TE bio 2016-L2 - cvut.czamber.feld.cvut.cz/17bbtel/files/TE_bio_2016-L2.pdf · $0 # 2' AB 0,5H 1k...

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Transcript of TE bio 2016-L2 - cvut.czamber.feld.cvut.cz/17bbtel/files/TE_bio_2016-L2.pdf · $0 # 2' AB 0,5H 1k...

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������� ����� ������ ����������� �

������ �� �������� �����

����� ���� � �� ��� ����

���� ����

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�� � ������ � ���� ��!�� �"#�$%

�� &'� $�� #!(�)��

*� +!(�)�� #� ����#��)�� �" ,�

-� +!(�)�� # ),�#� �" ,�

.� /"#�$� $�0���� �'$0 1 ��2�����

�� +�$���

3� � ���������� ��� # ��

4� � ���������� ,�0,

5� 6� ��'��� �"#�$� 7 $#�8(9 � :$ �$;

��� 6� ��'��� �"#�$� 7 $#�8"��� :<=� >/�?�=;

��� � ��� 2�� # ��� � � $�#��� "�$0 :� ! , ��' ;

��� �(��)� 7 �"#�$� ,� ,(��)

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������� ������� �� � ����

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���������

@ 7 ��2 ,��� 7 u = Ri

� 7 �(� ��� 7 Q = Cu

� 7 �$0���� 7 Φ = Li

0 7 2$��8 �(A�� 7 (��0$ i = (�$ � �$"A�0 : "�#� �!;

7 2$��8 (��0$0 7 �(A�� u = 0�)��� 2'�AB� :���� ���2���;

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����� ������ ������ � �����

�$�' �� 2$��8 �(A�� 0$�B08� � ,#!�� ,#���'�� �(A�� $�� ��$1

����0 u(t) #� #� ����� �� (�� #C����� ��B�� (��0$� �$#'$A��

:) $������ 2#'$A��; 2 :$�; 8��� ,#����� +� ) � u(t) (��$,�1

#08� (��$( , (�� �(A�� 2$��8�� �� 8� �� $� ��$����� �� �� $�

),�#!�� 2�A��

�$�' �� 2$��8 (��0$0 $� �"#�$0 2#'$� (��0$ (��$�(,�� ��$1

���� i(t)� �� 2 #C��� ��� ��,��� ��$� "�2 �� �$0 � #� ��,�

(� � �0 �(A�� � 8��� ,#���'���

i(t)u(t)

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�$��"�'��#! :��2'# , !; 2$��8 �(A�� (��0$0 �0,� "!� (�(,'�

),�#�0 �0���� u(t) i(t)� ��� ,�0$ 0� � ���� ��!�� �"#�$%

(�0B 8��� ��8),�A8 ���� 2$��8�D

• &$��8 ���,������� �(A�� :,�� ��'��� 0,�' ��! ,�# 7 ���;

u(t) = U = ���,��

• &$��8 ,����#��� �(A�� :(�����$�! $A8;

u(t) =

{0, t < 0U, t ≥ 0

• &$��8 ����� ����� �(A�� :����� ��! 0,�' ��! ,�# 7 E��;

u(t) = Um sin(ωt + ϕ),

�$� Um 8� �( �0$ , �0,�#��� , ��' 0 #� #� ����� ω = 2πf

8� ��0��#' ����#���� # �$ '���� 2 ,��0�$0 F�$�,−1G� f 8�

����#���� # ����2��� FE2G ϕ 8� (�)'��)�� �'2� , �0,�#���

(�%"A�0�

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@�2 ,��� :,�0)',�� # 2��2��� ,� "AB�A �2�)08� 8�� �$(��� ��B

���� 2�� ,(�'#�� 7 �$(�� 8� 8�8� # ,���,�� ��� ��' ��$����

��B� ,(�'#�A8C� � �' � (�0B�#�!� 8� �'2�# H�$(�����I D1;� @�1

2 ,��� 8� � ����� ��,���#)�! :# (�( ,0 2'# , �,� ��2 (��0$��

�(A��� ��#�,�0(08� ), 7 (��0$ ���08� � �(A�� ���B �A

��(�;� +2�� ��2 �(A��� (��0$�� (�( ,08� /��%# 2'���

u = Ri, i =u

R, R =

u

i,

�$� R 8� �$(�� # ������ FΩG�

u 8� �(A�� #� #� ���� F+G

i 8� (��0$ # �(����� FJG�

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K(� ��� :# 2��2��� 8� �� 2�#'� ��8),�A8 ���$��2'�����; 8�

� ����� ,���(�! �0�0 �#� �'"�8� 6'"�8 �0,� "!� $� �(� 1

���0 $�$'� (�����8���� (��0$��� ��B (��"��' # 0�) ��� ),��

$� ��$� � � ����� ,���#)�! #� #2��0 ��2 �(A��� (��01

$�� "0$� #�,�0(�#� ),� +2�� ��2 �'"�8�� �(A��� 0�)08�

��#� ��

q = Cu, u =q

C, C =

q

u,

�$� ��#A #�,�0(08�

q �'"�8 # ��0 ��"��� F�G

C �(� � #� ��$��� F?G�

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�#�$ 8,��� B� �'"�8 $�(�#� $� �(� ���0 (��0$ i� ���0$ "0$�

(��0$ i = I ���,����� �(� ��� � (�)'��0 #�" �� �'"�8�

"0$� (� "!#� ��'��A , ),�� :�'"�8 # ��0 ��"���� (��0$ #

�(����� ), # ,��0�$'��;

q(t) = I.t, ��B� u(t) =I.t

C.

K$�B ,� "0$� (��0$ # ),� �A� �� 8� �0��� (�� �'"�8 �����1

$A�! 2 ), T (�0B�� 2'( ,0 (����� �����' 0 :(��$(�� '$'���

B� # ),� t0 8� �'"�8 ��#�� q(t0);

q(T ) =∫ T

t0i(t)dt + q(t0), ��B� uC(T ) =

1C

∫ T

t0i(t)dt + uC(t0).

L�� #�� (��$���2� ��#� �� $��'B���� B� $� �(� ���0 #���'

(��0$ 8�� ���$�� �$� ,� � 8��� ,#���'�� �A�� �(A��D

i(t) = Cdu(t)dt

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��$0���� :# 2��2��� 8� �� 2�#'� ��8),�A8 ��#��0; 8� � �����

#��#'��8��� ����� ��! ���� #� ������ 8� 0 �B�� ����� � #��#�1

���' � ���� ��!� (��0$��� ����! #�$ )�� (����'2�� @�#�AB #2� �

����� ����� ���0 8� $A8 ,���#)�!� +2�� ��2 ����� ��!�

����� (��0$�� (�( ,08� ��#� ��

Φ = L.i, i =ΦL

, L =Φi,

�$� ��#A #�,�0(08�

Φ ����� ��! ��� #� M�"����� FN"G

L �$0�)��,� # ����� FEG�

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�����B� 8� #2� � �$0�)���� ���0 ,���#)�!� 8�#�� :���%B� ,�

�A� � ,������ (�$�"�A 8�� �'"�8 # �(� ���0;� �%B��� � �1

$� 8�! (��0$ (���)� �$0������ # ),� T � (��0$ 8� � 8���

,#���'�� 0�) �� :(���A���; �(A��

i(T ) =1L

∫ T

t0u(t)dt + i(t0).

���0$ � ,#���� �$0����0� ����!� � (�)'��0 ����)� B'$�!

(��0$� (� (�8��� 2$��8 ���,������� �(A�� U � (� "0$� (��0$

$� �$0����0 ��'��A , ),�� ��%,��� ���0$ �(A�� � 2$��8

(�� �,�� # ),� T � �0 0� "0$� $' �"#�$�� (����'2�� (��0$D

i(t) =U

Lt, i(T ) =

U.T

L.

+ ,0(�#�$ #�� �"#�$0 "0$� ,�0��)�A (��0$ ��# � (����'2��

�"#�$�� :8�� 8� ���0 $0' �A # (��($A ��# � 2��������� �1

(A�� � ,#���'�� $�' ���� �(� ���0;� + ��' �� ��#�� #C� (��1

��'2�8��� (��0$ ��2(�! � �����$A��0 ����� � �$(��0 # �0���

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6 ������� # �0�� ��#�� ���� B'$�� �(A��� (��0$ ,� # 8�8�� ��� �

���A�� ����� ��� (� �� J#C� (��0$ ,� ����� ��! ��� �A���

#2� �' � # �0�� � ���� ��� �(A�� :�(�� # �B$� � ����'��A;�

=�B� 2� (����� $�� #�� (��$���2��� #2��0 �(,�

u(t) =dΦ(t)

dt.

�,� B� 8� ����� ��! ��� 0�)�� (����'2�8���� (��0$��� (� 2�

2� 2�A� (��0$0 0�) � �(A�� � ,#���'�� �$0����0

u(t) = Ldi(t)dt

.

��

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/"#�$ �@ ,� ,(��)��

+ �"#�$0 , �$0������ �%B��� #��#�� � , �0� � �$� ,� �

,#���'�� �A����!�� ,�0)',��� �%B� �"8�# � �(A�� #�CC�� ��B �'

����!�� 2$��8 �(A��� =� #�0B�#'�� # �2#� ,(���!�� 2$��8���

���0 '������ �('8����� �(A���

�� �� ( ���#!�� �"#�$% 8� 2 �B�� � ,�0��)��,� � B� �$0����

����$� ����� # ����� ���� (� � ����� 8� #��#�����

(����'2�8���� (��0$��� ����� �' (����'2�8��� (��0$ ,���#)��

���#'��� ���0$ ,(��) ,����� 2�A�� �$(�� # �"#�$0�

(�����$�! $A8 "0$� #���'2�� 2 (�)'��)�� (�$������ $��

(��0$�� (��$ (��(�0���� � �� ��$�� 8�1 (��0$ (��$ ,�(�0���

0�)�� ���C�� �$(���� ��B 8� �$(�� (� (�8��! (� (��(�0���

#2� ��� � ,#���'�� (� (�8����� ��2 ,���0 ,��� �(A�� , #A�C��

�(A���� ��B �' 2$��8� ����! $� �$0����0 (��0$ 2#�$ ���

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�(��'�� �$0�� #�� 2'�AB�

0,5H1k�

10V

0 ms 10 ms 20 ms 30 ms 40 ms 50 ms

0 ms 10 ms 20 ms 30 ms 40 ms 50 ms

50 ms

150 mA

100 mA

0 mA

50 mA

0 ms 10 ms 20 ms 30 ms 40 ms

-225V

-150V

-75V

0

75V

150V

2�

sepnuto vypnuto

proud induktoru

napìtí na induktoru

��

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+ (��$���2��� #2�2��� 8,�� (��$(�� '$ � B� � ,#���'��

0#�$��!�� � �����% (�2��08��� �(A�� (��0$� 7 �"#�$�#�

#� ) ��� ���0$ �(A�� ����0 ���� (�'� � �'�� � $ ,(�2 �

� ���� ���0 ����� � + 0�) ��� ���B �0 ( ��� B� #!���

� ���� ����� (��0$0 8� $'� ,�0) ���

P (t) = u(t) · i(t), �$� #!��� 8� #� M����� FNG.

+!��� # ),� (��$,�#08� (�'�

W (T ) =∫ T

0P (t)dt =

∫ T

0u(t)i(t)dt,

�$� (�'�� :����� �; 8� # M��,��0�$'�� FN,G� ����� 8,�0

��# # ����� 8�0 %� FG :$�� ( ���� 2 � �M����$ �� F�N�G

��N�O *�����6 N,;�

��

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• @�2 ,��� # �"#�$0 ��2(�� 08� � ���� ���0 ����� 7 (���AP08�

8 � ��( � (����� B� 8� ��,���#)�!� (�#���� 0 ������� 8�

,�' � ,�0) � �(A�� (��0$0 � $�!

WR =∫ T

0u(t)i(t)�� =

∫ T

0

u(t)2

R

�� =∫ T

0Ri(t)2��

• K(� ��� 8� (�#�� �0�0 08��� ����� � �$�' �� �(� ��� ����1

� ����2(�� 08� 7 8� (�#��� "�2�2��'��#!��

WC =12C.U2

• ��$0���� 8� � �� ��� "�2�2��'��#!� (�#��� ����� � �1

����$A�' # ����� ���� ���0 8�

WL =12L.I2,

��

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&$��8 ����� ����� �(A�� # ),� (�( ,08� 28�$��$0C��A #2��

u(t) = Um sin(ωt),

�2�)08� ,� 8�� ,���$#� �(A��� ��� 8��� (�( , 8� $�Q��#'�

#� ) � �2!#�' ����� #�� ��$��� �(A�� Uef � 6(A�� Uef 8�

$�Q��#'�� 8�� Q�� #�� ,��8��,�A��� �(A��� ����� "� #

��2 ,���0 ��2(�! � ,��8��0 ����� 8�� ��2(�� 08� �(A��

, �0,�#��

U2ef =12π

∫ 2π0[Um sin(ωt)]2dt

Uef =1√2Um → Uef ≈ 0, 707Um

��

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� ��'��� ��2��� :�� #��; � ������ 7 $#�8"���

• 2$��8 (��0$0 ��2��! (��0$�� 7 (������� 8� "�2��2�A��'

���,���� (��0$�#! :2�, �#��; ) � �� :�2�)�� �(�� α�

• 2$��8 �(A�� ��2��! �(A��� 7 (������� 8� "�2��2�A��'

���,���� �(AR�#! (����, :2�,� ���; :�2�)�� �(�� A;

• �(A��� ��2��! 2$��8 (��0$0 7 (������� 8� ���,���$0�1

���� , ��2�A��� #�$ #�,� # , ����,��� :�2�)�� �(�� G;

• (��0$�� ��2��! 2$��8 �(A�� 7 (������� 8� ���,��2 ,�����

����' �' ��2�A� �$(��0 # ������ :�2�)�� �(�� Z;

�1 ��$ �� #� ) ��0 (��0$� (��$,�#08� #,�0(�� ,#���� 2���� #�

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�"#�$� � 8���B ,#���'�� ,� �A�� �(A���

��

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������ ��� 2�)�� ��2��!�� 2$��8%

i1

i1

u1

u1

i2 = α.i1

u2 = Z.i1 u2 = A.u1

i2 = G.u1

��

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�$�' �� �(��)�� 2�, �#) :�/&;

�/& 8� $#�8"�'��#! � ����� 7 ��2��! 2$��8 �(A��� 8���B #!,�0(��

"�'��0 8� $�' �� 2$��8 �(A��� +,�0(�� "�'��0 8,�0 ��2(�8���

,#���� :��#���' $� � �� B'$�! (��0$;� +!,�0(�� �(A�� 8� ��2���

��� "� � #,�0(���� ,#���'�� "� � �0 �#� �(A��� � 2��8���

B� ���#�0 (�$����0 2� ,( � � 8���� ��� B� �/& #!,�0(���

�(A��� �%B� ,(� 0 ,� #,�0(��� , ��' �� �# #P�#� ,#� #,�0(��

�(A�� 7 �"#�$ �0,� ��� 2#�$��0 2(A���0 #2"0�

0ZVu1

u2

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���� $ 0,(��'$'�� �"#�$0 , �/&

0u1u2

R1

R2

Rz

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/� ����� �"#�$�#!�� #� ) �

6(A�� (��0$ �0,��� $��� Q��#� ��8�� ��$���� #�

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/� ����� �"�0 �"#�$�#!�� #� ) � 8� #A�� $���$�D

• /� ���� �(A�� ��2 $#A� ,#���� �"#�$0 #� ��� #�1

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0 ����� C (� 2)��' 28 ,���� ��2 �' �A � $��0 (� � �0 �(A��

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�� K ������%# 2'���

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���B �0 �0 �#!�

R1

C

R2u(t)

i1(t) i2(t)

i3(t)i4(t)

N∑n=1

in(t) = 0.

��

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��#C ��A�� , � B� #�2�)��! (��0$ #C��� #A�#� ,�A�08� $�

02 0� >'1 "!� ,�0)�� �0 �#! : 8 ,�A ��8,�0 #C����� (��0$�

$��� ��� ��#�� �0 �;� (� �A����� ��$���� (��0$0 "0$�0

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,��8��,�A���� �"#�$0 8� ,�A� �$ � $�� � 2'(���� ,#�������

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�� K ������%# 2'���

��0)�� �(A�� (�$� "�#� �� ,��)�� # �"#�$0 8� # �B$��

���B �0 �0 �#!�

i

R1C

R2

u4(t)

u1(t)

u2(t)

u3(t)

n∑1

un(t) = 0.

��

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��� �#� ,(�8��� ��2 ,���%

R

i(t)i(t)

R1 R2

u(t)u(t)

u1(t) u2(t) ux(t)

⇐⇒

E �$'�� ��$���0 �$(��0 R ��2 ,���0� ����! 8� ��# # �����

,�� �#� ���" �� R1 R2� ��$� ���#��� ��2 ,���0� ����! #

�"#�$0 ,� 2$��8�� u(t) �,�#� (��0$ i(t)� ����B�! , ���� ����!

(����'2� �"#�$�� , ��2 ,���� R1 R2� & �� K �������# 2'���

�%B��� �$#�$ �

��

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u(t) = u1(t) + u2(t) = R1i(t) +R2i(t) = (R1 +R2)i(t)

u(t) = ux(t) = Ri(t) = (R1 +R2)i(t) −→ R = R1 +R2.

�1 2�2��� N ��2 ,���% # ,�� � 8� ��B�� 8� ���$ � 8�$���

��2 ,����� , �$(����R =

N∑n=1

Rn.

��

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�1 #!, �$�! �$(�� ,�0)��� �$(��% 8�$��� #!�� ��2 ,���%

(�����'1 #C�� ,��8�! (��0$� (� �%B��� 28 ,� �� 8�� 8,�0

��$���� �(A�� � �B$�� 2 � ���

i(t) =u(t)

R1 +R2u1(t) = R1

u(t)R1 +R2

u2(t) = R2u(t)

R1 +R2.

/"���A � n1��� ��2 ,���0 "0$� �(A��

i(t) =u(t)∑Nn=1Rn

un(t) = Rnu(t)∑Nn=1Rn

(n = 1 . . . N).

��

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�� � �� ,(�8��� ��2 ,���%

E �$'�� 8�$�� ��# # ����� ��2 ,��� :8��� ��$���0 R;� ����!

�$#�$� (��0$ 2� 2$��8� �(A�� ,��8�!� 8�� 2� 2$��8� �$#'$�

�$ (� � �A ,(�8��!�� ��2 ,���%�

R

i(t)

R1 R2u(t)u(t)

i1(t) i2(t) ix(t)

⇐⇒

i(t) = i1(t) + i2(t) =u(t)R1+

u(t)R2

ix(t) =u(t)R= i(t)→ 1

R=1

R1+1

R2→ R =

R1.R2R1 +R2

.

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���#�'����0 ��$���0 �$(��0 �2�)08��� 8�� #�$ #�,�

G = 1/R 0$'#'�� 8 # 8�$����'�� , ����, F�G :� O Ω−1;� �1

2�2��� N ��2 ,���% (� � �A� 8� ��B�� 8� ���$ � 8�$���

��2 ,����� , �$(���� R ��"� #�$ #�,�� G

1R=

N∑n=1

1Rn

G =N∑

n=1Gn.

�� 8,�0 ��$���� (��0$%� ����� �B$!� 2 ��2 ,���% (����'2�8�

28 ,���� (� �('8��� �(AR�#!� 2$��8��� ���0$ #C� 2�'��

8�� �� ��#! (��0$ i(t) #,�0(08��� $� 02 0� 2 ������� 8,�0 $�

,(� �)�� ,#���� 2(�8��� �%2�� ��2 ,����� (� 8� ��2$A ���

(��0$0 ��2 8�$��� #� ��2 ,���� $'��D

in(t) =i(t)Gn∑Nn=1Gn

(n = 1 . . . N).

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�� � �� ,(�8��� �(� ���%

CC1 C2UU

⇐⇒

6(A�� U � 0�)08� �'"�8� ����! 8� 0 �B�� # �B$�� �(� ���0�

+ �(� ���0 C1 8� 0 �B�� �'"�8 Q1 = C1U # �(� ���0 C2 8�

0 �B�� �'"�8 Q2 = C2U � + �(� ���0 C 8� (� ���B� �(A�� �'"�8

Q = CU � & ��# # ����� "0$��� �" �"#�$� (�#B�#�� (��0$

2$��8 0 �B� $� �"�0 �"#�$% �!B �'"�8� =�$�

Q = Q1 +Q2 −→ C = C1 + C2.

�1 2�2��� N �(� ���% (� � �A� 8� ��B�� 8� ���$ � 8�$1

��� �(� ����� , �(� ��0

C =N∑

n=1Cn.

��

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��� �#� ,(�8��� �(� ���%

II

u(T )

u1(T ) u2(T )

C

C1 C2

II

⇐⇒

6���R (� $�"0 T (����'2� 2� 2$��8� (��0$0 ���,����� (��0$ I�

+ ),� T ����R � �,�� (��0$ � �0 ��

���0$ I 0 �B� 2 ), T # �(� ���0 C1 �'"�8 q1(T ) = I.T =C1u1(T )�

=!B (��0$ 0� '$ �'"�8 ��� $� �(� ���0 C2� ��B� q2(T ) =I.T = C2u2(T )

# ��# # ������ �"#�$0 q(T ) = I.T = C.u(T )�

��

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� ��

u(T ) = u1(T ) + u2(T )→q(T )C=

q(T )C1+

q(T )C2

→ 1C=1

C1+1

C2

��B�

C =C1.C2

C1 + C2

(�� N �(� ���%

1C=

N∑n=1

1Cn

.

& 0#�$��� S#�� 8 B 2� �$#�$ �� 8� ,� � ,�� �#� ���" ��

�(� ���% ��2$A 08� �� ��#� �(A���

��� �#' ���" ��� �(� ���% �' #B$���� ���C� �� ��#�0 �(1

� �0� ��B �' ��8���C� 2� 2(�8��!�� �(� ���%�

��

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��� �#� (� � �� ,(�8��� �$0����% , ��2'# , !�

����� ��!� ����

��� ,�� �#�0 ���" �� N �$0����% 8� ��B�� �8�� 8�$��

��# # ����� �$0���� , �$0�)��,��

L =N∑

n=1Ln.

��� (� � �� ���" �� N �$0����% 8� ��B�� �8�� 8�$��

�$0���� , �$0�)��,��1L=

N∑n=1

1Ln

.

��

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��� �#� (� � �� ,(�8��� 2$��8%

&$��8� �(A�� 2� �$ � $� ,�� � #!, �$�� �(A�� 8� ,�0)���

�(A�� 8�$��� #!�� 2$��8%

u(t) =N∑

n=1un(t).

�� � �� ,(�8��� �� 2� � �$� (�0B��

&$��8� (��0$0 2� �$ � (� � �A 7 ���#� ,(�8��� $�$' $�

�"#�$0 (��0$ $�! ,�0)��� (��0$% 8�$��� #!�� 2$��8%

i(t) =N∑

n=1in(t).

��� �#� ,(�8��� �� 2� � �$� (�0B��

��

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LA ) �(A�� 7 ��' �! 2$��8 → ,(����" )

u0 uz

R0

Rz

,(����" )2$��8

��

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� � � �� ,�� �#� ,(�8��� ��2 ,���% # 8�$ �� ,��)�� ,� 2$��8��

�(A��� �� ��#! �$(�� # �"#�$0 8� R = R0 +Rz� &$��8 �(A��

$�$'#' $� �"#�$0 (��0$ i = u/R� ��$,���� #C� 8�� B�

uz = u0Rz

R0 +Rz.

6(A�� uz #2� � � ��2$A ���� �(A�� 2$��8� � $#A )',� � �

�(A�� � ��2 ,���0 R0 � Rz 7 #��#�� 8,�� $A ) �(A��� ,�

����!� ,� ,���'�� # ��,)����� ���B,�#� ��$� % ��' �!��

2��2����

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��' �! 2$��8 �(A�� 8� # ��88�$��$0CC�� (��($A �0���

��$� �#� $�' ��� 2$��8�� �(A�� ��2 ,����� ��(��2���08����

8��� #� ���� �$(��� <��� � �%B� ��� �(�� �(A�� �� + #� ����

�$(�� ��� �

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L�#'��1 ,� � 8��0�� $#�8 � ,#����� ����' �' (�, �0B � 8��

2$��8 �(A�� :�('8����� ,��8��,�A����� ��"� ,���$#���� )

�(0 ,����;� #B$� 2 �� �0,��� # $A� �"#�$ # ��88�$��$0CC��

(��($A ���$� �#�! �(A��� u0 �$(���� R0�

K ���0�� ��$� 0 ,� #'B� �', �$08��� (�8��D

6(A�� �(�'2$�� 8� �(A��� ����� � ,#���'�� ��' ���� 2$��8�

2�A����� �$�B ���� (� (�8�� ,(����" )� Rz → ∞, i → 0�

unaprazdno = u0

���0$ ���'��� 8� (��0$� ����! "����� ��A� � �$�"�����

,#���� 2$��8� 2����# :����$� 2� 8�� � ((���;� =��$�

Rz → 0inakratko =

u0R0

&8��#� (�0B �� �"�0 S$8% #�$� � #�8'$����

R0 =unaprazdno

inakratko,

��

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�#�$��! #2�� (�� #!(�)�� R0 2� #�0B�� � #!(�)�0 #� , �B ���

�"#�$0 ��"� � (��� ����0 �A����� (��0$ 2����#'�� ��#�$�

� $�,��0�� �A���� �(�'2$�� 2� $�,�0(�!� (��,���8 (��#�,��

L C� #!2���' S#� ,(�)�#' # ��$������ $%, �$�%� ����� �'

2�AB�#'�� 2$��8� �%2�!� 2'�AB�� :(��0$ �'� , �0�� $'#'

#� #� "A 2'�AB� #� ��,�;

�� ��B$08��� �� ��8#A�C� �(A�� :�(AR�#! ��2�� �;� �(�� �

#!,�0(0 (���0 (�)��)�� 6�8#A�C� ��B�� �(A�� 8� u0 (� �0 �1

#�� (��0$0 2'�AB� :�$�B 8� �$(�8��; ��"� �' #� � #� �!

�$(��� � ��,��0��� (��0$�� $� 2'�AB� :, � �,8���� 2�AB�1

#��� �$(����; #C� �(A�� ��' ���� 2$��8� � �,'�

�� ��B$08��� #� �! (��0$� �(�� (�� ��2,#����� �$ �)�� ��L�

+ ��' ��� 2$��8 #C� , ��,��0��� (��0$�� :� �,8���� �$(�1

��� 2'�AB�; � �,' �(A��� >T �' �� (��0$ 8� (��0$ ���'����

(� �0 �#�� �(A�� � 2'�AB �

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*� ��B$08��� �T �' �� #!��� �$�#2$�! $� ,(����" )� (��

$�� �(A�� U #� ������ 2$��8� $�! #� ���� �$(�� 2$��8�

R0� ��� #!��� � 2'�AB # 2'# , �,� � #� ��,� 2�AB�#����

�$(��0 Rz ( ��D

Pz = uz.i = URz

Rz +R0.

U

Rz +R0=

U2Rz

(Rz +R0)2=

Pzi

(Rz +R0)2,

�$� Pzi 8� $�' �� #!���� ����! "� 2$��8 U $�$ (� �0 �#��

#� ����� �$(��0 2$��8��

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+!��� # 2'�AB # 2'# , �,� � 8�8�� �$(��0

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

Pz/P

zi

log(Rz/R0)

6�8#A�C� #!��� �$�#2$' ��' �! 2$��8 ���$�� �$� ,� 2�AB�#��

�$(�� ��#�' 8��� #� �����0 �$(��0� $� � ��������

����������� � =��� �T �0� (��$,�#08� )�#�� �0 #!���0�

����! "� $� ��B� 2'�AB� $�$ 2$��8 , �0 �#!� #� �����

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���� � �� ���� ! ������ " # �������� �����!

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+ ,���,� ��' ���� 2$��8� 2� $��� Q��#� 2 �(A�� �(�'2$��

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R02R01 +R02

,

R′0 =

R02u0R01 +R02

u0R01

=R01R02

R01 +R02.

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i1 = g(u1)R2 = 20Ω

R2 = 20Ω

u1

u1

u2

u2

u[V]

UU

i[A]

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�� �� ( ,0(��(�2 ��

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i = 5mA

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R1 = 3000Ω

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R2 +R1 ‖ R3+ U2

R1 ‖ R2R3 +R1 ‖ R2

u = −5 + 10/3 + 20/3 = 5 V

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+!(�)�� (�� �� �#���08��� 2�, �#)

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�(AR�#! , �$�#)

u2 = u1

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/"���! $#�8"�� 7 ,����� ��' 2�)� #� ) �� � ,#���'��

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u1 = Z1,1 i1 + Z1,2 i2

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u1 = A1,1 u2 − A1,2 i2

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& �"�0 ��#� � (�( ,08����� $#�8"�� ( ���DZ1,1 =

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, když i1 = 0

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, když i1 = 0

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KB$! ��, �0,�#! (�� �$ ��! , ��' 2� ��2 �B � � �$0 , ��' %

, ����#����� � ����� 8,�0 $'�� �� �)�,� �!� �',�"�� 2'� $��

:��8� BC�; ����#���� (�� �$ ����� , ��' 0 7 ?�0� ���# �$� ���

�"���! ��(�� �$ ��! , ��' 7 ?�0� ���# ���,�������

��

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K(� ��� ,� , �0,�#!� 2$��8�� �(A��

C

u(t) = Um sin(ωt)

i(t)

<0$��� ���� (��$(�� '$�� B� u(t) = Um sin(ωt). �� , #�0B ���

$�� #�� :i(t) = Cdudt ; $�,����� #!�2 (�� (��0$ 7

i(t) = UmωC cos(ωt) = UmωC sin(ωt + π/2).

��

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����#)�� # ,���,� �(� ���0 #�$�0 � ���0� B� #

����� ���� 0,�' ���� ,�#0 8,�0 (��0$ (����'2�8���

�(� ����� �(A�� � 8��� ,#���'��� �'2�#A (�,0�0�� � π/2�

=� 2',$��� 2(%,�"�� 0�)08� # # �(� ���0 � �"#�$�#�

#� ) �� # 8����� �"#�$0�

+2� �' ��'2�� 8� ���#! # # � �'2 �"#�$�#!�� #� ) �

����� ��� (�(,�� >���� � $��'2 #,��0( � $� $�0����

��2�A�0 #��#������ �"��0 ���( �T���� )�,� �

+��������' ���� ���� ��!���,� ' - ��� ��. ��/��

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��� ����� ��! 0,�' ��! ,�# 8,�0 2#�$��� �'2��� 7 ���( �T��

)�, ��(��2���08��� �"#�$�#� #� ) �� , �0,�#��� (�%"A�0�

��� , �0,�#� �(A�� Um sin(ωt + ϕ) 2#�$��� # �"��0

���( �T���� )�,� ��(��2����� 7 �'2��

U = Re(U) + j Im(U) = Um (cosϕ + j sinϕ)

��$�"�A (�� (��0$ Im sin(ωt + ϕ) 2#�$��� # �"��0

���( �T���� )�,� ��(��2����� 7 �'2��

I = Re(I) + j Im(I) = Im (cosϕ + j sinϕ)

� �� ��$�

Um = |U|, ϕ = arctgImU

ReU

(�$�"�A (�� I

��

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��� �'2���#�0 � !20 �"#�$% 2#�$��� $ C� (�8�� 7

�(�$��

��� ��2 ,��� 7 ZR = R

��� �(� ��� 7 ZC =1

j ω C

��� �$0���� 7 ZL = j ω L

��� ��2��� 2$��8� ,� 8�$�' � (����,�#! ) � �� 7

X = Re(X) + j Im(X)

��

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J(�'� ���( �T�� � ���� �� �'� (�,����� (��,���$��� 8�

(�(,� �"#�$�#� #� ) �� , �'2� "�#� �A (����)���0 7 2��8�A

8�� ���( �T�� )�, , ��' ��0 �� �'��� , �B��0�

+ �" ,� #!(�)�% ����� ����� 0,�' ����� ,�#0 ,� (�� 2'( ,

�"#�$�#!�� ��#� � (�$ � K �������#!�� 2'���% (�0B�#'

�(�$��� 2$��8% ,��8�A 8�� # �"#�$��� , ��2 ,�����

J( �� �'2��% 2� $����,���#� � $��#� 0#�$���� �"#�$0 ,�

2$��8�� �(A�� �(� �����D

I =U

ZC= jωCU −→ Im = UmωC, ϕ = π/2

��

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�2� (�(,� �"#�$�#� #� ) �� , �'2� �%2�A (����)���0 7 2��8�A

8�� ���( �T�� )�, , ��' ��0 �� �'��� , �B��0� �� (��

���" ��� �%2�!�� �"#�$�#!�� (�#�%�

CR

Z

Z =R 1

jωC

R + 1jωC

=R

1 + jωRC=U

I

��

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+!(�)�� �"#�$% #B$� ,�A�08� � � �2��� # ,���,�� �"#�$�#!��

#� ) � :�(A�� # 02 ���� (��0$% #� ,��)�'��;� �$�B 8� �"#�$

"02�� �A8�!� 2$��8�� �(A�� ��"� (��0$0� ��0B ��

K �������#!�� 2'���% �� #�$� �� #2��0� ����! �' �"���A �#�

Y2 = HY1,

�$� Y2 8� � �$�! �'2�� �A����� �"#�$�#� #� ) ��� Y1 8� �'2��

2$��8� H 8� �0���� (�( ,08��� S) ��� 2$��8� � 2#� ���0

�"#�$�#�0 #� ) �0� :J� �� ��� 8,�� # ��,���#)�!��

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(��0$ 8,�� 2� "0$ �� (��0$ ��"� �(A��;�

=��#� #�8'$���� ( �� # (��,���0 �'2��%� 2� #C� ,�$�� 2�,��

(�( , ),�#��� (�%"A�0D

y1(t) = Y1m sin(ωt + ϕ1) y2(t) = Y2m sin(ωt + ϕ2),

��

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�$�

Y2m = Y1m|H| = Y1m

√(Re(H)

)2+(Im(H)

)2

ϕ2 = ϕ1 + ����

(Im(H)

Re(H)

).

�#A� �0)�!�� ,��"� % ,� ,���C�� 7 �'2���#!� 8� 8 �! ,#A� ��B

,#A� ),�#!�� (�%"A�% u(t) i(t) 8 � �$(�#�$8����� $�� #��

�����' %� >� #C� #���� B� ,� $' 8�$�� 2 $�0���� #�(�)����

� $� 8�$�� ��#� �� � �$� ���� �%2�� ,��"� � ���(�C�����

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������)�� @� �"#�$ #� ����#��)�� �" ,�

R

C

U1 U2

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�#�$ 8,��� B� , �(�$���� (��08��� 8�� , �$(���� =�B�

#�0B 8��� 2� �,�� � $A ) �(A�� �(�C���

U2 = U1ZC

R + ZC= U1

1/jωC

R + 1/jωC= U1

11 + jωRC

.

& 0#�$����� �"������ #2��0 �%B��� 0B�� 2'( ,0

H =1

1 + jωRC=1− jωRC

1 + (ωRC)2.

�����B� ,� 8�$�' � ����� ��� #�8'$���� (��A�0 �'2��0

#!,�0(���� �(A�� � �'2��0 #,�0(���� �(A��� ��$� � (�( , 8�

,� #,�0(�� �(A�� �"#�$�� �# #��� 8�1 (����,��� � #!,�0(�

�2!#' ,� H :�(AR�#!�; (����,�� �"#�$0� $� �

"�2��2�A���0 ���( �T�� �0��� �� ��)�0 ω� ��B "!#' �A�$�

#�8'$���� 2'( ,�� H ≡ H(jω)�

��

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�����,�#� # ,���,� �"#�$% #� ����#��)�� �" ,� (�( ,08���

(����� � !2� �0���� H� ��'B��� �� � 0#�$����

�����)��� �"#�$0�

��#�� #!2���' ������� ,� �!�' # #0 �"#�$0 � �( �0$0

#,�0(���� �(A��� �#�$ 8,��� B� U2m = U1m|H|� ��B�

U2m = U1m1√

1 + (ωRC)2.

+ $���� B� �( �0$ , �0,�#��� (�%"A�0 8� 2'# , ' � �� ��)�0

, ��,��0��� �� ��)��� "0$� � �,�� � 2#���� �0�� 2'# , �,�

2�"�2 � # ���0 �0���� |H|� #� ������ ��2'# , � (���A���0

:#�$���#�' �,; 8� ��� ��0, �� ��)�0 � ,# , � �,� 8� A� �$�

A = 20 ��|H|.

��� 8�8� ��$���� ,� 0#'$A8� H 8�$�����I $�� "� � F$<G� ��� ,�

�2�)08� 8�� <�$��� ������� ,� �

��

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6 �"�'2�0 8� <�$��� ������� ,� � �����)���� �"#�$0�

������� �"#�$0 "� � 2#� ��� ����D R = 100 k٠C =�� �?�

1 10 100 1K-20.0

-15.0

-10.0

-5.0

0.0

5.0

A���

log f ����

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��� �����)�� �"#�$ 28 ,����� B� #!,�0(�� �(A�� H,� �(�BV08�I

2 �(A��� #,�0(���� & #!�20 (�� (����, ( ��� (� ϕ1 = 0

ϕ2 = ����

(

��(H)

@�(H)

)= −���� (ωRC) .

4,0 6,0 8,0 10,0-1,0

-0,5

0,0

0,5

1,0

1,5

u1(t)

u2(t)

u ��

t �����

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?'2�#' ����#��)�� ������� ,� � �����)���� @� �"#�$0

-100

-75

-50

-25

0

25

1 10 100 1K

ϕ◦�

log f ����

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J�( �0$�#' ����#��)�� ������� ,� �D |H| = 1√1+(ωτ )2

• ω → 0� ��� ��2�� �� ��)�� :ωτ � 1) ,� (����, �"#�$0 " �B� �

8�$� )��� ��� � 2�A�� ���B �� ��$���� ,���$#��� �(A��

#�$�0 � ���0� B� ,� ���$��2'��� �"�8� #�"�8� � !� ��B 1

�!� ��$���� (��0$0 � ��2 ,���0 ,� #��#'�� � ! S"����

�(A��� 6(A�� � �(� ���0 H,���'I , �$�#� #,�0(�� �(A���

+ ���0 (����,�#� ������� ,� �� ,� 8�8� (�%"A� ,��(��� ���

" �B� � #�$���#�� (����� �$(�#�$8��� ��$���A � $<

• ω → ∞� ��� #�,��� �� ��)�� :ωτ � 1) � �,' (����, �"1

#�$0 , ��,��0��� �� ��)��� |H| ≈ 1/ωτ � + ���� �" ,� �� 1

��)�% 2� (�� (����,�#�0 ������� ,� �0 # $�� "� ��� �(,�

A = −20 ��(2πfτ )� ��B 2����'� B� �B$� 2#!C��� �� ��)�0 �

$�,�� �',�"�� #�$� � (�� �,0 (����,0 � �� $<� �1 # ���0

�� ��)�� 2�"�2�� ��� �� ���� " �B� ,� (�%"A� ������1

� ,� �� ,��(��� ��� � (����� ,� ,� ���� 1�� $< � $��'$0

�� ��)�0

��

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• 0#�$��� $#A ,��(���� ,� (�����0 � �,� �� ��)�0 # "�$A

ωτ = 1 ��,(� ω = 2πf =1τ.

=��0�� �� ��)�0 ,� ���' !��� �!���-�� ,�0��)�! (�%1

"A� ������� ,� �� ,� � �A� ��8#��� #2$ 08� �$ ,��(����

L�,2���� 28 ,����� B� � ��2��� �� ��)�0 8�

|H| = 1√2=

√22

≈ 0, 707, ��,(� A ≈ −3$<.

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?'2�#' ����#��)�� ������� ,� �D ϕ = −����(ωτ )

• ω → 0� ��� ��2�� �� ��)�� :ωτ � 1) ,� �'2� (����,0 �"#�$0

" �B� � �0 �� B 8,�� 0#�$ � B� �(A�� � �(� ���0 H,���'I

, �$�#� #,�0(�� �(A��� + ���0 (����,�#� ������� ,� �� ,�

(�%"A� �'2� ,��(��� ��� " �B� � #�$���#�� (����� , ��$��1

��0 ϕ = 0◦�

• ω → ∞� ��� #�,��� �� ��)�� :ωτ � 1; � �,' (����, �"1

#�$0 (����� B� H(�� !I �(� ��� ��,�)� ����#� � ���� �

2�A�� ���B �� ��$���� #,�0(���� �(A��� �����B� 8$� � ��1

��� ��! 0,�' ��! ,�#� 8� �(A�� � �(� ���0 , �0,�#�� �

(�� �,� , 8��0 �(� ��� $�#� 08� �A� � � ,#!�� ,#���'��

�(A�� 2(%,�"�� B� ,� �'2� 2(�BV08�� ��� ��,��0�� �� ��)��

,� ,��(��� ��� " �B� � −90◦ ��,(� −π/2�

• L�,2���� $� #!C� 0#�$����� #2��0 28 ,����� B� � ��2���

�� ��)�0 8� ϕ = −����(ωτ ) = −����(1) = −45◦ = −π/4.

��

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L�� #)�� @� �"#�$ #� ����#��)�� �" ,�

R

C

U1 U2

U2 = U1R

R + ZC= U1

R

R + 1/jωC

H =jωRC

1 + jωRC

|H| = ωRC√1 + (ωRC)2

ϕ = ����( 1

ωRC

)

��

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1 10 100 1K

-48

-36

-24

-12

0

0

1212

1 10 100 1K.

25

50

75

100

125

A���

log f ��

log f ��

ϕ◦�

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L�� #)�� �"#�$ #� ����#��)�� �" ,� �' �', �$08���

,��(��� ��� # ,���,� D

• ω → 0� ��A��� � ��2�!� �� ��)�%� :ωτ � 1) ",� 0��� ��$1

��� (����,0 � �,'� J,��(��� �' ,� �� W�� $< � $��'$0

�� ��)�0 �'2� (����,0 �"#�$0 ,� " �B� � +90◦� ��$� #!,�0(��

�(A�� H(��$"��'I �(A�� #,�0(���

• ω → ∞� ��� #�,��� �� ��)�� :ωτ � 1; ,� ",� 0��� ��$���

(����,0 �"#�$0 ,��(��� ��� " �B� � 8�$� )�� �'2� � �0 ��

• 6 ��2��� �� ��)�0 :τ = 1/ω; 8�

|H| = 1√2=

√22

≈ 0, 707, ��,(� A ≈ −3$<.

ϕ = ����(1/ωτ ) = ����(1) = 45◦ = π/4.

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/"#�$� @�

�����)�� �"#�$ $�� #)�� �"#�$

R

R L

L

U1U1 U2U2

�����)�� H(jω) =1

1 + jωLR

, $�� #)�� H(jω) =jωL

R

1 + jωLR

.

��

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• ��( , �"#�$0 # ����� ���� 0,�' ���� ,�#0 8� (��� ���

#!2���! (����� B� ��(��2���08� # ,���,� �"#�$0 (�� C ��1

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( ��� (� "02��� ,����� �(A�� ��"� �(0 ,�� $��� ��� #2���

8�� (�� ) '��� @��

[,�#' ���,��� 8� τ =L

R

��

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@�' �! �(0 ,�� , ��'

u

tr

ti

tf

td

t

#���� �(0 ,0 :ui;

(�

ui

��Yui

.�Yui

5�Yui

)� � �!

��

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• tr 8� $�" ��#'�� )� :�'"A�0; �(0 ,0 :� ,� � ��; �A��

,� 8�� ),� ����! �(0 ,�� �(A�� (����"08� � (�����$0 ��2

��Yui 5�Yui�

• tf 8� $�" ��#'�� �! 0 :(�� �,0; �(0 ,0 :� � ��; �A��

,� 8�� ),� ����! �(0 ,�� �(A�� (����"08� � (�����$0 ��2

5�Yui ��Yui�

• td 8� $�" 2(�B$A�� )� �(0 ,0 :$� � � ��; �%B� "!�

#2�B�� � 8���0�� ),�#��0 ���B �0� �"#�� � (��$ (��1

���$�� )� � /"���A �%B� "!� #2�B�� � ���B �0 (�21

$A8C��0� (� �' 2'(���� 2������� ���0$ ,� #2��08� � 8 1

���0 �(0 ,0� "!#' �A��� ��#�AB #%) ���B �0� �$� �����

�(0 , (����'2� S��#�� .�Yui�

��

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• ti $�" ��#'�� �(0 ,0

• 0 (�� �$ ��� ,� �(�08����� �(0 ,% ,� 0#'$�

" �� ��)�� ��"� (�� �$ �(��#'�� �(0 ,0

" ,���$ :$0�� ��� �;� �8�� (��A� $�"� ��#'�� �(0 ,0 � $�"A

��#'�� (��� �(A� �A���� # S��#� .�Yui

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"���� ������ #���

� !��������� � ����� �����$�

• \ ����(',��#! �"#�$

• @�2���)�� �"#�$

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/"#�$ $�0���� �'$0 7 E��

7 C ����(',��#! �"#�$ :28�$��$0C��! ��$� ���,����'���0;

R1 L1

R2L2U1 U2

L1 7 ��2(�� �#' �$0�)��,�

L2 7 � #�� �$0�)��,�

R1 7 �$(�� # �0��

R2 7 (�����,�����#�! �$(�� 2'�AB� 2��'�� # 8'$��

��

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Z1 = R1 + jωL1,

Z2 =jωL2R2

R2 + jωL2,

U2 = U1Z2

Z1 + Z2 �$�0$

U2 = U11(

1 + R1R2+ L1

L2

)− j

(1

ωτd− ωτi

).

�$� τd =L2R1

τi =L1R2

8,�0 ),�#� ���,���� (�� $�� #)��

�����)�� ������� ����#��)�� 2'# , �,� (����,0�

���

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U2m = U1m1√(

1 + R1R2+ L1

L2

)2+(1

ωτd− ωτi

)2.�,� B� "0$��� (�#B�#� �'2 #,�0(���� �(A�� 2 �0 �#�0�

(� �%B��� (�� �'2 #!,�0(���� �(A�� �8�� ����#��)��

2'# , �,� #� �#�0

ϕ(ω) = arctg

(Im(H(jω))Re(H(jω))

)=

(1

ωτd− ωτi

)(1 + R1

R2+ L1

L2

).

K# ��� ���,����'��� #��208� R1 � R2 L1 � L2� ����� #

0�) ��� (',�0 �� ��)�%� �$� ϕ ≈ 0 "0$� U2m → U1m�

��� ���#! ���,����'��� ( �� τi � τd 2� (� " B�A 0�) � C���0

(���'C����� (',�� ��8� ����#��)�� ����# � #� ������ 8� �'2�

ϕ = ±45◦

Δω ≈ 1/τi − 1/τd

���

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6�2������#��)�� :0$ �; ���,����'���

10 100 1K 10K 100K-20

-15

-10

-5

0

5

db(v(1)) F (Hz)

10 100 1K 10K 100K-100

-50

0

50

100

150

ph(v(1)) (Degrees) F (Hz)

L1 = 10mH� L2 = 200mH� R1 = 50Ω� R2 = 1kΩ, Δf ≈ 16kHz

���

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��� �#! ��2���)�� �"#�$ @�� #� ����#��)�� �" ,�

I

U

R

L �����

C �μ�

I =U

R + jωL + 1jωC

= UjωC

(1− ω2LC) + jωRC

�2�)���

1LC= ω2r,

�$� ωr =1√LC= 2πfr 8� ��2���)�� �� ��)���

���

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U↔ Um sinωt

I↔ Im sin(ωt + ϕ)

Im = |I| = UmωC√

(1− ω2LC)2 + (ωRC)2

ϕ = arctg

(1− ω2LC

ωRC

)

�$�B ω = ωr = 1√LC

, "0$� �"#�$ � $A� $� ��2�����D

Im = |I| = UmωC√

(1− ω2LC)2 + (ωRC)2=

Um

R

ϕ = arctg

(1− ω2LC

ωRC

)= 0

���

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@�� �"#�$ 7 ����#��)�� ������� ,� �

-90

-75

-60

-45

-30

-15

10 100 1K

-100

-50

0

50

100

logf [Hz]

log(

i/i 0)[d

B](i0=1A)

Q = 4 Q = 40Q < 0, 5

ϕ[◦]

���

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6(A�� � �$0����0 UL = jωLI = U −ω2LC(1−ω2LC)+jωRC

�� ��2���� UL = jU 1ωrRC = jQ U

�$� Q = 1ωrRC =

1R

√LC 8� ) � �� 8��,� �"#�$0

ULm = Um Q ϕ = π/2

70 Hz 100 Hz 140 Hz0 V

10 V

20 V

30 V

40 V

50 V

log(f)

UL

m

Um = 1V Q = 40

���

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\��� (',� 7 ��2�� �� ��)�� ϕ = ±45◦, URm/Um = −3dB

UR = UωRC

jωRC−ω2LC+1

ϕ = ±45◦ =⇒ 1−ω2LCωRC = ±1

Pro Q > 5 ω1,2 = ωr(1± 12Q) =⇒ Q = ωr

ω1−ω2= fr/Δf(3dB)

100 Hz

-30,0

-22,5

-15,0

-7,5

0,0

7,5

-3 dB

98,75Hz 101,25 Hz

Δ=2,50 Hz

UR

m[dB]

f

���

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������$�! $A8 (� , �0,�#�� "02��� ��2���)���� �"#�$0

2$��8�� �(A��

Um = 1V, f = 100Hz, Q = 40.

0m 100m 200m 300m 400m 500m-40

-20

0

20

40

60

v(3) (V) T (Secs)

RLC-HUS.cir

R1

39.5

V1L1

2.533

C11u

6(A�� � ,#���'�� �$0����0

��

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/"#�$ $�0���� �'$0 7 �$�2# � ,���

7 C ����(',��#! �"#�$ :28�$��$0C��! ��$� ���,����'���0;

R1 pL1

R2pL2U1(p) =1pU U2(p)

L1 7 ��2(�� �#' �$0�)��,�

L2 7 � #�� �$0�)��,�

R1 7 �$(�� # �0��

R2 7 (�����,�����#�! �$(�� 2'�AB� 2��'�� # 8'$��

��

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� (�0B ��� �"�20 (�� u1(t) #� �#�0 U1(p) = U 1p $�,�����

�( ��%# �"�2 (�� U2(p)

U2(p) = UR2L1

1

p2 + p(

R1L1+ R2

L2+ R2

L1

)+ R1R2

L1L2

U2(p) =R2L1

U

(p − λ1)(p − λ2)

u2(t) =R2L1

U

(λ1 − λ2)

(eλ1t − eλ2t

)

��� $�� ��$���� L1 = 10mH, L2 = 100mH, R1 = 100Ω, R2 = 1kΩ,

$�,����� ��$���� λ1 = −119160, λ2 = −840

���

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U�C��� ������$ ������ ' �� ��#� �� , ���,������ ���Q� ���� ,

(�0B ��� �( ���#� ���,������ � �2 � #2���� (�( ,08���

),�#�0 �0��� #�"��� �"#�$�#� #� ) ��� ��$� ��C���

# � �� ���� �#�0�

� �0 )�� (������ (�� $�� (������ �"#�$�#!�� (�#�%

#�0B�#8� �0��� ��� ��C��� �"#�$�#!�� ��#� � #�$�0 � 2�,�'��

(�%"A�0 �(A�� (��0$% # ���������� �"#�$0� � B 8� (����"

2�'� � �� ��� #!�2�� ����� 8� (�( ,08��

���

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6 �"�'2�0 8� ,��#�'�� #!,�0(0 2 >� "0 > ����(0

0 1 2 3

x 10 -4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.u 60.u 120.u 180.u 240.u 300.u0.

0.25

0.50

0.75

1.Matlab MicroCap

L1 = 0.01; L2 = 0.1; R1 = 100; R2 = 1000; t = [0 : 1e− 6 : 3e− 4]; U = 1;lambda = roots([1 R1/L1 + R2/L2 + R2/L1 R1 ∗ R2/(L1 ∗ L2)]);

u2 = (R2/L1) ∗ (1/(lambda(1)− lambda(2))) ∗ (exp(t ∗ lambda(1))− exp(t ∗ lambda(2)));plot(t, u2)

���

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��� �#! ��2���)�� �"#�$ @�� # ),�#� �" ,�

i(t)

U0

R

L �����

C �μ�

���

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/"#�$ (�( ,08� ������$ ������ ' �� ��#� ��

Ri + Ldidt+1C

∫ t

0i(τ )dτ = U0 pro t > 0 a uC(0+) = 0

(� (���P�� ��2���)�� �� ��)�� ) � �� 8��,�

1LC= ω2r, Q =

1R

√L

C=

ωrL

R=

1ωrRC

,

����� $�( ���� � ) � �� � 0����α =

R

2L=

ωr

2Q

���

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�( ��%# �"�2 ��#� �� (�( ,08��� (��0$ # �"#�$0 (� �0 �#!��

(�)'��)���� (�$����'�� 8�

I(p)

(R + pL +

1pC

)=

U0p

I(p) =U0L

1(p2 + 2αp + ω2r)

=U0L

1(p − λ1)(p − λ2)

���0$ λ1 �= λ2� (� 8� ),�#! (�%"A� $'� #2����

i(t) =U0

L(λ1 − λ2)

(eλ1t − eλ2t

)

�$� λ1,2 = −α ±√

α2 − ω2r

���

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���0$ α = ωr, 8� I(p) =U0L

1(p2 + 2αp + α2)

=U0L

1(p + α)2

���0 �$(�#�$8��� ),�#! (�%"A�

i(t) =U0L

t e−αt

���0$ λ1,2 8,�0 $#A ��' �' )�, :α > ωr;� "0$� ),�#! (�%"A�

i(t) =U0e−αt

2L√

α2 − ω2r

⎛⎝et

√α2−ω2r − e−t

√α2−ω2r

⎞⎠

���0$ λ1,2 8,�0 $#A ���( �T�A ,$�0B��' )�, :α < ωr;� "0$�

i(t) =U0e−αt

2jL√

ω2r − α2

⎛⎝ejt

√ω2r−α2 − e−jt

√ω2r−α2

⎞⎠ = U0e−α t

L√

ω2r − α2sin(t√

ω2r − α2)

���

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/$�2# @�� �"#�$0 :R = 3500Ω; � ,��� �(A�� :U0 = 10V;

t=[0:0.0001:0.02];

U=10;

R=3500;

C=1e-6;

L=2.533;

alfa=R/(2*L)

omegar=1/sqrt(L*C)

fr=omegar/(2*pi)

i=(U*exp(-alfa*t)/(2*L*sqrt(alfa^2-omegar^2))).

*(exp(t*sqrt(alfa^2-omegar^2))-exp(-t*sqrt(alfa^2-omegar^2)));

plot(t,i)

0 20ms

0

0.5

1

1.5

2

2.5mA

ωr = 628.32 [rad/s] (fr = 100 [Hz]) α = 691 [1/s]

���

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/$�2# @�� �"#�$0 :R = 400Ω; � ,��� �(A�� :U0 = 10V;

t=[0:0.0001:0.1 ];

U=10;

R=400;

C=1e-6;

L=2.533;

alfa=R/(2*L)

omegar=1/sqrt(L*C)

fr=omegar/(2*pi)

i=(U*exp(-alfa*t)/(2*L*sqrt(alfa^2-omegar^2))).

*(exp(t*sqrt(alfa^2-omegar^2))-exp(-t*sqrt(alfa^2-omegar^2)));

plot(t,i)

100ms-4mA

-3

-2

-1

0

1

2

3

4

5

6mA

ωr = 628.32 [rad/s] (fr = 100 [Hz]) α = 79 [1/s]

��

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��� α = ωr 8� �"#�$ � ��2 (�� �$ � ��� /$�2# ���' (���� �

(��0$0 �()�� (� � ��� ������$�! $A8 8� ��8���C� ��B�!D

i(t) =U0L

t e−αt

0 20ms

0

25mA

��

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�����

���

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E�������� #�$���

R/2R/2

C

C

L/2L/2

L/2L/2

G

vedení se ztrátami

bezeztrátové vedení

���

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>�$� "�2�2��'��#��� #�$���

CCCCCC

L/2L/2 LLLLL

���

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L �0�� #�$��� ,� ���#' � �"�0 ������� 8�� �"#�$

, �(�$��� Z0� $� #C� � �"#�$� ����!� ,� C��� # �� ����'

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^( �! �"#�$ , $ �0�!� #�$����

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��� (�( , ���#'�� �"#�$0 2#�$��� $# ���Q� ���� �$�20

ρA =R0 − Z0R0 + Z0

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.

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uA(0) = UZ0

Z0 +R0, uB(0) = 0.

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uA(4td) = uA(0)(1 + ρB + ρBρA + ρBρAρB + ρBρAρBρA)· · ·

uB(∞) = uA(∞) = URz

Rz +R0.

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0ns 5ns 10ns 15ns 20ns 25ns 30ns 35ns 40ns0

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/"#�$ , (�� �$ ��!� �(0 ,��� , ��' �� Z0 = 100Ω td = 100nsu0 = 5V R0 = 30Ω ti = 250ns tp = 500ns Rz = 2kΩ

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��� # �0 C����� ,� (� "�2�2��'��#�� #�$��� 2� �8�� # ��#�0

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• )�#��# �� #�$��� l = λ/4 Zvst = Z20 / Zz

• (% # �� #�$��� l = λ/2 Zvst = Zz

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iLs =u2R

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t2

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⇒ UL(p) = pLIL(p)− LiL(0+)

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∫ t

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UL(p) +iL(0+)

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∫ t

0iC(τ )dτ + uC(0+) ⇒ UC(p) =

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⎛⎝ 1

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i(t) =U − uC(0)

Re−tτ , �$� a = 1/τ τ = RC

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ω(a2+ω2)sin(ωt)

α = 0, β = 0, γ = Umωτ , ω ≡ ω

uC(t) =Umωτ(1+ω2τ2)

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(1+ω2τ2)cos(ωt) + Um

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