·I1 + h22·V2

h11 = input impedance (output shorted), h21 = current gain, h12 = reverse voltage ratio (input open), h22 = output admittance. Used for BJT models.

4. Transmission (ABCD) parameters

 V1 = A·V2 + B·I2'      (I2' = -I2, current leaving port 2)
 I1 = C·V2 + D·I2'

Cascaded networks multiply: [ABCD]total = [ABCD]1 · [ABCD]2. Reciprocal → AD − BC = 1. Example: a series impedance Z has A = 1, B = Z, C = 0, D = 1.

5. Scattering (S) parameters – the microwave model

At microwave frequencies the opens and shorts needed above are hard to make, so the network is described by waves. With reference impedance Z0:

 an = (Vn + Z0·In)/(2√Z0)   incident wave
 bn = (Vn - Z0·In)/(2√Z0)   reflected wave

 b1 = S11·a1 + S12·a2
 b2 = S21·a1 + S22·a2

• S11 = b1/a1 at a2 = 0 (port 2 matched): input reflection coefficient.
• S21 = b2/a1 at a2 = 0: forward transmission coefficient.
• S22 = b2/a2 at a1 = 0: output reflection coefficient.
• S12 = b1/a2 at a1 = 0: reverse transmission coefficient.

Derivation of S from Z: substitute V = √Z0(a + b) and I = (a − b)/√Z0 into V = Z·I:

 Z0(a + b) = Z(a - b)
 (Z + Z0·U) b = (Z - Z0·U) a
 [S] = (Z - Z0·U)(Z + Z0·U)^-1

For a two-port with ΔZ = (Z11 + Z0)(Z22 + Z0) − Z12·Z21:

 S11 = [(Z11 - Z0)(Z22 + Z0) - Z12·Z21]/ΔZ
 S21 = 2·Z21·Z0/ΔZ

Properties used in design

Useful derived quantities: input reflection Γin = S11 + S12·S21·ΓL/(1 − S22·ΓL); return loss = −20 log|S11|; insertion loss = −20 log|S21|; transducer gain (matched) = |S21|².

Example: a series impedance Z between two Z0 lines gives S11 = S22 = Z/(Z + 2Z0) and S21 = S12 = 2Z0/(Z + 2Z0); for Z = 50 Ω and Z0 = 50 Ω, S11 = 1/3 and S21 = 2/3.

Usefulness of S-parameters in microwave network analysis

• They relate incident and reflected waves, which are measurable at microwave frequencies, while voltage and current are not (and are not unique in waveguides).
• They are measured with matched loads, avoiding open/short terminations that are hard to realise and that make active devices unstable.
• |S11| and |S21| directly give return loss, VSWR, insertion loss and gain; reciprocity and losslessness show as symmetry and unitarity of [S].
• They are measured accurately with a vector network analyzer and are the natural language of filter and amplifier design.

S-parameters of a two-port network

 b1 = S11·a1 + S12·a2        a = incident wave
 b2 = S21·a1 + S22·a2        b = reflected wave

• S11 = b1/a1 (a2 = 0): input reflection coefficient.
• S21 = b2/a1 (a2 = 0): forward transmission coefficient.
• S22 = b2/a2 (a1 = 0): output reflection coefficient.
• S12 = b1/a2 (a1 = 0): reverse transmission coefficient.

Butterworth and Chebyshev responses in the insertion loss method

In the insertion loss method, a filter is specified by its power loss ratio:

 PLR = Power available from source / Power delivered to load
     = 1/|S21(ω)|² = 1/(1 - |Γ(ω)|²)

For a lossless filter, |S21|² + |S11|² = 1.

Why these forms are used:

1. |Γ(ω)|² of a passive, lossless, reciprocal LC network is an even function of ω, so it can be written as a ratio of polynomials in ω²:

 |Γ(ω)|² = M(ω²)/(M(ω²) + N(ω²))
 PLR = 1 + M(ω²)/N(ω²)

2. Any PLR of this form (PLR ≥ 1, even, rational) is physically realisable as a ladder of L and C elements. Taking N = 1 gives an all-pole low-pass prototype, the simplest ladder.
3. The designer then chooses the polynomial M to give the desired passband shape:

Butterworth (maximally flat):

 PLR = 1 + k²(ω/ωc)^(2N)

All derivatives of PLR are zero at ω = 0, so the passband is as flat as possible; the stopband falls at 20N dB/decade. k = 1 gives the 3 dB point at ωc.

Chebyshev (equal ripple):

 PLR = 1 + k²·T_N²(ω/ωc)
 T_N(x) = cos(N·cos⁻¹x),    |x| ≤ 1
 T_N(x) = cosh(N·cosh⁻¹x),  |x| > 1

Passband ripple = 1 + k²; the cutoff is much sharper than Butterworth for the same order N (stopband ≈ k²(2ω/ωc)^(2N)/4).

4. Both are polynomial (all-pole) responses, so the same ladder prototype structure (g₁, g₂, … gₙ tables) serves both. Through impedance and frequency scaling, low-pass to high-pass/band-pass/band-stop transforms, and Richards' transformation and Kuroda identities, the same prototype becomes a microwave filter of stubs or coupled lines.

Hence the Butterworth and Chebyshev responses are the standard (common) responses of prototype microwave two-port filters: they are the simplest realisable PLR functions, one optimizing flatness and the other optimizing selectivity, and both map directly onto S21 = 1/√PLR of the two-port.

Why S-parameters are used at high frequencies

At microwave frequencies voltages and currents are hard to measure (and not unique in waveguides), and the open/short terminations needed for Z, Y or h parameters are hard to realise and may make active devices oscillate. S-parameters use incident and reflected waves measured with matched loads, they are measured directly by a network analyzer, and they give reflection, transmission and power flow directly.

Given

 S11 = 0.4 + j0.5    S12 = j0.6
 S21 = j0.6          S22 = 0.4 - j0.5

a) Is the network reciprocal?

A network is reciprocal if [S] is symmetric, S12 = S21. Here S12 = j0.6 = S21, so the network is reciprocal.

b) Is the network lossless?

A lossless network has a unitary S-matrix, so each column must satisfy |S11|² + |S21|² = 1.

 |S11|² = 0.4² + 0.5² = 0.16 + 0.25 = 0.41
 |S21|² = 0.6²        = 0.36
 |S11|² + |S21|²      = 0.77  ≠ 1

(The second column also gives 0.41 + 0.36 = 0.77. The orthogonality condition S11·S12* + S21·S22* = 0 happens to hold, but the column sums fail.) So the network is not lossless: 23% of the power fed at port 1 is absorbed in the network.

c) Return loss at the input

 |S11| = √0.41 = 0.6403
 RL = -20 log|S11| = -20 log(0.6403) = 3.87 dB

Answer: return loss at the input = 3.87 dB

d) Reflected power for 5 W input

 Pref = |S11|² · Pin = 0.41 × 5 = 2.05 W

Answer: reflected power = 2.05 W

(For interest: power transmitted to a matched port 2 = 0.36 × 5 = 1.80 W, and power lost in the network = 5 − 2.05 − 1.80 = 1.15 W.)