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Dependent source (power supply)

Definition

The voltage or current of the dependent power supply is controlled by the voltage (current) somewhere in the circuit.

Dependent voltage source

Symbol in a circuit:

Controlled voltage source

reference direction

GB (national standard)

widely adopted internationally

four-terminal element

control element

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four-terminal element

control element

reference direction

GB (national standard)

widely adopted internationally

Dependent source (power supply)

- Dependent current source

Symbol in a circuit:

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Dependent source (power supply)

Definition

According to whether the control quantity and the dependent quantity are voltage or current, the controlled source can be divided into four types, i.e., voltage controlled current source, voltage controlled voltage source, current controlled current source, current controlled voltage source

current controlled current source (CCCS):

i₁

u₁

u₂

i₂

βi₁

Input:

control part

Output:

dependent part

i₂=βi₁

β is the scale parameter

In general, the element can

be replaced by a general element

i₁

u₁

u₂

i₂

βi₁

Input:

control part

Output:

dependent part

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Dependent source (power supply)

voltage controlled current source (VCCS):

i₁

u₁

u₂

i₂

gu₁

Input:

control part

Output:

dependent part

i₂=gu₁

g is the scale parameter,

and can be called transfer

conductance.

i₁

u₁

u₂

i₂

gu₁

Input:

control part

Output:

dependent part

It can be a general

circuit element

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Dependent source (power supply)

voltage controlled voltage source (VCVS):

i₁

u₁

u₂

i₂

μu₁

Input:

control part

Output:

dependent part

u₂=μu₁

μ is the scale parameter

i₁

u₁

u₂

i₂

Input:

control part

Output:

dependent part

It can be a general

circuit element

μu₁

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Dependent source (power supply) – 受控电源

current controlled voltage source (CCVS):

i₁

u₁

u₂

i₂

ri₁

Input:

control part

Output:

dependent part

u₂=ri₁

r is the scale parameter,

and can be called transfer resistance.

u₂

i₂

Output:

dependent part

It can be a general

circuit element

ri₁

i₁

u₁

Input:

control part

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Dependent source (power supply)

Example

i_b

i_c

i_b

: base current

i_c

: collector current

i_c= β i_b

i_b

u₁

u₂

i_c

βi_b

Input:

control part

Output:

dependent part

circuit model

NPN triode

i₁

5i₁

u₁=6V

u₂

3Ω

i₁=u₁/3=2A

u₁ = u₂+10V

u₂ = -4V

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Dependent source (power supply)

Comparison between dependent source and independent source

The voltage or current of an independent source is determined by the power supply itself and is independent of other voltages or currents in the circuit.

The voltage or current of the dependent source is determined by other control quantities in the circuit.

The independent source plays an excitation role in the circuit, producing voltage and current. The dependent source reflects the control relationship between the voltage or current in one part of the circuit and the voltage or current in another part, and cannot be used as an excitation.

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Kirchhoff‘s law

Seven circuit elements (components) so far. Which seven?

These elements’ properties can be commonly described by voltage-current relationship (VCR), i.e., the voltage-current function of each element’s terminals.

Kirchhoff’s law including Kirchhoff current law (KCL) and Kirchhoff voltage law (KVL), which reflect the basic laws of voltage and current in all sub-circuits in the circuit. KVL reflects the relationship between voltages of all sub-circuits, and KCL reflects the relationship between currents of all sub-circuits.

It is a basic law for analyzing lumped parameter circuits (L<λ). It forms the basis of circuit analysis together with element’s VCR.

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Kirchhoff‘s law

Branch (two definitions)

any element in a circuit that has two terminals (5 branches)
a branch of a circuit through which the same current goes (3 branches)

Node (two definitions)

connection point of two-terminal elements (4 nodes)
connection points with more than three branches (2 nodes)

Path

a collection of all branches between two nodes

Loop

closed path consisting of branches, where any

node is traversed only once.

Mesh

for planar circuit, the loop without any branch inside.

i₂

u_s₁

R₃

u_s2

R₂

R₁

i₁

i₃

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Kirchhoff‘s law

Kirchhoff current law (KCL) In a lumped parameter circuit, for any node at any time, the algebraic sum of the currents flowing into (or out of) the node is equal to zero.

⅀i(t)= 0 or ⅀i_in= ⅀i_out

i₁

i₅

i₄

i₃

i₂

If set the flowing-in current as positive,

i₁+i₂+i₃-i₄-i₅ =0

i₁+i₂+i₃=i₄+i₅

Note: all currents take the reference direction, although the actual direction

might be different. KCL applies to the currents labeled with the actual directions.

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Kirchhoff‘s law

Example

i₆

i₃

i₂

i₁

i₄

i₅

i₁+i₄+i₆=0

If set the out-coming current as positive, we have

i₃-i₅-i₆ =0

-i₂-i₄+i₅=0

By summing up the three equations, we have

i₁-i₂+i₃=0

Thus, KCL can be applied to a loop (dotted ellipse)

that contains multiple nodes. The current coming into the loop

equals the current coming out of the loop.

The loop can be viewed as a general node in circuit.

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Kirchhoff‘s law

Note

KCL is the reflection of the principle of charge conservation (电荷守恒) and current continuity (电流连续性) at any node in the circuit.

KCL is a constraint added to the branch current at the node, which is independent of the component characteristics (元件特性) on the branch, such as the linear or nonlinear characteristics of the component.

KCL equation is based on the reference direction of current.

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Kirchhoff‘s law

Kirchhoff voltage law (KVL) In a lumped parameter circuit, the algebraic sum of all branch voltages along any loop at any time is equal to zero.

⅀u(t)= 0 or ⅀u_inc= ⅀u_dec

u_s₂

u_s₁

i₁

i₂

i₃

i₄

R₄

R₃

R₂

R₁

u_R₃

u_R2

u_R₁

Set the reference direction for each branch voltage
Set the reference direction for the loop

-u_s1- u_R1+ u_R2+ u_R3+ u_R4+ u_s2 = 0

u_s1+ u_R1= u_R2+ u_R3+ u_R4+ u_s2

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Kirchhoff‘s law

Example

u_s₁

u₁

u₂

u₃

u₄

u₄- u₁– u_s1– u₂ =0

u₄ = u₁+ u_s1+ u₂ = u_ba

Note:

KVL can also be applied to a virtual loop as shown above. Implicitly, voltage is independent of paths in a circuit.
KVL is a constraint on the voltage of the branch on the circuit, independent of the component type on the circuit.
KVL equation is based on the reference direction of voltage.

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Kirchhoff‘s law

Example

i=?

u_s

R=5Ω

10v

u_s₂

u_s₁

i₁

i₂

i₃

i₄

R₄

R₃

R₂

R₁

u_R₃

u_R2

u_R₁

R₅

u_R5=?

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i₆

i₃

i₁

i₄

i₅

Kirchhoff‘s law

Example

i₂

i₃=?

20V

10V

10-?-5-20=0

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Kirchhoff‘s law

Example

R=3Ω

-4V

i=?

i=u_R/3 = (5-4)/3 =1/3A

R=3Ω

i=1A

u=?

u-3*1-4=5V

10V

-10V

R=10Ω

i=?A

u_R+10=-10 🡪 u_R=-20V (KVL)

i_R=u_R/10 = -2A 🡪 i_R=i+1🡪 i=i_R-1=-3A (KCL)

i_R

10A

2Ω

i_2Ω

i_2Ω =10-3=7A (KCL)

4+u = u_2Ω= 2*7=14 🡪 u=10V (KVL)

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Kirchhoff‘s law

Example

i₁=-i₂ = -1A 🡪 i₃=0A 🡪 i₄=2i₂=2A (KCL)

5V = -3+u+10 🡪 u=-2V (KVL)

10V

3i₂

i₁

i₂

2i₂

5Ω

i=0

i₃

i₄