User:Yonderling/Mutual Impedance

Mutual Impedance
Mutual impedance is the impedance associated to two coupled circuits, via magnetic fields. Impedance is the extension of real properties of circuits to represent resistance, inductance, and capacitance into complex numbers that can easily be synthesized. All non-idealized materials have an associated impedance and thus an associated resistance, capacitance, and inductance that is innate to the material. Mutual impedance occurs naturally in straight wires, but also can be found in transformers where mutual impedance is utilized. In a speaker with a resistance 8 Ohms, if a transformer, with 10:1 ratio of windings, is connected in circuit before the speaker is connected to the voltage or current source, the mutual impedance will give the speaker an apparent 800 Ohms resistance to the AC source (Figure 1). [1]

Mutual Impedance of Transformer
$$N= \ winding \ number \ of \ coil $$

$$I=current $$

$$V=voltage $$

$$Z_F=impedance \ of \ secondary \ coil \ after \ coupled \ to \ primary $$

$$Z_L=principle \ impedance \ of \ secondary \ coil $$

Current Relation for Transformer:
$$N_1 I_1=N_2 I_2 $$ solving for $$I_1=(N_2 I_2)/N_1 $$

Voltage Relation for Transformer:
$$N_1 V_2=N_2 V_1 $$ solving $$V_1=(N_1 V_2)/N_2 $$

Applying Ohm's Law for AC Circuits: V = IZ yields
$$Z_F=V_1/I_1 = Z_L (N_1/N_2 )^2 $$