R. W. Erickson. Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder

Transcription

1 R. W. Erickson Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder

2 Graphical construction of transfer functions 8.3. Graphical construction of impedances and transfer functions Series impedances: addition of asymptotes Series resonant circuit example Parallel impedances: inverse addition of asymptotes Parallel resonant circuit example Voltage divider transfer functions: division of asymptotes 8.4. Graphical construction of converter transfer functions 2

3 R 10 Z(s) C 1 F 80 db 10 k 60 db 1 k 40 db db 10 0 db 1-20 db 1 Hz 10 Hz 100 Hz 1 khz 10 khz 100 khz 1 MHz 0.1

4 R 1 k Z(s) L 1 mh C 0.1 F 80 db 10 k 60 db 1 k 40 db db 10 0 db 1-20 db 1 Hz 10 Hz 100 Hz 1 khz 10 khz 100 khz 1 MHz 0.1

5 R 10 Z(s) L 1 mh C 0.1 F 80 db 10 k 60 db 1 k 40 db db 10 0 db 1-20 db 1 Hz 10 Hz 100 Hz 1 khz 10 khz 100 khz 1 MHz 0.1

6 Z(s) R L C 10 1 mh 0.1 F 80 db 10 k 60 db 1 k 40 db db 10 0 db 1-20 db 1 Hz 10 Hz 100 Hz 1 khz 10 khz 100 khz 1 MHz 0.1

7 Z(s) R L C 1 k 1 mh 0.1 F 80 db 10 k 60 db 1 k 40 db db 10 0 db 1-20 db 1 Hz 10 Hz 100 Hz 1 khz 10 khz 100 khz 1 MHz 0.1

8 Z(s) R C 1 16 µf 80 db 10 k R 2 10 C µf L 2 16 mh 60 db 1 k 40 db db 10 0 db 1-20 db Hz 10 Hz 100 Hz 1 khz 10 khz 100 khz 1 MHz

9 Impedance graph paper 80dB 100pF 10k 60dB 10H 1nF 1k 40dB 20dB 0dB 20dB 40dB 1H 100mH 10mH 1mH 100µH 10µH 1F 1µH 100mF 100nH 10mF m 10m 60dB 1m 10Hz 100Hz 1kHz 10kHz 100kHz 1MHz 10nH 1mF 10nF 100nF 1µF 10µF 100µF 1nH 88

10 Voltage divider transfer function Division of asymptotes Voltage divider H(s) Output impedance Input impedance L L Z in v 1 (s) C v 2 (s) R L C R Z out Z out Z in C R Z 1 Z 2 Z 1 Z 2 Z 1 Z 2 Two ways to construct the transfer function H(s): v 2 (s) v 1 (s) = Z 2 Z 1 Z 2 = Z 2 Z in v 2 (s) v 1 (s) = Z 2Z 1 Z 1 Z 2 1 Z 1 = Z out Z 1 91

11 Voltage divider H(s) v1 (s) Z in L C v 2 (s) R Z out 80 db 10 k 60 db 1 k Z 1 Z 2 40 db db 10 0 db 1-20 db Hz 10 Hz 100 Hz 1 khz 10 khz 100 khz 1 MHz

12 Construction of transfer function v 2 (s) v 1 (s) = Z 2Z 1 Z 1 Z 2 1 Z 1 = Z out Z 1 R 1 C Z 1 = L Q = R /R 0 f 0 R 0 Z out L L =1 Q = R /R 0 H = Z out Z 1 f 0 1/ C L = 1 2 LC 92

13 Transfer functions predicted by canonical model H e (s) e(s) d(s) 1 : M(D) L e v g (s) j(s) d(s) v e (s) Z in C v(s) R Z out Z 1 Z 2 89

14 Output impedance Z out : set sources to zero L e C R Z out Z 1 Z 2 Z out = Z 1 Z 2 90

15 Graphical construction of output impedance 1 C Z 1 = L e R Q = R / R 0 f 0 R 0 Z out 91

16 Graphical construction of filter effective transfer function L e L e =1 Q = R / R 0 f 0 1/ C L e = 1 2 L e C H e = Z out Z 1 92

17 Boost and buck-boost converters: L e = L / D 2 1 C R Q = R / R 0 f 0 R 0 increasing D L D' 2 Z out 93

18 8.4. Measurement of ac transfer functions and impedances Injection source Network Analyzer Measured inputs Data v z magnitude v z frequency v y v x 17.3 db Data bus to computer v z output v x input v y input v y v x

19 Swept sinusoidal measurements Injection source produces sinusoid frequency of controllable amplitude and Signal inputs and perform function of narrowband tracking voltmeter: Component of input at injection source frequency is measured Narrowband function is essential: switching harmonics and other noise components are removed Network analyzer measures v x v y v z v y v x and v y v x 95

20 Measurement of an ac transfer function DC blocking capacitor Injection source v z magnitude v z output v z frequency Network Analyzer Measured inputs v x v y input input v y v x v y v x Data 4.7 db Data bus to computer v y (s) v x (s) = G(s) Potentiometer establishes correct quiescent operating point Injection sinusoid coupled to device input via dc blocking capacitor DC bias adjust V CC input G(s) Device under test output Actual device input and output voltages are measured as v x and v y Dynamics of blocking capacitor are irrelevant 96

21 Measurement of an output impedance Z(s)= v(s) i(s) DC bias adjust V CC input Device under test G(s) output i out current Z out probe v z Z s DC blocking capacitor R source Z out (s)= v y(s) i out (s) amplifier ac input =0 voltage probe v y v x 97

22 Measurement of output impedance Treat output impedance as transfer function from output current to output voltage: Z(s)= v(s) i(s) Z out (s)= v y(s) i out (s) amplifier ac input =0 Potentiometer at device input port establishes correct quiescent operating point Current probe produces voltage proportional to current; this voltage is connected to network analyzer channel v x Network analyzer result must be multiplied by appropriate factor, to account for scale factors of current and voltage probes 98

23 Measurement of small impedances Grounding problems cause measurement to fail: Impedance under test i out injection source return connection R source Network Analyzer Injection source Injection current can return to analyzer via two paths. Injection current which returns via voltage probe ground induces voltage drop in voltage probe, corrupting the measurement. Network analyzer measures Z(s) Z (1 k) Z probe = Z Z probe Z rz voltage probe voltage probe return connection For an accurate measurement, require (1 k) i out ki out Z probe iout Zrz (1 k) i out Z probe v z Measured inputs v x v y Z >> Z probe Z rz 99

24 Improved measurement: add isolation transformer Injection current must now return entirely through transformer. No additional voltage is induced in voltage probe ground connection Impedance under test Z(s) voltage probe voltage probe return connection i out 0 i out injection source return connection Z probe Z rz 0V 1 : n R source Network Analyzer Injection source v z Measured inputs v x v y 100