# The bisection theorem.

Essentially, the bisection theorem implies that all network parameters may be derived from [Z.sub.e] and [Z.sub.o].

If a circuit has two ports and is symmetrical about a central plane, the bisection theorem may be used to calculate [S.sub.22] = [S.sub.11] and [S.sub.21] = [S.sub.12]. The circuit must also be passive and reciprocal as are many lumped and distributed networks. This article applies the bisection theorem to determine both the lumped elements of a power divider and the optimum transmission line spacing for the series-connected PIN diodes in an SPST switch. This technique allows the important [S.sub.21] term insertion loss L to be determined quickly as

L = 20 [log.sub.10] [absolute value of [S.sub.21]] dB

where

[S.sub.21] = [Z.sub.e] - [Z.sub.o]/([Z.sub.e] + 1) ([Z.sub.o] + 1)

[S.sub.11] = 1 - [Z.sub.e][Z.sub.o]/([Z.sub.e] + 1) ([Z.sub.o] + 1)

where

[Z.sub.e] = normalized input impedance into the half network with an open circuit across the plane of symmetry

[Z.sub.o] = normalized input impedance into the half network with a short circuit to ground across the plane of symmetry

Essentially, the bisection theorem implies that all network parameters may be derived from [Z.sub.e] and [Z.sub.o]. [Z.sub.e] results from even excitation where two equal voltages excite the two-port network. [Z.sub.o] results from odd excitation, as when two equal and opposite sign voltages excite the two-port network. Forms of this theorem are true for n-port networks possessing symmetry about one or more planes. Essentially, any excitation of the two-port network may be expressed as a linear combination of an even voltage plus an odd voltage excitation. Even and odd excitations are the fundamental excitations of a two-port network with symmetry about one central plane. An analogy can be drawn using the fundamental unit vectors [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. Any vector in the xy plane may be expressed as the sum of projected lengths in the [Mathematical Expression Omitted] and [Mathematical Expression Omitted] directions, respectively.

This solution method involving calculating [Z.sub.e] and [Z.sub.o] of the half networks is much quicker than calculating the transfer function by multiplying ABCD matrices or many other techniques. The [Z.sub.e] and [Z.sub.o] terms simplify finding [S.sub.21] and [S.sub.11] into the combinations of two easily derived component parts. Examining the [S.sub.21] and [S.sub.11] terms gives

[S.sub.21] = [Z.sub.e] - [Z.sub.o]/([Z.sub.e] + 1) ([Z.sub.o] + 1)

and

[S.sub.11] = 1 - [Z.sub.e][Z.sub.o]/([Z.sub.e] + 1) ([Z.sub.o] + 1)

These equations show that for high insertion loss [S.sub.21] is small and [Z.sub.e] = [Z.sub.o] is necessary. This characteristic is true in filter reject bands, for isolated ports in power dividers and for high switch isolation. If [Z.sub.e] = [Z.sub.o] = 1, then high isolation is obtained with a good match since [absolute value of [S.sub.11]] is small as well, that is,

[absolute value of [S.sub.11]] = [Epsilon]/([greater than] 2)

where

[Epsilon] = a small positive number

The symmetrical network N is shown bisected in Figure 1.

A TWO-WAY LUMPED POWER DIVIDER ANALYSIS

A 15 MHz two-way power divider comprising lumped components is shown in Figure 2. Figure 3 shows the lumped two-way power divider as a symmetrical network about the output ports. To solve for the lumped elements, [Z.sub.e] = [Z.sub.o] = 1 at 15 MHz, that is, high isolation with a good match. The even excitation puts an open across the plane of symmetry, giving

[Mathematical Expression Omitted]

After cross multiplying and equating real and imaginary parts at [Omega] = [[Omega].sub.o],

L/C = 2500

and

[f.sub.o] = 1/2[pi][square root of LC]

where

[[Omega].sub.o] = 2[pi][f.sub.o]

[f.sub.o] = the frequency at which [Z.sub.e] = [Z.sub.o] = 1

Odd excitation puts a short circuit to ground across the plane of symmetry, giving

[Mathematical Expression Omitted]

at [Omega] = [[Omega].sub.o]

After cross multiplying and setting the real and imaginary parts equal at [f.sub.o], it is determined that

[C.sub.a] = L/50[R.sub.a]

and

[R.sub.a] = 50 [Omega]

Note that the [Z.sub.e] = [Z.sub.o] = 1 conditions were implied by the form of [S.sub.21] and [S.sub.11] as a general requirement independent of the particular network.

OPTIMUM SPACING FOR THE SPST SWITCH SERIES PIN DIODES

The symmetrical network of an SPST microwave switch using two series PIN diodes in their back-biased slate is shown in Figure 4. At 1 to 2 GHz, back-biased, low capacitance PIN diodes are essentially small capacitors. The highest isolation is required at the high frequency end of the band. For this case,

[absolute value of [S.sub.21]] = [absolute value of [Z.sub.e] - [Z.sub.o]]/[absolute value of ([Z.sub.e] + 1)] [absolute value of ([Z.sub.o] + 1)]

is to be minimized at 2 GHz.

With open and short circuits to ground across the plane of symmetry,

[Z.sub.e] = 1/j[Omega]C[Z.sub.o] - j cot [Theta]/2

[Z.sub.o] = 1/j[Omega]C[Z.sub.o] - j tan [Theta]/2

where

[Omega] = 2[pi]f

[Theta] = electrical length of the transmission line separating the two PIN diodes

Subtracting the terms in the numerator of [absolute value of [S.sub.21]] gives

[absolute value of [S.sub.21]] = [absolute value of (tan [Theta]/2 + 1/tan [Theta]/2)]/[absolute value of [Z.sub.o] + 1] [absolute value of [Z.sub.e] + 1]

With the numerator simplified and the denominator terms entered, the equation becomes

[Mathematical Expression Omitted]

Observe that the numerator is a slowly varying function of [Theta] for [Theta] near 90 [degrees]. The numerator stays close to 2 for a [+ or -]13 percent variation of [Theta] about 90 [degrees]. The denominator's dominant term is the capacitive reactance for [Theta] near 90 [degrees]. Therefore, for minimum [absolute value of [S.sub.21]] (maximum isolation), [Theta] = 90 [degrees] at the maximum frequency. At 2 GHz, consider these two cases: for C = 0.1 pf, [([Omega]C[Z.sub.o]).sup.-1] = 15.9 and for C = 0.05 pf, [([Omega]C[Z.sub.o]).sup.-1] = 31.8. The tan[Theta]/2 and cot[Theta]/2 terms each equal 1. For [Theta] = 90 [degrees], the numerator is at its minimum of 2. With the denominator terms included, the isolation may be calculated easily. At 2 GHz with [Theta] = 90 [degrees],

[absolute value of [S.sub.21]] = 2/(14.9)(16.9), for C = 0.10 pf

then

20 [log.sub.10] [absolute value of [S.sub.21]] = -42 dB isolation

and

[absolute value of [S.sub.21]] = 2/(30.8)(32.8), for C = 0.05 pf

then

20 [log.sub.10] [absolute value of [S.sub.21]] = -54 dB isolation

For a two-throw switch, the passing path provides an additional -6 dB to the isolation from the input to the isolated output. This increase in isolation occurs because the passing path effectively reduces the voltage at the first PIN diode to one-half the generator voltage. This switch is shown schematically in Figure 5.

The series diodes are spaced 90 [degrees] apart at the highest operating frequency. This optimum value for [Theta] is apparent from the simple form of [absolute value of [S.sub.21]] as determined by applying the bisection theorem to this symmetrical two-port network.
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Author: Printer friendly Cite/link Email Feedback Silverman, Larry Microwave Journal Feb 1, 1997 1289 Acronyms used by the RF/microwave industry. A low cost RF transceiver for data communications. Formulas (Mathematics) Mathematics Microwave communications