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Electrical Engineering

Two-port networks, mutual inductance and the ideal transformer

Z, Y, hybrid and transmission parameters, reciprocity, coupled inductors with proof that k ≤ 1, the ideal transformer and impedance transformation.

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Many components and subcircuits have an input port and an output port: an amplifier, a filter, a transmission line, a transformer. To connect them you do not need to know what is inside: the relations between voltages and currents at the two ports are enough. This is the two-port model, which in this lesson we also apply to coupled inductors and to the ideal transformer.

The two-port

A two-port has two pairs of terminals, the ports. At each port the current entering one terminal leaves through the other. There are four quantities: v_1, i_1 at the input port and v_2, i_2 at the output port, with both currents taken as entering.

Two-port with voltages and currents at both ports, and ideal transformer
On the left a generic two-port: voltage and entering current at each port. On the right the ideal transformer, with dots marking the corresponding terminals and the voltage ratio equal to the turns ratio.

If the internal circuit is linear and contains no sources, two of the four quantities are linear functions of the other two. The possible choices give the different representations.

Parameter matrices

Impedance parameters (currents as inputs):

    \[ \begin{pmatrix} \bar V_1 \\ \bar V_2 \end{pmatrix} = \begin{pmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{pmatrix}\begin{pmatrix} \bar I_1 \\ \bar I_2 \end{pmatrix} \]

Each parameter is measured with one port open: z_{11} = \bar V_1/\bar I_1 with \bar I_2 = 0, z_{21} = \bar V_2/\bar I_1 with \bar I_2 = 0, and so on.

Example. A T network with R_a in the input branch, R_b in the output branch and R_c to ground: with the output open, z_{11} = R_a + R_c and z_{21} = R_c; by symmetry z_{22} = R_b + R_c and z_{12} = R_c.

The other representations:

  • admittance Y = Z^{-1} (voltages as inputs, short-circuit measurements);
  • hybrid h: \bar V_1 and \bar I_2 as functions of \bar I_1 and \bar V_2. These are the ones in bipolar transistor datasheets: h_{fe} is the current gain;
  • transmission (ABCD): input as a function of output. Their advantage is that for two two-ports in cascade the overall matrix is the product of the two matrices.

Reciprocity. In a circuit made only of resistors, capacitors and inductors (coupled ones included) z_{12} = z_{21} holds: the effect of a current at port 1 on the voltage at port 2 equals the reverse one. In the T example you can see it directly. Circuits with controlled sources, like transistor models, are generally not reciprocal.

Coupled inductors

Two nearby coils share flux: the current in one induces a voltage in the other. The equations are

    \[ v_1 = L_1\frac{di_1}{dt} + M\frac{di_2}{dt},\qquad v_2 = M\frac{di_1}{dt} + L_2\frac{di_2}{dt} \]

where M is the mutual inductance. Its sign depends on the winding direction, and schematics mark it with a dot: if both currents enter the dotted terminal, the fluxes add and M appears with a plus sign. In sinusoidal steady state it is a two-port with z_{11} = j\omega L_1, z_{22} = j\omega L_2, z_{12} = z_{21} = j\omega M.

Theorem. M^2 \le L_1 L_2, that is the coupling coefficient k = M/\sqrt{L_1L_2} is at most 1.

Proof. The energy stored in the magnetic field is

    \[ W = \tfrac12 L_1 i_1^2 + M i_1 i_2 + \tfrac12 L_2 i_2^2 \]

and it must be non-negative for any pair of currents, because a passive component cannot give back more energy than it received. With i_2 \ne 0 and x = i_1/i_2, W/i_2^2 = \tfrac12 L_1 x^2 + Mx + \tfrac12 L_2 is a quadratic in x with positive leading coefficient: it is non-negative for every x only if the discriminant is not positive, M^2 - L_1L_2 \le 0. ∎

k = 1 means all the flux of one coil passes through the other; power transformers with iron cores reach 0.99, while for two distant air coils k is close to zero.

The ideal transformer

It is the limit of two coils with perfect coupling (k = 1), no losses and very large inductances. Wound on the same core, with n_1 and n_2 turns, they are linked by the same flux \Phi, and by Faraday’s law v_1 = n_1\,d\Phi/dt, v_2 = n_2\,d\Phi/dt. Hence

    \[ \frac{v_2}{v_1} = \frac{n_2}{n_1},\qquad \frac{i_2}{i_1} = -\frac{n_1}{n_2} \]

The second relation follows because the ideal transformer neither absorbs nor stores energy: v_1 i_1 + v_2 i_2 = 0. The minus sign says that, with entering currents, when current enters the primary it leaves the secondary.

Impedance transformation. With a load \bar Z_L on the secondary, the impedance seen from the primary is

    \[ \bar Z_{in} = \frac{\bar V_1}{\bar I_1} = \left(\frac{n_1}{n_2}\right)^2 \bar Z_L \]

because the voltage is divided by n_2/n_1 and the current multiplied by the same factor. It is the classic way to match a load to a source: an 8 Ω loudspeaker, through a transformer with a 10:1 turns ratio, looks like 800 Ω to the amplifier.

Real transformers. A real transformer is modelled by adding to the ideal one a resistance for the windings, an inductance for the leakage flux and a branch for the magnetising current and the iron losses. It works only with AC: with a DC voltage the flux does not change and nothing is induced in the secondary. That is why the power grid is AC: transformers step the voltage up for transmission (at 380 kV the currents, and so the RI^2 losses, are small) and back down for use.

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