Lecture 14: Transformers. Ideal Transformers

Transcription

1 White, EE 3 Lecture 14 Page 1 of 9 Lecture 14: Tranforer. deal Tranforer n general, a tranforer i a ultiort ac device that convert voltage, current and iedance fro one value to another. Thi device only erfor thi tranforation for tie varying ignal. Here, we will conider the tranforer circuit hown below: (t) ψ (t) V (t) V (t) "Priary" μ r large ψ "Secondary" The tie varying current ( ) a agnetic flux denity B ( ) the econdary coil roduce B ( ) denity i the u B() t B ( t) B ( t) t in the riary circuit roduce t in and around coil. Siilarly, t. The total agnetic flux =. We will aue that the tranforer core ha a very large relative ereability μ r. Conequently, B( t ) will alot excluively be contained within the core. Thi B() t for cloed loo within the core. (We can think of thi a a elfhielded core.) 006 Keith W. White

2 White, EE 3 Lecture 14 Page of 9 The agnetic flux over a cro ection of the core: ψ () t = B() t d = B( t) B( t) d (1) = ψ, () t ψ, () t With B() t contained excluively within the core, ψ () t will be the ae throughout the tranforer (though it will vary with tie). ψ i defined ily a the integral of B( t ) The agnetic flux will be roortional to the nuber of coil turn, the geoetry of the coil and the current in the coil: ψ t = A t [Wb/turn] (6.5),() l () ( ), j j l j A i the inductance contant [H/turn ] of the core and j =,. Thi A l i rovided by the anufacturer of the core that you ue for your tranforer (and inductor). Table D. in your text lit A l for variou core ued in the orcal 40A.

3 White, EE 3 Lecture 14 Page 3 of 9 ote that A l can be a very trong function of frequency. nduced Voltage A we know, a tie varying agnetic field through a coil of wire roduce a voltage between the end of the coil. Thi iraculou henoenon wa dicovered by Michael Faraday and i atheatically tated in Faraday law a dψ ef = [V] (3) dt where i the nuber of (identical) turn of the coil. Thi ef i a net uh around a circuit that caue electron to ove. Voltage and ef are cloely related concet. We can deterine the induced voltage V( t ) uing the following equivalent circuit: V ef Alying thi equivalent circuit to each ide of the tranforer hown on the firt age give:

4 White, EE 3 Lecture 14 Page 4 of 9 Priary B d < 0 Secondary B d < 0 V ef V ef The olarity of the lued ef ource i et by the direction of the current: a voltage ource ha current entering the negative terinal. The ign of the ef ource i due to the direction of d (by the RHR) and the aued direction for ψ (and hence B ). Fro thee circuit and alying (3), the inuoidal teady tate voltage at the riary and econdary are both of the for V = jω ψ (6.6),(4) where V and ψ are now haor. Secifically, the riary and econdary voltage are V = jω ψ (6.9),(5) V = jω ψ (6.10),(6) Dividing thee two equation give V = = n (7) V where n i called the turn ratio. nteretingly, we ee here that the outut voltage V can be different in alitude than the inut voltage V V = V (6.11),(8)

5 White, EE 3 Lecture 14 Page 5 of 9 ote that if >, the econdary voltage i larger in alitude than the riary voltage. Very intereting. f >, called a teu tranforer, f <, called a tedown tranforer. Priary and Secondary Current ext we will conider the electrical current in the riary and econdary of the tranforer. Fro (1), the agnetic flux i the u of the two agnetic fluxe fro each coil ψ = ψ, ψ, (9) Uing () and noting that ψ, will be negative ince the direction of the current i aued OUT of the econdary, then ψ = A l A l (6.1),(10) Solving for we find that ψ = (6.13),(11) Al The agnetic flux i not a circuit quantity. To derive an equivalent circuit for the tranforer we need to exre ψ in ter of electrical circuit quantitie. To accolih thi, we ue (5) in the firt ter of (11) yielding ψ V = (1) Al jω Al L

6 White, EE 3 Lecture 14 Page 6 of 9 where L i the inductance of the riary coil. Subtituting thi reult back into (11) give V = [A] (6.14),(13) jω L agnetization tranforer Thi lat reult i extreely illuinating. We ee that the current in the riary i the u of two art: (1) agnetization current and () tranforer current. (1) Magnetization Current. The firt ter in (13) doe not involve the econdary in any way. n other word, thi i the current the tranforer would draw regardle of the turn ratio of the tranforer. () Tranforer Current. The econd ter in (13) directly deend on the econdary becaue of the ter. Thi coonent of the riary current i a tranfored econdary current, in a anner iilar to the voltage in (7), though inverely. deal Tranforer f the agnetization current V/( jω L) in (13) i very all in agnitude relative to the tranforer current ( / ) then uch a device i called an ideal tranforer. The equation for an ideal tranforer are fro (7) and (13):

7 White, EE 3 Lecture 14 Page 7 of 9 and V V = (6.15),(14) = (6.16),(15) The circuit ybol for an ideal tranforer i V V : Dicuion 1. We can urie fro (15) that for an ideal teu tranforer <. Therefore, while fro (14) the voltage increae by /, the current ha decreaed by /. n the orcal 40A, the tranforer T1 i ued to te u the current fro the Driver Alifier to the Power Alifier. For T1, = 14 and = 4 o that = ( ) = 7/. Becaue of thi current behavior, the ower inut to the riary equal the ower outut fro the econdary: P t = V t t (16) () () ()

8 White, EE 3 Lecture 14 Page 8 of 9 P t V t t V t t V t t () = () () = () () = () () (14) Therefore, the inut ower P ( ) P ( t ), a would be exected. (15) (17) t equal the outut ower. With an iedance Z connected to the econdary, then V = Z Subtituting for V and in thi equation uing (14) and (15) ( ) V = Z ( ) or V = n other word, the effective inut iedance Z,eff at the riary terinal (the ratio V / ) i Z,eff = Z [Ω] (6.19),(18) The ideal tranforer tranfor the load iedance fro the econdary to the riary. (Reeber that thi i only true for inuoidal teady tate ignal.) Z

9 White, EE 3 Lecture 14 Page 9 of 9 3. For axiu ower tranfer, we deign a circuit o that the load i atched to the outut reitance. We can ue tranforer a atching network. For exale, in the orcal 40A, T3 i ued to tranfor the outut iedance fro the RF Mixer (3 kω) to atch the inut iedance of the F Filter (00 Ω). Uing (18): 6 Z = Z = 3000 = 04. Ω 3 which i very cloe to the deired 00 Ω.