# A pole, a zero and a transformer

For this one project, I needed to make a single pole / single zero transfer function that would go from a particular value of low frequency gain to a reduced value of high frequency gain. The corner frequencies of the pole and the zero that I needed were not critical. It was the gain values at frequencies above and below the sloped region of the roll-off curve that I needed to control.

This MultiSim SPICE analysis shows how it got done.

The low frequency gain is set by the ratio of the resistors, R1 and R2 in the conventional non-inverting fashion. The high frequency gain is set by the inverse of the turns ratio of the transformer. In this example, the transformer's 25:1 turns ratio sets the high frequency gain to 1/25 or -28 dB.

This circuit dates back to way back when. My T1 was a "50L6 transformer" as they used to be called. It was an audio frequency transformer that was designed to drive a 3.2Ω radio speaker's voice coil and to present a 2000Ω load impedance to the plate of a type 50L6GT vacuum tube. Such a transformer has a 25:1 turns ratio and a magnetization inductance of approximately 1.7 mHy on the voice coil winding.

The transformer's magnetization inductance, modeled here as L1, sets both the pole frequency and the zero frequency. If L1 varies, both the pole frequency and the zero frequency will vary too, but the two gain levels will not change.

I actually caught some flack for using T1 this way instead of making a resistor capacitor design of some kind. However, this gain stage was already a part of a switchmode power supply feedback arrangement that was intermittently oscillatory. Physical hardware was already in place and to do the RC approach would have meant having to redo the circuit board or having to make and install a piggyback circuit board, both messy processes under the circumstances.

Also, these power supplies were special purpose items and only two of them, the two units sitting there on my workbench, were ever going to be delivered. I needed something I could install easily.

It occurred to me that I could accomplish the loop stabilization by just cutting two circuit foils of the existing board and inserting the transformer. T1 was then physically mounted on a nearby chassis wall and that was that. It was actually the least intrusive way to fix the problem.

It is instructive to idealize the above circuit a little bit and to consider the following:

Note how the 100:1 transformer takes the high frequency gain to -40 dB.

The following is the algebraic derivation of the transfer function:

The following GWBASIC code uses the above algebraic result, but includes provision for a finite value of R3:

10 CLS:SCREEN 9:COLOR 15,1:YSTART=170:XSTART=40:PI=3.14159265#

20 PRINT "save "+CHR$(34)+"polezero.bas"+CHR$(34):PRINT

30 PRINT "save "+CHR$(34)+"a:\polezero.bas"+CHR$(34):PRINT:PRINT

40 C$=" ##### Hz ###.### dB"

50 F=10:FOR HDB=-40 TO 10 STEP 5:GOSUB 160:XHOLD=X:X=X+2

60 HH=ABS(HDB):IF ABS(HH-10*INT(HH/10))<.01 THEN X=X+3

70 GOSUB 170:X=XHOLD:GOSUB 170:NEXT HDB:KK=0:HDB=0

80 FOR FK=1 TO 6:FOR FF=1 TO 10:F=FF*10^FK:GOSUB 160:YHOLD=Y:Y=Y+4

90 IF ABS((FF-1)*(FF-10))<.001 THEN Y=Y+4

100 GOSUB 170:Y=YHOLD:GOSUB 170:NEXT FF:NEXT FK:KK=0:GOTO 200

110 XHOLD=X:YHOLD=Y:X=XHOLD+9:GOSUB 170:X=XHOLD-9:GOSUB 170:X=XHOLD:GOSUB 170

120 Y=YHOLD+9:GOSUB 170:Y=YHOLD-9:GOSUB 170:Y=YHOLD:GOSUB 170:RETURN

130 W=2*PI*F:NR=R1+R2:NI=W*L*(N^2*(R1+R2)/R3+1):NUM=SQR(NR^2+NI^2)

140 DR=R1:DI=W*L*(N^2*R1/R3+N):DEN=SQR(DR^2+DI^2):H=NUM/DEN

150 HDB=20*LOG(H)/LOG(10):RETURN

160 Y=HDB*3:X=20*LOG(F)

170 CC=XSTART+1.33*X:DD=(320-Y-YSTART):IF KK<>0 THEN LINE (AA,BB)-(CC,DD)

180 AA=CC:BB=DD:KK=1:RETURN

190 REM

200 READ L,N,R1,R2,R3:DATA .001,100,1000,1000,1e12:REM R3 is negligibly large.

210 REM

220 FOR FK=1 TO 6:FOR FF=1 TO 10:F=FF*10^FK:GOSUB 130:GOSUB 160:NEXT FF:NEXT FK

230 F=32230:GOSUB 130:KK=0:GOSUB 160:PRINT USING C$;F,HDB:KK=0:GOSUB 110

Note in line 200 that the value of R3 is set negligibly large at 10^{12} ohms. You can play with this a bit by lowering R3 and you will then see a diminished lessening of gain from below the pole frequency to above the zero frequency. This is why I let R3 be open.

For the 100:1 transformer case with L1 = 1 mHy, we see the following:

Note too that the mid-frequency result for 32.23 kHz agrees with the SPICE simulation result above.

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