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All-Pass Network, First-Order (AP1)

All-Pass Network, First-Order (AP1)

Properties	Description	Unit	Default	Range
R	Image impedance	Ohm	Required	³ 0
F	Resonant frequency	Hz	Required	³ 0
M	Shape factor	None	Required	³ 0
Q	Quality factor	None	¥	> 0

Notes

1. The model is described by the following:

where:

2. RI is the image impedance, and corresponds to R in the table of parameters, and

3. F₀ is the resonant frequency, and corresponds to F in the table of parameters.

4. AP1 is useful for finding values of M, F (F₀), and possibly R (RI) through optimization.

5. The quality factor Q is assumed to be infinite (R_d=0 W) unless a non-zero positive value is supplied.

6. The lattice impedances of the first-order all-pass network are:

where:

f is the frequency. In terms of the AP1 parameters:

According to the Bartlett bisection theorem [Ref. 1], the circuit shown below is an exact equivalent of the AP1 lattice structure.

The TEE section containing the negative inductance, -L/2, as the shunt element can be replaced with perfectly coupled coils, k=1, to obtain the circuit shown below:

Note: The self-inductance, L_S, is given by L_S=L/2.

In terms of the parameters for AP1, the design formulas are:

As an illustration, the circuits CKT1 and CKT2 defined below are equivalent:

.PARAM F0=1MHZ
.PARAM M1=0.5
.PARAM RI=75
*.SUBCKT CKT1 1 2
AP1: 1 2 R=RI F=F0 M=M1
.ENDS

*.PARAM LS=2.98416UH
.PARAM C1=1.06103NF
*
.SUBCKT CKT2 3 4
MUIK: 1 3 3 2 L1=LS L2=LS K=1
CAP: 3 C=(2*C1)
.ENDS

Netlist Form

AP1:NAME n1 [n2] [n3] [n4] R=val F=val M=val [Q=val]

Netlist Example

Quality factor ¥:
AP1:APQI 1 2 R=500 F=1MHZ M=1

Quality factor 50:
AP1:APQ501 2 R=500 F=1MHZ Q=50

References

1. Ernest A. Guillemin, Communications Networks, Vol. II: The Classical Theory of Long Lines, Filters, and Related Networks, (New York: John Wiley and Sons, Inc., 1935), p. 439.

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