The Mesh-Current MethodThe Analysis of a Linear Transformer CircuitThe Ideal TransformerThe Mesh-Current Method We can use the mesh-current method in frequency domain circuits as using the same methods in section 4.5-4.7. Example: +- V1 V3 1Ω 12Ω -j16Ω 39Ix 150∠0° V2 Ix I1 I2 j2Ω 1Ω j3Ω For mesh 1: ()150I)1612(I)162(12121=−−−++ jj ()150I)1612(I)14(1321=−−− jj (1) For mesh 2: ()0I39I)1612(I)163(12112=+−−−++xjj ()0I39I)1313(I161221=+−+−−xjj and 21III −=x()0)II(39I)1313(I16122121=−+−+−− jj ()0I)1326(I162721=+−− jj (2) Using Equation (1) and (2), to find I1 and I2 Aj5226I1−−= Aj5824I2−−= Ajx62I +−= The voltages at nodes Vjj 10478I)21(V11−=+= Vjjx10472I)1612(V2+=−= Vjj 130150I)31(V23−=+= Vjx23478I93+−= 4-19.10 The Transformer Two Topics • The Sinusoidal steady-state behavior of the linear transformer • Ideal Transformer The Analysis of a Linear Transformer Circuit R2jwMR1V Zs ZLjωL1 jωL2 I1 I1 a c b d R1 : The resistance of the primary winding R2 : The resistance of the secondary winding L1 : The self-inductance of the primary winding L2 : The self-inductance of the secondary winding M: The mutual inductance Vs : The sinusoidal voltage source Zs : The internal impedance of the source Vs ZL : The internal impedance of the load connected to the secondary winding Ii : The primary current I2: The secondary current Lets write two mesh-current equation 2111II)(V MjLjRZssωω−++= 2221I)(I0 LjZRMjLωω+++−= Let say that 1111LjRZZsω++= 2222LjZRZLω++= Therefore we can find sMZZZVI222211221ω+= 1222222112IVIZMjMZZMjsωωω=+= The impedance between node a and b 4-2s222211s22222211s1ZZZ- −+=−+==ZMZZMZZIVZsabωω or 222211LjZRMLjRZLabωωω++++= The third term in Zab is called reflected impedance 2222LjZRMZLrωω++= We can see that if M becomes zero then Zr becomes zero. The Ideal Transformer Determining the Voltage and Current Ratios The primary coil is wound so that it has N1 turns. The secondary coil has N2 turns. The turn ratio is 12NNN = The relationship between primary and secondary current in ideal transformer is is 122VVNN= and 1212IINN= idealNN12 2211VVNN=, 2211II NN−= idealNN12 2211VVNN−=, 2211II NN= - - ++I1 I2 V1 V2 - - ++ V2 V1 I1 I2 4-3idealNN12 2211VVNN=, 2211II NN =idealNN12 2211VVNN−=, 2211II NN−= + 4-4 I1 I2 - - I1 I2 VV1 V2 V1 2 +++-