Module 4. Superposition theorem for circuits. Transformers
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 transformer – Perfect transformer – Ideal transformer
2 Circuit Theory / Fundamental theorems / Principle of superposition Principle of superposition
• Any linear circuit 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: – Voltage 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 voltages 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 jt = N ( e jt + e jt )= jN ( + )e jt 1 1 dt 1,C 2,C 1 1,C 2,C jt d jt jt jt V2e = N2 (2,Ce + 1,Ce )= jN2 (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 inductor
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 = jL1 I1 − I2 a V V 1 2 V2 = jL2 (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 =V11− is called the isolation voltage a
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