Ideal Transformer
In general, a transformer is a two-port AC device that converts
time varying voltages and currents from one amplitude to
another between its two ports. This also has the effect of
transforming impedance levels. This device only performs this
transformation for time varying signals. Here, we will consider the transformer circuit shown below:
We will analyze this physical transformer as a time varying
magnetic circuit (ignoring flux leakage):
From this magnetic circuit we find
Vm1((t)- -Vm2((t) o y m((t)
Where
.
Substituting these into (1) gives
In an ideal transformer:
. the core permeability m is linear,
. m ® ¥, and
. the windings are perfect conductors.
From the second of these ideal transformer assumptions, the
RHS of (2) vanishes leaving
N1I1(t)-N2I2(t)= 0
Or
Furthermore, by Faraday’s law we know that
For the transformer geometry above
(because a transformer does not “transform” at DC)
Equations (3) and (6) are the basic equations of an ideal
transformer. Discussion
1. From (6), the voltage from the so-called “secondary” of the
transformer is
Note that if N2> N1, the secondary voltage is larger than the
primary voltage! Very interesting.
. If N2 < N1, called a step-up transformer,
. If N2 N1, called a step-down transformer.
2. From (3), the secondary current is
We can surmise from (8) that for a step-up transformer,
I2 (t)< I1 (t).Therefore, while the voltage increases by
N2 / N1, the current has decreased by N1 / N2 .
Because of this property, the power input to the primary
equals the power output from the secondary
Therefore, the input power P1(t)equals the output power P2 (t )
3. With a resistance RL connected to the secondary, then
Substituting for V2 and I2 from
Or
In other words, the effective input resistance R1,eff at theprimary terminals (the ratio V1/I1) is
The transformer “transforms” the load resistance from the
secondary to the primary. (Remember that this is only true for
time-varying signals.) For sinusoidal steady state and load impedance ZL, equation (10) becomes
4. For maximum power transfer, we design a circuit so that the
load is matched to the output resistance. We can use
transformers as “matching networks.”
5. Notice that the primary has the source connection so that the
ground occurs at the “-” Vs terminal. However, the secondary is not grounded. This secondary is said to be “balanced.” (Exception to this is the autotransformer.)
6. Remember that only time varying signals are “transformed”
by a transformer.
Example N10.1: Design the transformer shown below so that
maximum power is delivered to the load RL for fixed Rs and RL
This transformer “transforms” the load resistance to the primary
according to (10). An equivalent circuit at the primary terminal
can be constructed using this effective primary resistance:
From
(As an aside, note that Rp,eff ®¥ as RL ®¥, which is an open
circuit. In practical transformers, it’s not uncommon for ( ) 1 I1(t) » some small fraction of rated I for an open load.)
For maximum power transfer Rp,eff= Rs . Consequently
The “turns ratio” N1/N2 is adjusted to this value for maximum power transfer from the source to RL, even when RL ¹ Rs .
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