Lecture 01�Circuit model and circuit law

• Circuit and Circuit Law
• Reference direction of current and voltage *
• Electric power and energy
• Circuit elements
• Resistor element *
• Voltage and current sources *
• Dependent source
• Kirchhoff's law *

Circuit and Circuit model

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• Real-world electric circuit - The path of current formed by the connection of electrical equipment and electrical devices according to the intended purpose

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• Examples: electronic and communication system, automatic control system and computer information system

Circuit and Circuit model

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• main functions of real-world electric circuit
• energy transmission, distribution and conversion
• information transmission, control and processing

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• Circuit model

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• Def: ideal circuit elements and their combinations reflecting the main electromagnetic properties of actual circuit components

Circuit and Circuit model

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• Def: ideal circuit elements and their combinations reflecting the main electromagnetic properties of actual circuit components

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• The main electromagnetic properties in circuit analysis include current, voltage at each circuit element. Excluding heat, light, or some other emittance.

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• Ideal circuit elements: for which only the major electromagnetic properties are considered, nothing else.
• Resistor element - energy consuming components
• Inductive element - Generating magnetic field and storing magnetic field energy

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Circuit and Circuit model

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• Idea circuit elements: for which only the major electromagnetic properties are considered, nothing else.

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• Five types of ideal circuit elements
• Resistor element - energy consuming components
• Inductive element - generating magnetic field and storing magnetic field energy
• Capacitance element - Generating electric field and storing electric field energy
• Voltage and current sources - element that converts other forms of energy into electrical energy

Circuit and Circuit model

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• Note1 -Properties
• Two terminals
• Describable by a math expression

(a relationship of current and voltage)

• Minimal circuit element,

not decomposable

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Circuit and Circuit model

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• Note2 – The actual circuit components with the

same main electromagnetic characteristics can be represented by the same circuit model under certain conditions (e.g., two resistors of 5 ohm and 10 ohm).

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• Note3 – The circuit model of the same actual circuit component has different forms under different conditions.

Inductance coil circuit model

Reference direction of current and voltage

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• The main physical quantities in the circuit include voltage, current, charge, magnetic flux, energy and electric power, etc. Voltage, current and power are the main concerns in linear circuit analysis (especially for this course).
• Current (i): regular directional motion of charged particles
• Current intensity: the amount of charges passing through the cross section of a conductor in unit time
• Unit: Ampere (A), kA(kilo-A), uA (micro-A), mA (milli-A)
• Current direction: direction of positive charge flow
• Reference direction: assumes a direction of positive charge flow

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Reference direction of current and voltage

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• Two ways of representation

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• Potential (φ): the amount of work by the electric field force when a unit positive charge moves from a point in the circuit to a reference point.
• Voltage (u): the amount of work by the electric field force when a unit positive charge moves from a point in the circuit to another point.
• Voltage direction (actual): the direction along which the potential decreases
• Unit: Volt (A), kV, uV, mV

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і

A

B

A

B

B

і_AB

Reference direction of current and voltage

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• Example

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It is known that a particle with 4C positive charge moves uniformly from point A to point B, and the electric field force does 8 joules of work, and the electric field force does 12 joules of work from point B to point C.

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1. If B is taken as the reference point, calculate the potential of points A, B and C, and the voltage of u_AB and u_BC

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φ_B = 0v, Since (φ_A-φ_B) x q = (φ_A-0) x 4 = 8, we have φ_A= 2v

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Since (φ_B-φ_C) x q = (0-φ_C) x 4 = 12, we have φ_C= -3v

u_AB = φ_A-φ_B =2v, u_BC = φ_B-φ_C =3v

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(2) If C is taken as the reference point, calculate the potential of points A, B and C, and the voltage of u_AB and u_BC

_{Try it yourself after class}

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Reference direction of current and voltage

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• Ref. direction of voltage

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• Three representations of reference direction of voltage

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A

B

A

B

+

-

arrow representation

pos. neg. sign representation

A

B

subscript representation

u_AB

Reference direction of current and voltage

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• Associated reference direction: If u and i of a component have the same reference direction, they are associated reference directions. Otherwise, they are not.

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A

B

non-associated reference directions

+

-

і

In terms of A and B, whether are they

associated reference directions?

Reference direction of current and voltage

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• Associated reference direction: If u and i of a component have the same reference direction, they are associated reference directions. Otherwise, they are not.

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• Note:
• The reference direction must be selected before analyzing the circuit.
• Once the reference direction is selected, the direction and symbol must be marked at the corresponding position.
• When the reference direction is different, the expression differs by a negative sign, but the actual direction of current and voltage remains the same.

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Electric power and energy

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• Electric power: The work that the electric field force does per unit time, unit: Watt (W)

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• Judge whether the circuit element is absorbing power or emitting power

P = dw/dt, u=dw/dq, i=dq/dt

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P = dw/dt = dw/dq * dq/dt = u*i

P = u*I to represent absorbing power

If P>0, absorbing power (actually)

If P<0, emitting power (actually)

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an arbitrary

circuit component

P = u*I to represent emitting power

If P>0, emitting power (actually)

If P<0, absorbing power (actually)

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an arbitrary

circuit component

Example:

u₁=1v

u₂=-3v

u₃=8v

u₄=-4v

u₅=7v

u₆=-3v

i₁=2A

i₂=1A

i₃=-1A

positive values: the reference

direction is the same as actual

direction.

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negative values: the reference

direction is the opposite of actual

direction.

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p₁=u₁xi₁=2w (emitting)

p₂=u₂xi₁=-6w (emitting)

p₃=u₃xi₁=16w (absorbing)

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p₄=u₄xi₂=-4w (emitting)

p₅=u₅xi₃=-7w (emitting)

p₆=u₆xi₃=3w (absorbing)

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Circuit element (component)

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• Circuit element: Ideal circuit element, not actual ones.
• Five types of ideal circuit elements
• Resistor element - energy consuming components
• Inductive element - generating magnetic field and storing magnetic field energy
• Capacitance element - Generating electric field and storing electric field energy
• Voltage and current sources - element that converts other forms of energy into electrical energy

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Note: If the mathematical relationship characterizing the characteristics at both ends of the element is linear, the element is linear. Otherwise, it is a nonlinear element. E.g., resistor, inductive, and capacitance elements.

Circuit element (component)

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• Lumped-parameter circuit: Circuit that is composed of lumped elements.

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• Lumped element: the electromagnetic process takes place inside the element, i.e., we only care about the electromagnetic process inside the element, which means we care about current and voltage mostly.

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• What condition should be satisfied so that the element is viewed as a lumped elements (condition of lumped element)? the element size is far smaller than the wavelength of the electrical signal being considered. E.g., for a signal of 50Hz, its wavelength is 300000km/50 = 6k km.

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• Note: In a lumped circuit, current and voltage can be functions of time, but not of space coordinates. That is to say, the current flowing into and out of the same lumped element at each time is the same, and the voltage at both ends is constant.

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Resistor element (component)

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• Resistor element: Elements that exhibit resistance to current, whose characteristics can be described by a curve in the u-i plane

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• Linear time-invariant resistive element: Resistor whose value does not change over time, and it satisfies R=u/i anytime.

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• Symbol:

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R

GB (national standard)

R

widely adopted internationally

Resistor element (component)

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• In the u-i plane, Ohm‘s law R=u/i, or u=Ri or i=u/R=uG (G=1/R)

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• Unit: R (Ohm, Ω)，G(Siemens, S)

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volt-ampere characteristic curve

Resistor element (component)

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Resistor element (component)

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• Note

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Ohm‘s law

• only applicable to linear time-invariant resistance

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• if non-associative reference direction, the formula is up to a negative sign, i.e., R=-u/i

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• Thus, associative reference direction of current and voltage is necessary in calculation.

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• linear resistive elements are memoryless and bidirectional

Resistor element (component)

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• Power

p=u*i = u*u/R = i*R*R

p=u*i = -u*u/R = -i*R*R

For both cases, a resistor element always absorbs power.

Resistor element (component)

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• Energy

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• open and short circuit

R

i

u

+

i

short circuit

u=0,i=?

R=0,G= ∞

volt-ampere characteristic curve

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u

i

u

i

Voltage and current sources

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• Ideal voltage source: a circuit element across which the voltage remains constant or as a function of time, independent of the magnitude of the current flowing through the element.

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• Symbol in a circuit:

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• Note:
• The voltage at both ends of the ideal voltage source is

independent of the magnitude and direction (actual direction)

of the current flowing through the element and the external

circuit

GB (national standard)

widely adopted internationally

Voltage and current sources

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• Note:

2. The current passing through the voltage source is jointly determined by the voltage source itself and the external circuit.

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• Power of ideal voltage source

i=u_s/R

R = ∞, i=0

R = 0, i= ∞

Voltage source cannot be short circuited!

+

-

P=i*u_s>0,

emitting power

u,i take non-associated reference direction

Physical meaning: the current (positive charge) moves from low potential to high potential, voltage source overcomes the electric field force to work, and generates power.

i>0 and u_s>0,

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i<0 and u_s<0,

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Voltage and current sources

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• Power of voltage source

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• Example

i

u,i take associated reference direction

P=i*u_s>0,

aborbing power

acting as a load

Physical meaning: the current (positive charge) moves from high potential to low potential, the electric field force works, and voltage source absorbs power.

i>0 and u_s>0,

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i<0 and u_s<0,

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u_R

u_R = 10-5 = 5V

i = u_R/R = 1A

P_R = 5*1 = 5W (absorbing)

P_10V = 10*1 = 10W (emitting)

P_5V = 5*1 = 5W (absorbing)

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Note: voltage source can

absorb power. For example, rechargeable batteries

Voltage and current sources

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• Ideal current source: a circuit element across which the current remains constant or as a function of time, independent of the magnitude of the voltage at both ends the element.

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• Symbol in a circuit:

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• Note
• The current across the ideal current source is

independent of the magnitude and direction (actual

direction) of the voltage at both end of it and the

external circuit

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GB (national standard)

u,i_s can take either associated or non- associated reference direction

widely adopted internationally

Voltage and current sources

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• Note

2. The actual voltage at the two ends of the ideal current source is jointly determined by the current source itself and the external circuit.

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R

u=R*i_s

R = ∞, u= ∞

R = 0, u= 0

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Ideal current source cannot be open circuited!

Current transformer

Voltage and current sources

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• Generation of actual current source

It can be generated by current stabilizing electronic devices. For example, the collector current of the transistor is independent of the load, and the photocell generates current under the irradiation of light, etc.

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• Power of ideal current source

P=i_s*u>0,

emitting power,

acting as a power supply

u,i_s take non-associated reference direction

P=i_s*u>0,

absorbing power,

acting as a load

u,i_s take associated reference direction

Voltage and current sources

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• Example

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• Actual power supply

i

u

u=5V,

i=-2A

P_2A = 5*2 = 10W (emitting power)

P_5V = 5*(-2) = -10W (absorbing power)