Electronics Guide

Ports and the Port Condition

A port is a pair of terminals through which energy enters or leaves a network. The port condition requires that the current entering one terminal of the pair equal the current leaving the other. A network with two such pairs is a two-port, with an input, port 1, and an output, port 2.

The condition is not automatic, because four terminals could carry four different currents that merely sum to zero. It holds when each port connects only to a source or load across that port. A three-terminal network, such as a common-emitter stage or a grounded ladder filter, satisfies it by construction, because its common terminal carries both return currents. It fails when an external path, such as a second ground connection, lets current enter at port 1 and leave through port 2.

Port Variables and Sign Convention

The four port variables are V₁ and I₁ at port 1 and V₂ and I₂ at port 2. Throughout this article, each voltage is the potential of the upper terminal of its port with respect to the lower one, and I₁ and I₂ both flow into the upper terminal. A load Z_L on port 2 therefore draws I₂ = −V₂/Z_L.

Six Parameter Sets

A linear two-port without internal sources imposes two linear relations on its four port variables. When any two variables are taken as independent, the relations give the other two through four coefficients, and the six possible pairs give six parameter sets for the same network.

A set exists only if its independent variables can be chosen freely. Internal independent sources would add a constant term to each equation, as in a Thevenin equivalent. The inverse ABCD set, which gives the port 2 variables in terms of the port 1 variables, is not treated further here.

Impedance Parameters

The impedance parameters, or Z-parameters, take the port currents as independent:

V₁ = Z₁₁I₁ + Z₁₂I₂

V₂ = Z₂₁I₁ + Z₂₂I₂

A zero port current means an open circuit, so these are the open-circuit impedance parameters, measured by driving each port in turn with a current source while the other port is open.

All four are in ohms; Z₂₁, for example, is the open-circuit voltage at port 2 per ampere driven into port 1.

Example: A Resistive T Network

A T network has 10 Ω in series with port 1, 20 Ω in series with port 2, and 30 Ω from their junction to the common terminal. With port 2 open, no current flows in the 20 Ω arm, so I₁ sees 10 + 30 = 40 Ω, and V₂ is the drop across the 30 Ω arm. By inspection, [Z] = [40, 30; 30, 50] Ω. Both transfer impedances equal the shared 30 Ω arm, and such equality holds in any network of resistors, capacitors, inductors, and transformers.

When Z-Parameters Do Not Exist

A single impedance in series between the upper terminals forces I₂ = −I₁, so it has no Z-parameters, although its Y- and ABCD parameters exist. An ideal transformer fixes both its current ratio and its voltage ratio, so it has neither Z- nor Y-parameters. These gaps belong to ideal models; a real transformer, with finite magnetizing inductance and some leakage, has both.

Admittance Parameters

The admittance parameters, or Y-parameters, take the port voltages as independent:

I₁ = Y₁₁V₁ + Y₁₂V₂

I₂ = Y₂₁V₁ + Y₂₂V₂

A zero port voltage means a short circuit, so these are the short-circuit admittance parameters, measured by driving one port with a voltage source while the other is shorted.

All four are in siemens, and [Y] = [Z]⁻¹ whenever both sets exist. The inverse applies to the whole matrix: Y₁₁ is not 1/Z₁₁, because one is measured with port 2 shorted and the other with port 2 open.

For the T network above, the determinant of [Z] is 40 × 50 − 30 × 30 = 1,100 Ω², so

[Y] = [50, −30; −30, 40] / 1,100 = [0.04545, −0.02727; −0.02727, 0.03636] S

A short-circuit test agrees. With port 2 shorted, the 20 Ω and 30 Ω arms are in parallel, so port 1 sees 10 + 12 = 22 Ω and Y₁₁ = 1/22 = 0.04545 S. The fraction 30/(20 + 30) = 0.6 of the input current leaves through the short, so Y₂₁ = −0.6/22 = −0.02727 S, negative only because of the reference direction.

Y-Parameters and Nodal Analysis

Nodal analysis produces Y-parameters naturally: eliminating the internal nodes leaves two equations relating port currents to port voltages. SPICE-family simulators use modified nodal analysis, an extension of the method, as circuit simulation (SPICE) describes. A shunt element, with the upper terminals of the two ports joined, forces V₁ = V₂, so it has no Y-parameters, while all four of its Z-parameters equal its impedance.

Hybrid Parameters

The h-parameters mix the two kinds of variable, taking the input current and the output voltage as independent:

V₁ = h₁₁I₁ + h₁₂V₂

I₂ = h₂₁I₁ + h₂₂V₂

The mixture suits a device with a low input impedance and a high output impedance, such as a bipolar transistor in the common-emitter connection: a large capacitor easily shorts the high-impedance output for signals, and a large series impedance easily opens the low-impedance input. The equations also read as a circuit: port 1 is h₁₁ in series with a voltage source h₁₂V₂, and port 2 is h₂₂ in parallel with a current source h₂₁I₁.

Inverse Hybrid Parameters

The g-parameters take the input voltage and the output current as independent:

I₁ = g₁₁V₁ + g₁₂I₂

V₂ = g₂₁V₁ + g₂₂I₂

Here g₁₁ is the input admittance with port 2 open, g₁₂ the reverse current ratio with port 1 shorted, g₂₁ the forward voltage ratio with port 2 open, and g₂₂ the output impedance with port 1 shorted. When both sets exist, [g] = [h]⁻¹. The g-parameters fit a voltage amplifier. An ideal one, which draws no input current and holds its output at μ times its input voltage under any load, has g₂₁ = μ and A = 1/μ, with every other g- and ABCD parameter zero. It has no Z-, Y-, or h-parameters, because its input current is always zero and its output voltage is tied to its input voltage.

Transmission Parameters

The transmission, ABCD, or chain parameters express the input quantities in terms of the output quantities. With both currents flowing into their ports, they are

V₁ = AV₂ − BI₂

I₁ = CV₂ − DI₂

The minus signs appear because −I₂ is the current flowing out of port 2, which is the current that enters the next network in a chain. This article writes the ABCD matrix as [T] = [A, B; C, D].

Every ABCD measurement drives port 1 and terminates port 2. Because the equations put the port 1 and port 2 variables on opposite sides, cascaded networks multiply their ABCD matrices.

The Direction of I₂

Some texts define the port 2 current as flowing out of the port and write V₁ = AV₂ + BI₂ and I₁ = CV₂ + DI₂. Both forms give the same A, B, C, and D. Mixing them does not: plus signs combined with a current flowing into port 2 reverse the signs of B and D. Confirm the direction before combining ABCD formulas or data from different sources.

ABCD Matrices of Common Building Blocks

For a series impedance Z, an open output leaves V₁ = V₂ with no current, so A = 1 and C = 0, and a shorted output puts the whole input voltage across Z, so B = Z and D = 1.

The T and Π rows are products of the first two rows, and the line is derived below. For the transformer, N:1 means N primary turns per secondary turn, so V₁ = NV₂ and the current out of port 2 is N times I₁.

Converting Between Parameter Sets

Every set that exists carries the same information, and rearranging the defining equations converts one into another. Solving the first h-parameter equation for I₁, for example, gives Y₁₁ = 1/h₁₁ and Y₁₂ = −h₁₂/h₁₁ directly. The tables below collect all such results, using the determinants

Δ_Z = Z₁₁Z₂₂ − Z₁₂Z₂₁, Δ_Y = Y₁₁Y₂₂ − Y₁₂Y₂₁, Δ_h = h₁₁h₂₂ − h₁₂h₂₁, Δ_T = AD − BC

A conversion fails where it would divide by zero, which is exactly where the target set does not exist. Near such a point it magnifies error: converting a nearly isolating network to ABCD parameters divides by a tiny forward transfer term, so small measurement errors become large ones.

The g-parameters follow from [g] = [h]⁻¹, or from the Z-parameters as g₁₁ = 1/Z₁₁, g₁₂ = −Z₁₂/Z₁₁, g₂₁ = Z₂₁/Z₁₁, and g₂₂ = Δ_Z/Z₁₁. The determinants are related as Δ_Y = 1/Δ_Z, Δ_h = Z₁₁/Z₂₂, and Δ_T = Z₁₂/Z₂₁, relations that reappear in the tests for reciprocity and symmetry.

Worked Example: One Network in Four Forms

A network has [Z] = [10, 2; 40, 20] Ω, with a forward transfer impedance twenty times its reverse transfer impedance, as an amplifier stage might have.

1. Determinant: Δ_Z = 10 × 20 − 2 × 40 = 120 Ω².
2. Y-parameters: Y₁₁ = 20/120 = 0.1667 S, Y₁₂ = −2/120 = −0.01667 S, Y₂₁ = −40/120 = −0.3333 S, and Y₂₂ = 10/120 = 0.08333 S.
3. h-parameters: h₁₁ = 120/20 = 6 Ω, h₁₂ = 2/20 = 0.1, h₂₁ = −40/20 = −2, and h₂₂ = 1/20 = 0.05 S.
4. ABCD parameters: A = 10/40 = 0.25, B = 120/40 = 3 Ω, C = 1/40 = 0.025 S, and D = 20/40 = 0.5.
5. Check by converting back: Δ_h = 6 × 0.05 − 0.1 × (−2) = 0.5 gives Z₁₁ = Δ_h/h₂₂ = 10 Ω, and Δ_T = 0.25 × 0.5 − 3 × 0.025 = 0.05 gives Z₁₂ = Δ_T/C = 2 Ω.

The ABCD determinant, 0.05, equals Z₁₂/Z₂₁ = 2/40, and its departure from 1 marks the network as nonreciprocal.

Reciprocity and Symmetry

By the reciprocity theorem, in a network of linear, bilateral elements, a current driven into port 1 produces the same open-circuit voltage at port 2 as the same current driven into port 2 produces at port 1, so Z₁₂ = Z₂₁. Resistors, capacitors, inductors, coupled inductors, and ideal transformers form reciprocal networks. Transistors and other controlled sources generally do not, nor do gyrators or the magnetized ferrite circulators and isolators of microwave systems.

A reciprocal two-port needs only three independent parameters. It is also symmetric when its ports can be interchanged without changing any terminal voltage or current, which adds the condition Z₁₁ = Z₂₂ and leaves two independent parameters. Mirror symmetry guarantees electrical symmetry, but electrical symmetry does not require it.

The conversions explain the less obvious entries. Because h₁₂ = Z₁₂/Z₂₂ and h₂₁ = −Z₂₁/Z₂₂, equal transfer impedances make them equal and opposite; Δ_T = Z₁₂/Z₂₁ equals 1 under reciprocity, and Δ_h = Z₁₁/Z₂₂ equals 1 under symmetry. The resistive T network is reciprocal but not symmetric.

Lossless Networks

A network of ideal inductors, capacitors, and transformers dissipates no power, so at real frequencies its Z- and Y-parameters, where they exist, are purely imaginary. Its ABCD parameters have real A and D and imaginary B and C.

Interconnecting Two-Ports

Two networks can share both ports, joined at each end either in series, so that one current flows through both and their voltages add, or in parallel, so that one voltage appears across both and their currents add. The parameter set whose independent variables are the shared quantities then adds. In a cascade, ABCD matrices multiply instead.

A feedback amplifier is a two-port amplifier joined to a two-port feedback network, and the four topologies of negative feedback theory are these four addition rules.

The Port Condition and Brune's Tests

The addition rules hold only if both networks still satisfy the port condition after connection. Two three-terminal networks in parallel, with their common terminals joined, always do; series connections often do not. Stacking two T networks connects the common terminal of the upper network to both upper terminals of the lower one, which shorts the lower network's input terminal to its output terminal. With the 10, 20, and 30 Ω T above a 40, 50, and 60 Ω T, whose [Z] = [100, 60; 60, 110] Ω, the sum predicts [140, 90; 90, 160] Ω, but nodal analysis of the stack gives [122.2, 112.2; 112.2, 132.2] Ω. Inverting the lower network, so that the common terminals join each other, restores the port condition and makes the sum exact.

Brune's tests, named for Otto Brune, check a proposed connection. For a series–series connection, join the input ports in series, apply a voltage, leave the outputs unconnected, and measure the voltage between the output terminals that the connection would join; then repeat from the output side. For a parallel–parallel connection, join the inputs in parallel, short each output separately, apply a voltage, and measure between the two shorted outputs, again from both sides. Zero voltage in both tests permits the connection. When a connection fails, an ideal 1:1 isolating transformer at one port restores the port condition.

Cascade Connection

In a cascade, port 2 of network a feeds port 1 of network b, so V₂ and the outgoing current of a are V₁ and I₁ of b. Substituting one set of ABCD equations into the other gives

[T] = [T_a][T_b]

The matrices must be multiplied in signal order, since matrix multiplication does not commute. A cascade of three-terminal networks on a common ground preserves the port condition automatically.

Worked Example: A 6 dB Attenuator

A symmetric T pad for a 50 Ω system with 6 dB of attenuation has series arms of R₁ = 16.61 Ω and a shunt arm of R₂ = 66.93 Ω, rounded to four significant figures. It is a cascade of a series resistor, a shunt conductance of 1/66.93 = 0.014941 S, and a second series resistor.

1. Multiply the first two matrices: [1, 16.61; 0, 1][1, 0; 0.014941, 1] = [1.2482, 16.61; 0.014941, 1].
2. Multiply by the third, [1, 16.61; 0, 1]: A = 1.2482, B = 1.2482 × 16.61 + 16.61 = 37.34 Ω, C = 0.014941 S, and D = 0.014941 × 16.61 + 1 = 1.2482.
3. Check: A = D, so the pad is symmetric, and AD − BC = 1.5579 − 0.5579 = 1.0000, so it is reciprocal.

Terminated Two-Ports

In use, a two-port sits between a source of open-circuit voltage V_s and internal impedance Z_s at port 1 and a load Z_L at port 2, with Y_s = 1/Z_s and Y_L = 1/Z_L. The terminations add two equations, V₁ = V_s − Z_sI₁ and V₂ = −Z_LI₂, to the network's own two. Solving all four gives the input impedance Z_in = V₁/I₁, the output impedance Z_out seen back into port 2 with V_s set to zero, the voltage gain A_v = V₂/V₁, and the current gain A_i = I₂/I₁.

The load reaches back to the input only through the product Z₁₂Z₂₁. A unilateral network, with Z₁₂ = 0, has Z_in = Z₁₁ for any load and Z_out = Z₂₂ for any source, which is why simplified amplifier models neglect reverse transmission, at a cost that grows with it. Port 2 also acts as a Thevenin source of open-circuit voltage V_s/(A + CZ_s) behind Z_out, and the overall voltage gain is

V₂/V_s = Z_L/(AZ_L + B + CZ_sZ_L + DZ_s)

Power Gain

With RMS phasors, the load absorbs P_L = |I₂|² Re(Z_L), and the input absorbs P_in = |I₁|² Re(Z_in). The operating power gain G_P = P_L/P_in depends on the load only. The transducer power gain G_T = P_L/P_avs compares the load power with the source's available power, P_avs = |V_s|²/(4 Re Z_s), and depends on both terminations. The available power gain G_A, the power available at port 2 divided by P_avs, depends on the source only. All three are equal when both ports are conjugately matched.

Worked Example: The Attenuator Between Two Loads

Terminate the 6 dB pad in its design load, Z_L = 50 Ω:

1. Input impedance: Z_in = (1.2482 × 50 + 37.34)/(0.014941 × 50 + 1.2482) = 99.75/1.9953 = 49.99 Ω, a match within the rounding of the resistors.
2. Voltage gain: A_v = 50/99.75 = 0.501, or −6.00 dB.

Now use a 100 Ω load, whose reflection coefficient in a 50 Ω system is (100 − 50)/(100 + 50) = 1/3:

1. Input impedance: Z_in = (124.82 + 37.34)/(1.4941 + 1.2482) = 162.16/2.7423 = 59.13 Ω.
2. Input reflection coefficient: (59.13 − 50)/(59.13 + 50) = 0.0837.

The reflection falls by a factor of 0.251, the square of the pad's voltage transmission of 0.501, because a reflected wave passes through the pad twice. Return loss improves by twice the attenuation, from 9.5 dB at the load to 21.5 dB at the input, which is why engineers pad a poorly matched port when they can spare the signal level.

T and Π Equivalent Circuits

A reciprocal three-terminal two-port can be modeled at its terminals by a T of three impedances if its Z-parameters exist, or by a Π of three admittances if its Y-parameters exist. For the T, place Z_a in series with port 1, Z_b in series with port 2, and Z_c from their junction to the common terminal. Open-circuit tests give Z₁₁ = Z_a + Z_c, Z₂₂ = Z_b + Z_c, and Z₁₂ = Z_c, so

Z_a = Z₁₁ − Z₁₂, Z_b = Z₂₂ − Z₁₂, Z_c = Z₁₂

For the Π, place Y_a across port 1, Y_b across port 2, and Y_c between the upper terminals. Short-circuit tests give

Y_a = Y₁₁ + Y₁₂, Y_b = Y₂₂ + Y₁₂, Y_c = −Y₁₂

From its Y-parameters, the resistive T network becomes a Π with 55 Ω across port 1, 110 Ω across port 2, and 36.67 Ω between the upper terminals. These are the values that the wye–delta conversion of DC circuit analysis gives, since a T is a wye and a Π is a delta. The equivalents hold only at the terminals and only at the frequency of the parameters, unless each element is kept as a function of frequency.

Coupled Inductors

Two coupled coils with self-inductances L₁ and L₂ and mutual inductance M, sharing a common terminal with their dotted ends at the upper terminals, have Z₁₁ = jωL₁, Z₂₂ = jωL₂, and Z₁₂ = Z₂₁ = jωM. Their T equivalent is three uncoupled inductors, L₁ − M and L₂ − M in the series arms and M in the shunt arm, so ordinary series and parallel rules apply. When M exceeds L₁ or L₂, as it can for tightly coupled windings with unequal turns, one series arm is a negative inductance, which is harmless in analysis. Magnetically Coupled Circuits develops the coupled-coil equations behind this equivalent.

Nonreciprocal Networks

A nonreciprocal network needs one controlled source besides the three arms. The T adds, in series with the port 2 arm, a current-controlled voltage source (Z₂₁ − Z₁₂)I₁ with its positive reference toward the port 2 terminal. The Π adds, across port 2, a voltage-controlled current source (Y₂₁ − Y₁₂)V₁ directed from the upper terminal to the common terminal. That Π form is the structure of the hybrid-pi transistor model.

Transistor Small-Signal Models as Two-Ports

A transistor is nonlinear, but for small signals about a fixed operating point it is a linear two-port, named for the terminal that input and output share, as in common emitter or common source. Its parameters hold only near the bias point where they were found and say nothing about compression or distortion. Biasing and stage design belong to small-signal amplifier analysis.

h-Parameters on Data Sheets

Bipolar h-parameters use letter subscripts: i for input (11), r for reverse (12), f for forward (21), and o for output (22), followed by the common terminal, so the common-emitter set is h_ie, h_re, h_fe, and h_oe. The opens and shorts apply to the signal only, while the bias supply holds the operating point. The onsemi data sheet for the 2N3903 and 2N3904, revision 9 of August 2021, gives these ranges for the 2N3904 at a collector current of 1.0 mA, a collector–emitter voltage of 10 V, and 1.0 kHz: h_ie from 1.0 to 10 kΩ, h_re from 0.5 to 8.0 × 10⁻⁴, h_fe from 100 to 400, and h_oe from 1.0 to 40 µmhos (microsiemens).

Worked Example: Exact h-Parameter Analysis

Take h_ie = 2.6 kΩ, h_re = 2 × 10⁻⁴, h_fe = 100, and h_oe = 20 µS, values inside the 2N3904 ranges, with an AC load R_L = 2 kΩ and a source resistance R_s = 1 kΩ.

1. Determinant: Δ_h = 2,600 × (20 × 10⁻⁶) − (2 × 10⁻⁴) × 100 = 0.052 − 0.020 = 0.032.
2. Current gain: A_i = h_fe/(1 + h_oeR_L) = 100/1.04 = 96.15.
3. Input impedance: Z_in = (h_ie + Δ_hR_L)/(1 + h_oeR_L) = 2,664/1.04 = 2,562 Ω.
4. Voltage gain: A_v = −h_feR_L/(h_ie + Δ_hR_L) = −200,000/2,664 = −75.08.
5. Output admittance: Y_out = h_oe − h_reh_fe/(h_ie + R_s) = 20 µS − 5.56 µS = 14.44 µS, so Z_out = 69.2 kΩ.
6. Power gains: G_P = |A_v||A_i| = 7,219, or 38.6 dB. With V₂/V_s = A_vZ_in/(Z_in + R_s) = −54.0, the transducer gain is G_T = |V₂/V_s|² × 4R_s/R_L = 5,831, or 37.7 dB, lower because the source is not matched to the input.

The simplified model, with h_re = h_oe = 0, predicts Z_in = 2.6 kΩ and A_v = −h_feR_L/h_ie = −76.9, errors under 3 percent, but an infinite output impedance. Even h_oe alone would give 50 kΩ; the tiny reverse ratio raises it to 69.2 kΩ, because h_reh_fe = 0.02 is not negligible.

The Hybrid-Pi Model as a Π Network

The hybrid-pi model places r_π in parallel with C_π from base to emitter, C_μ from base to collector, r_o from collector to emitter, and a current source g_mV_be from collector to emitter. Neglecting the base spreading resistance, its common-emitter Y-parameters are

Y₁₁ = 1/r_π + jω(C_π + C_μ)

Y₁₂ = −jωC_μ

Y₂₁ = g_m − jωC_μ

Y₂₂ = 1/r_o + jωC_μ

Y₂₁ and Y₁₂ differ by g_m, the controlled source of a nonreciprocal Π. At low frequencies, conversion gives h_ie = r_π, h_fe = g_mr_π, h_oe = 1/r_o, and h_re = 0; a real transistor's small h_re comes from effects the simple model omits, such as base-width modulation.

The model also explains the transition frequency. The short-circuit current gain h₂₁ = Y₂₁/Y₁₁ equals β = g_mr_π at low frequencies, falls at 20 dB per decade above f_β = 1/[2πr_π(C_π + C_μ)], and reaches 1 near f_T ≈ g_m/[2π(C_π + C_μ)]. With g_m = 38.5 mS, r_π = 2.6 kΩ, C_π = 16 pF, and C_μ = 2 pF, f_β is 3.40 MHz, and f_T is 340 MHz by the approximation and 343 MHz exactly. Between f_β and f_T, the product of |h₂₁| and frequency is nearly constant, so f_T can be found from a measurement made well below it; the 2N3904 data sheet specifies a minimum f_T of 300 MHz, measured at 100 MHz with a collector current of 10 mA and a collector–emitter voltage of 20 V.

Field-Effect Transistors and y-Parameters

A field-effect transistor's gate draws almost no current at low frequencies, so Y₁₁ approaches zero and h₁₁ = 1/Y₁₁ grows without bound. FETs are therefore described by common-source Y-parameters, written y_is, y_rs, y_fs, and y_os. The basic model is the same Π without r_π: Y₁₁ = jω(C_gs + C_gd), Y₁₂ = −jωC_gd, Y₂₁ = g_m − jωC_gd, and Y₂₂ = g_ds + jωC_gd, where g_ds is the output conductance. The onsemi MPF102 JFET data sheet, revision 3 of January 2006, specifies |y_fs| as 2,000 to 7,500 µmhos at 1.0 kHz, with 15 V from drain to source and zero gate–source voltage, and plots all four admittance parameters against frequency. Field-effect transistors describes the devices.

Transmission Lines as ABCD Networks

A uniform transmission line with characteristic impedance Z₀, propagation constant γ = α + jβ, and length d is a two-port whose parameters come from the telegrapher's equations, which describe coaxial cable and, approximately, printed-circuit traces. The voltage and current along the line are sums of forward and backward waves, and evaluating them at both ends gives

V₁ = V₂ cosh γd + Z₀(−I₂) sinh γd

I₁ = (V₂/Z₀) sinh γd + (−I₂) cosh γd

so A = D = cosh γd, B = Z₀ sinh γd, and C = (sinh γd)/Z₀. The line is symmetric, since A = D, and reciprocal, since AD − BC = cosh² γd − sinh² γd = 1. For a lossless line, γ = jβ, and

A = D = cos βd, B = jZ₀ sin βd, C = j(sin βd)/Z₀

with A and D real and B and C imaginary, as for any lossless reciprocal network. For a given Z₀, the electrical length βd sets everything, where β = 2π/λ and λ is the wavelength on the line.

Input Impedance and Special Lengths

Substituting the line's parameters into Z_in = (AZ_L + B)/(CZ_L + D) and dividing by cosh γd gives the familiar result

Z_in = Z₀(Z_L + Z₀ tanh γd)/(Z₀ + Z_L tanh γd)

A lossless quarter-wave line, with βd = 90°, has A = D = 0, B = jZ₀, and C = j/Z₀, so Z_in = Z₀²/Z_L, the principle of the quarter-wave transformer in impedance matching. A lossless half-wave line has A = D = −1 and B = C = 0, so it repeats the load impedance and inverts the voltage.

A line much shorter than a wavelength has cos βd ≈ 1 and sin βd ≈ βd. With inductance L′ and capacitance C′ per unit length, β = ω√(L′C′) and Z₀ = √(L′/C′), so B ≈ jωL′d and C ≈ jωC′d: the short line is its total series inductance and shunt capacitance, the lumped model of ordinary circuit analysis.

Worked Example: An Eighth-Wave Line

A lossless 50 Ω line one-eighth of a wavelength long, so that βd = 45°, feeds a 100 Ω load.

1. Line matrix: A = D = cos 45° = 0.7071, B = j50 × 0.7071 = j35.36 Ω, and C = j0.7071/50 = j0.01414 S.
2. Numerator and denominator: AZ_L + B = 70.71 + j35.36 Ω, and CZ_L + D = 0.7071 + j1.414.
3. Input impedance: Z_in = (70.71 + j35.36)/(0.7071 + j1.414) = 40 − j30 Ω.

The reflection coefficient has magnitude 1/3 at the load and |−10 − j30|/|90 − j30| = 1/3 at the input: a lossless line rotates the phase of a reflection but cannot change its size.

Image Impedance and Transfer Constant

Any symmetric reciprocal two-port terminated in its image impedance, Z_I = √(B/C), presents Z_I at its input, just as a line terminated in Z₀ presents Z₀, and its image transfer constant θ, defined by cosh θ = A, plays the role of γd. The 6 dB attenuator has Z_I = √(37.34/0.014941) = 49.99 Ω and θ = cosh⁻¹ 1.2482 = 0.6907 neper, or 6.00 dB. Image-matched sections cascade by adding transfer constants, as lines add lengths, and classical constant-k and m-derived filter design, covered in passive filter networks, rests on these image parameters.

From Z and Y to S-Parameters

Why Radio-Frequency Work Uses S-Parameters

Every set so far is defined by open circuits, short circuits, or both, and at radio and microwave frequencies those terminations stop being practical. Richard W. Anderson set out the problem in "S-Parameter Techniques for Faster, More Accurate Network Design," published in the Hewlett-Packard Journal in February 1967 and later revised as Hewlett-Packard Application Note 95-1. Lead inductance and capacitance spoil shorts and opens, higher frequencies required tuning stubs adjusted at each frequency, and a stub across a transistor's input or output could make it oscillate. A device measured between a 50 Ω source and load, by contrast, has very little chance of oscillating.

Waves have two further advantages. On a lossless line, a traveling wave keeps its magnitude while terminal voltages and currents vary with position, so a device can be measured through cables. In a waveguide, voltage and current have no unique definition, but incident and reflected power remain measurable. S-parameters describe a network by those waves, and network analyzers measure them directly.

Waves and the Reference Impedance

S-parameters require a reference impedance at each port. This article uses the same real, positive Z₀ at both ports, usually 50 Ω. The incident and reflected waves at port n are

a_n = (V_n + Z₀I_n)/(2√Z₀) and b_n = (V_n − Z₀I_n)/(2√Z₀)

with I_n flowing into port n. With RMS phasors, |a_n|² − |b_n|² is the average power delivered into the port. The S-parameters relate reflected to incident waves:

b₁ = S₁₁a₁ + S₁₂a₂

b₂ = S₂₁a₁ + S₂₂a₂

Setting a₂ = 0 requires V₂ = −Z₀I₂, a termination in Z₀. S₁₁ is thus the input reflection coefficient and S₂₁ the forward transmission coefficient with port 2 terminated in the reference impedance, which replaces the open and the short. A reciprocal network has S₁₂ = S₂₁, a symmetric one also S₁₁ = S₂₂, and a lossless two-port has |S₁₁|² + |S₂₁|² = 1. Complex reference impedances need a more careful definition, such as the power waves of K. Kurokawa's paper "Power Waves and the Scattering Matrix," in the March 1965 IEEE Transactions on Microwave Theory and Techniques.

Converting to and from Z and Y

Substituting the Z- or Y-parameter equations into the wave definitions gives, with [U] the 2 × 2 identity matrix and Y₀ = 1/Z₀,

[S] = ([Z] − Z₀[U])([Z] + Z₀[U])⁻¹

[Z] = Z₀([U] + [S])([U] − [S])⁻¹

[S] = (Y₀[U] − [Y])(Y₀[U] + [Y])⁻¹

[Y] = Y₀([U] − [S])([U] + [S])⁻¹

The factors in each product commute. Written out, the Z relations give the formulas below, with K_Z = (Z₁₁ + Z₀)(Z₂₂ + Z₀) − Z₁₂Z₂₁ and K_S = (1 − S₁₁)(1 − S₂₂) − S₁₂S₂₁. The Y relations have the same form with Y for Z, Y₀ for Z₀, and every S-parameter negated.

Examples and Connections

A 50 Ω series resistor has no Z-parameters, but its Y-parameters are Y₁₁ = Y₂₂ = 0.02 S and Y₁₂ = Y₂₁ = −0.02 S. In a 50 Ω system, the Y relations give S₁₁ = S₂₂ = 1/3 and S₁₂ = S₂₁ = 2/3. Directly, port 1 sees 100 Ω when port 2 is terminated in 50 Ω, and (100 − 50)/(100 + 50) = 1/3; the reflected and transmitted powers, 1/9 and 4/9, leave 4/9 dissipated in the resistor. In a 75 Ω system, the same resistor has S₁₁ = 0.25 and S₂₁ = 0.75, so an S-parameter is incomplete without its reference impedance. An ideal 2:1 transformer, which has neither Z- nor Y-parameters, has S₁₁ = 0.6, S₂₁ = S₁₂ = 0.8, and S₂₂ = −0.6 for any equal reference impedances, and 0.6² + 0.8² = 1 confirms that it is lossless.

Data files therefore record the reference impedance. The Touchstone format, whose version 2.0 the IBIS Open Forum ratified on April 24, 2009, states a reference resistance on its option line, 50 Ω by default, and lets a two-port file carry S-, Y-, Z-, H-, or G-parameter data. S-parameters also connect to the ABCD set: S₂₁ = 2/(A + B/Z₀ + CZ₀ + D), which for the 6 dB pad in a 50 Ω system is 2/(1.2482 + 0.7468 + 0.7471 + 1.2482) = 0.501. To cascade S-parameter blocks, engineers convert to ABCD or to scattering transfer parameters and multiply; some texts call the latter T-parameters, a name others give the ABCD set. Balanced circuits extend the framework to mixed-mode S-parameters.

Summary

A two-port that obeys the port condition is characterized at its terminals by four parameters. Open-circuit tests give Z-parameters and short-circuit tests give Y-parameters; h- and g-parameters mix the two, and ABCD parameters multiply in cascades. The sets interconvert wherever they exist, reciprocity and symmetry each remove one independent parameter, and series and parallel connections add Z, Y, h, or g matrices when Brune's tests show that the port condition survives. Terminated two-port formulas give impedances and gains directly. Transistors and transmission lines fit the same framework, and at radio frequencies S-parameters, defined with terminations in a stated reference impedance instead of opens and shorts, convert to and from Z and Y.

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