CIRCUITS and SYSTEMS part I Prof dr hab

CIRCUITS and SYSTEMS – part I Prof. dr hab. Stanisław Osowski Electrical Engineering (B. Sc. ) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

Lecture 1 The basic laws of electrical circuits

Basic notions • Carriers of electricity: electrons and protons of atom • Electric current: the ordered movement of electrical charges q in time, measured as i=dq/dt. Denoted by the letter i. Its unit is amper (A) • Electric voltage: the difference of potentials between two points of the conducting media (circuit). Denoted by the letter u. The unit of voltage is volt (V). • Electric circuits: the connection of electrical elements enabling the flow of current in such connection.

Basic notions (cont. ) • Branch – one or more circuit elements connected together of two external terminals accessible for connection to other elements. • Node – the terminal of the circuit enabling to connect the next branches. The nodes are separated by the branch. • Mesh – the set of circuit branches forming closed way (loop) for the current. • Element – the smallest part of the circuit of strictly defined function.

Basic notions (cont. ) • Passive elements - the electric elements able to either accumulate or dissipate the energy. They don’t generate energy. To this set belong: resistor, capacitor and inductor. • Active element (sources): the elements generating the electrical energy. It is usually generated by converting from other types of energy (mechanical, solar, nuclear, etc). This set is formed by independent and controlled sources of either voltage or current type. • Linear element – the circuit element described by the linear relation between its voltage and current signals. • Nonlinear element - the element described by the nonlinear relation between its voltage and current signals.

Resistor Graphical symbol of linear resistor Mathematical description (Ohm’s law) R – resistance G = 1/R – conductance The unit of resistance is om ( ) and of conductance siemens (S).

Inductor Graphical symbol of inductor Mathematical description • – flux linkage (unit: Weber = Vs) • L – self-inductance (inductance) , (unit: Henr = Ωs)

Capacitor Graphical symbol of capacitor Mathematical description • q – charge (unit kulomb = As) • C – capacitance (unit: farad = As/V)

Independent sources Graphical symbols of a) voltage, b) current source Current-voltage characteristics of : a) voltage source, b) current source

Independent sources (cont. ) • Voltage on the terminals of the ideal voltage source is independent on the current flowing through it. • Internal resistance of the ideal voltage source (R=dv/di) is equal zero (short circuit). • Current of the ideal current source is independent on the voltage (load of the source). • Internal resistance of the ideal current source (R=dv/di) is equal infinity (open circuit).

Controlled sources - description • Voltage controlled voltage source • Current controlled voltage source • Voltage controlled current source • Current controlled current source

Controlled sources – circuit structures Graphical symbols of controlled sources

Kirchhoff’s laws • Current law (KCL) • Voltage law (KVL)

Example 1 KCL equations: KVL equations:

Example 2 Determine the currents and voltages in the circuit at following values of parameters: R 1=1 , R 2=2 , R 3 = 3 , R 4 = 4 , e = 10 V, iz 1 = 2 A, iz 2 = 5 A.

Equations of the circuit • KCL and KVL equations • Equations including Ohm’s law • Solution: i 1 = 3, 187 A, i 2 = 0, 875 A, i 3 = 3, 812 A oraz i 4 = -2, 062 A.

Series connection of resistors Resistors connected in series Circuit equation Equivalent resistance

Parallel connection of resistors Resistors connected in parallel Circuit equation Equivalent conductance Equivalent resistance for 2 resistors

Wye and delta connections Connection of resistors a) delta i b) wye

Wye-delta transformation

Delta-wye transformation

Example Determine the equivalent input resistance Rwe of the circuit. Assume: R 1=2Ω, R 2=4Ω, R 3=3Ω, R 4=2Ω, R 5=4Ω, R 6=5Ω, R 7=5Ω, R 2=10Ω.

The succeeding stages