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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


[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)


[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)


[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)


[Epsilon] = a small positive number

The symmetrical network N is shown bisected in Figure 1.


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


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


[[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]


[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.


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


[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


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


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


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:Silverman, Larry
Publication:Microwave Journal
Date:Feb 1, 1997
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