Ideal transformer notes and tutorials
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
V_{m1}((t)- -V_{m2}((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
N_{1}I_{1}(t)-N_{2}I_{2}(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 N_{2}> N_{1}, the secondary voltage is larger than the primary voltage! Very interesting.
. If N_{2} < N_{1}, called a step-up transformer,
. If N_{2 } N_{1}, called a step-down transformer.
2. From (3), the secondary current is
We can surmise from (8) that for a step-up transformer, I_{2} (t)< I_{1 }(t).Therefore, while the voltage increases by
N_{2 }/ N_{1}, the current has decreased by N_{1} / N_{2 }. Because of this property, the power input to the primary equals the power output from the secondary
Therefore, the input power P_{1}(t)equals the output power P_{2 }(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 R_{1,eff} at theprimary terminals (the ratio V_{1}/I_{1}) 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 “-” V_{s} 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 R_{L} for fixed R_{s} and R_{L}
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 R_{p,eff} ®¥ as R_{L} ®¥, which is an open circuit. In practical transformers, it’s not uncommon for ( ) 1 I_{1}(t) » some small fraction of rated I for an open load.) For maximum power transfer R_{p,eff}= R_{s} . Consequently
The “turns ratio” N_{1}/N_{2} is adjusted to this value for maximum power transfer from the source to R_{L}, even when R_{L} ¹ R_{s}