Module 4. Superposition theorem for circuits.

Circuit Theory, GIT 2018-2019

Philip Siegmann Departamento de Teoría de la Señal y Comunicaciones

1 Circuit Theory / Fundamental theorems / Index Fundamental theorems

• Superposition theorem – Principle of superposition – Multiplication by a constant • Transformers – Real – Perfect transformer – Ideal transformer

2 Circuit Theory / Fundamental theorems / Principle of superposition Principle of superposition

• Any can be solved by considering each independent source acting by it’s own: – Solve the circuit for each independent source – The solution of the global circuit is then the superposition of de solutions obtained by each source

Example 1:

 iX e1(t) = E1 sin(1t +1),

i1(t) = I1 sin(1t +2 ), e1(t) e2(t) i1(t) e2 (t) = E2 sin(2t +2 ) iX

3 Circuit Theory / Fundamental theorems / Principle of superposition Principle of superposition

• One circuit for each source of the same temporal behavior (e.g. the same frequencies) • Independent sources of different temporal behavior are annulled: – sources replaced by s.c. – Current sources replaced by o.c. • (Dependent sources are not annulled)

4 Circuit Theory / Fundamental theorems / Principle of superposition Principle of superposition

• Example 1: Two different kind of sources (1, 2)

Circuit with sources of frequency =1

 iX e1(t) = E1 sin(ω1t +1), i (t) = I sin(ω t + ), s.c. 1 1 1 2 e (t) 1 e (t) = 0 i1(t) 2 iX

Circuit with sources of frequency =2

 iX e1(t) = 0, i (t) = 0, s.c. e (t) 1 o.c. 2 e2 (t) = E2 sin(ω2t +2 ) iX 5 Circuit Theory / Fundamental theorems / Principle of superposition Principle of superposition • Each circuit is solved independently for each temporal behavior (e.g. same frequency) • For the global circuit: – The solution for the currents and are added in the time domain: i(t) = i(t) + i(t) +... This means obtained 1 2 in the circuit with v(t) = v(t) + v(t) +... only sources of 1 2 frequency 2 – Te power delivered or dissipated by a dependen source is the addition of the power calculated in each circuit with a specific frequency: P = P + P +... dependen source dependen source 1 dependen source 2

– The power of an independent source of frequency 1 is calculated only for the circuit with sources of the same frequency: P = P independen source of frequency 1 independen source 1 6 Circuit Theory / Fundamental theorems / Principle of superposition Principle of superposition

• Example 1: Two different kind of sources (1, 2)

Circuit with sources of frequency =1 i (t) = I sin( t + ) i (t) R R 1 1 1 1 R 1  iX 1 2 P = I R Superposition: R 1 2 R 1 e (t) s.c. 1 P iR (t) = I1 sin(1t +1) + ix i1(t) 1  iX  I2 sin(2t +2 ) Pe1 = Pe1  1 P = P + P  R R  R  Circuit with sources of frequency =  1 2 2 P = P + P  ix ix ix i (t) = I sin( t + ) 1 2 i (t) R R  2 2 2 2  R 2  iX  1 2 PR = I R R 2 2  2 It can be used to s.c. e2(t) P separately analyze o.c. ix 2 the DC and AC i X P = P 7 e2 e2  2 behavior of a circuit Circuit Theory / Fundamental theorems / Multiplication by a constant Multiplication by a constant

• In a linear circuit, if all the independent sources with the same temporal behavior are multiplied by a constant, k, then the current (or voltage) along any wire (or between any two points) in the circuit is multiplied by the same constant.

Example 1. Circuit with sources of frequency =1 are multiply by k

You only need to multiply by k the previous result i (t) R R 1  iX

i (t) = k I sin( t + ) + I sin( t + ) s.c. R 1 1 1 2 2 2 k·e1(t) k·i2(t) (The previous calculated powers vary accordingly) iX

8 Circuit Theory / Transformers

Transformers

9 Circuit Theory / Transformers / Real transformers Real transformer

• The transformers transfer electrical energy from one circuit to another one trough magnetically coupled coils

i i  R1, R2  0 1 2   K 1  v (t) N1, N2 are finite 1 v2(t) N a = 1 is the turn ratio. N2 Primary winding Secondary winding   d  d  v1(t) =  R1 + L1 i1(t) + M12 i2 (t)   dt  dt  d  d  v (t) = M i (t) +  R + L i (t)  2 12 1 2 2 2  dt  dt  10 Circuit Theory / Transformers / Perfect transformers Perfect transformer

• The coupling coefficient is one and the number of turns are finite

i1 i2

 R1, R2 = 0  v (t) v2(t) 1  K =1→ S  0  N1, N2 are finite

 d v1(t) = N1 (1,C + 2,C ) v (t) N L dt  1 = 1 = 1 = a.  d L v (t) = N ( +  ) v2 (t) N2 2  2 2 dt 2,C 1,C

11 Circuit Theory / Transformers / Ideal transformer Ideal transformer

• The coupling coefficient is one and the number of turns are infinite (but not the turn ratio)

i1 i2

R1, R2 = 0  d v (t) = N ( +  ) v (t)   1 1 1,C 2,C v1(t) 2  K =1 dt  d N , N →  v (t) = N ( +  )  1 2  2 2 dt 2,C 1,C

• For Sinusoidal Steady State: 0   d V e jt = N ( e jt +  e jt )= jN ( +  )e jt  1 1 dt 1,C 2,C 1 1,C 2,C  jt d jt jt jt V2e = N2 (2,Ce + 1,Ce )= jN2 (2,C + 1,C )e  dt 12 Circuit Theory / Transformers / Ideal transformer Ideal transformer

• The magnetic flux within the core becomes cero

I1 I2 C = 1C + 2C → 0

N1 N2  I1 + I2 = 0 V1 V 2 C C

N1  I2 = − I1 N2

The mutual induced current in the secondary winding generates a magnetic flux that opposes and cancels the magnetic flux generated by the primary winding  The intensity exiting the dot at the secondary

winding (I2) is a-times the intensity entering trough the dot of the primary winding (I1). 13 Circuit Theory / Transformers / Perfect to ideal transformer Perfect to Ideal transformer

• Perfect transformer has an equivalent ideal transformer in parallel with an

I1 I2 푉1 = 푗푤퐿1 퐼1 − 푗푤푀12 퐼2 푉2 = −푗푤퐿2 퐼2 + 푗푤푀12 퐿1 1 V1 V2 푎 = , 푀12 = 퐿1퐿2 푀12 = 퐿1 = 퐿2푎 퐿2 푎

I -1 I 1 a I2 2

V1 V2 I 1 aI1 I2   1  V1 = jL1 I1 − I2     a  V V  1 2  V2 = jL2 (aI1 − I2 )

14 Circuit Theory / Transformers / Ideal transformer Operating with ideal transformers

• We can “move” elements form one side of the ideal transformer to the other side as follows

IA I1  I = aI  2 1 V  1 1 V2 V = V  2 a 1

I2 I which is equivalent to: A

V Note: If the dot’s are 1 V2 crossed the direction of

the electrical sources I2 change

15 Circuit Theory / Transformers /Autotransformer Autotransformer

• It is a perfect transformer with only one winding. Portions of the same winding act as both the primary and secondary winding (k=1)

I1 I2 I I 1 2 I1 I2

V1 L1 V1 V2 V1 V2 L2 V2

 1 VAC =V1 −V2 =V11−  is called the isolation voltage  a 

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