Practical Transformer

Transcription

1 Practical Transformer φ c i P tructure and dot convention ymbol and polarity Dot convention: the primary and secondary currents flowing into the winding terminals marked produce a mutually additive magnetic flux. Lenz`s law: an electromagnetic induction occurs in such a way that the magnetic flux produced as the outcome of the magnetic induction opposes the magnetic flux that initiated the induction process. 35

2 Example of Transformer Waveforms i P sint 1

3 Modeling of Practical Transformer ip i φ c l m N P N φc : magnetic flux inside the core lm : mean magnetic path length λ P = φ : primary flux linkage ù c Perfect coupling λ = Nφc : secondary flux linkageú û 1) Faraday`s law dλ P() t dφ c() t vp() t = = dt dt dλ () t dφ c() t v() t = = N dt dt vp() t N = v () t = vp() t v() t N 36

4 Modeling of Practical Transformer ip i H c l m N P N Hc : magnetic field intensity inside the core lm : mean magnetic path length 2) Ampere`s law ip() t N i() t = Hc() t lm N () () H () c t l i m P t = i t N i () P t = i() t im() t with im() t = Hc() t lm Magnetizing current i m() t : the current required to create H c() t that couples the primary and secondary windings through magnetic induction. 37

5 Modeling of Practical Transformer ip i H c l m N P N 3) Magnetizing inductance Magnetizing current through primary winding () m () Hc t l i t = m N p Magnetic flux linkage at primary winding λp() t = φc() t = μμh r o c() t Magnetizing inductance Lm associated with im() t and λp() t λ () () () () P t μμh r o c t λp t = Lmim t Lm = = = μμ r o N 2 i () () P m t Hc t lm lm N p 38

6 Circuit Model of Practical Transformer 4) Circuit model ip i m n 1:n i L m Voltage equation Turnsratio () t = n() t Current equation N n = ip() t = ni() t im() t λ () P r o c() c() m with () P t N μμh t H t l im t = = = Lm N μμ p r o N 2 l P m Hc() t lm ip() t = ni() t N p 39

7 ummary of Transformer Modeling Circuit model ip N P H c N i ip L m i m n 1:n i l m Lm = μμ r o N2 P n = lm Voltage equation N v () () with t = nvp t n = Current equation 1 ip() t = ni() t im() t with im = vp() t t L ò d m When μr is assumed infinite, Lm becomes infinitelylarge and the circuit model reduces to the ideal transformer. N 40

8 Example of Transformer Circuit i P N 2V 5ms 10ms Transformer parameters μr = = 1 cm2 lm = 4π 101cm = 10 N = 20 Circuit parameters 2V 5ms 05A. 10ms N 20 n = = = 2 10 Lm μμ o r N P π 4 = 2 = = 5mH lm 4π A. 41

9 Example of Transformer Circuit i P i m 2i 1:2 vp L m= 5mH Circuit waveforms 2i i m i P 2V 5ms 10ms 1A 1A 2A 2V 1A 1A 2A 4V 5ms 05A. 10ms 4V 05A. v D i P m = Lm 2 = = 2 A ip = 2i im 42

10 olenoid Drive Circuit olenoid: an inductor fabricated by winding copper coil around an iron core Conceptual ( faulty) solenoid drive circuit witching Circuits in Practice i L v L L i L v L Closed Open 0 olenoid drive circuit must be designed so that the remnant energy is safely removed from the solenoid inductance. 43

11 Dissipative olenoid Drive Circuit i L Closed DT s Open witching Circuits in Practice Closed Closed Open Energy buildup Energy removal Energy buildup T s 44

12 Dissipative olenoid Drive Circuit Closed Closed Open V R il R R L il L il L i D V V witching Circuits in Practice i L i D DT s T s 45

13 Example of Dissipative olenoid Drive Circuit witching Circuits in Practice V i Lpeak = DTs L 90 = = 5A E 1 m L æ i ö 2 1 = ç Lpeak = = J 2 çè ø 2 90V i D 20 W i L The stored energy v L E m 180 mh D = ms i t il ()(A) t vl( t )(V) i( t)(a) D ()(A) is dissipated in the resistor. 46