1. Review of Circuit Theory Concepts

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1 1. Review of Circuit Theory Concepts Lecture notes: Section 1 ECE 65, Winter 2013, F. Najmabadi

Circuit Theory is an pproximation to Maxwell s Electromagnetic Equations circuit is made of a bunch of elements connected with ideal (i.e., no resistance) wires.

2 Circuit Theory is an pproximation to Maxwell s Electromagnetic Equations circuit is made of a bunch of elements connected with ideal (i.e., no resistance) wires. Circuit Theory is an pproximation to Maxwell s Electromagnetic Equations by assuming o Speed of light is infinite (or dimension of the circuit is much smaller than wave-length of voltage/current waveforms). o Electric and magnetic fields are confined within each element: 1) Internal of an element manifests itself as an iv characteristic eq. 2) Elements communicates with each other only through the wires! Since the rest of the circuit only sees the iv characteristics of an element, different physical elements with similar iv characteristics are identical! F. Najmabadi, ECE 65, Winter2013, Intro (2/15)

Linear circuits have many desirable properties linear circuit element has a linear iv characteristic equation, v + B i + C = 0 (either in time or frequency domain) If all elements in a circuit are

3 Linear circuits have many desirable properties linear circuit element has a linear iv characteristic equation, v + B i + C = 0 (either in time or frequency domain) If all elements in a circuit are linear, the circuit would be linear and has many desirable properties (e.g., proportionality and superposition) which are essential for many functional circuits. Circuit theory has symbols for ideal linear elements: o Five two-terminal elements : resistors, capacitors, inductors, independent voltage and independent current sources o Four four-terminal elements : controlled voltage and current sources. It is essential to remember that the above ideal elements are NOT representative of physical devices. Rather they are representative of elements with a certain iv characteristic equation. F. Najmabadi, ECE 65, Winter2013, Intro (3/15)

4 Practical elements are only approximated by ideal circuit theory elements Real resistor i i v v s the current increases, resistor heats up and its resistance increases t high enough current, the resistor burns up Lab resistor can be approximated as an ideal circuit theory resistor for a range of current or voltage (identified by its rated maximum power) F. Najmabadi, ECE 65, Winter2013, Intro (4/15)

5 Ideal circuit theory elements are NOT representatives of physical devices! Is a symbol for Is NOT representative of this F. Najmabadi, ECE 65, Winter2013, Intro (5/15)

6 Ideal circuit theory elements are representative of elements with a certain iv characteristic equation. Can be approximated with this Can be approximated with this (for small signals) In fact, in integrated circuit we usually configure transistors to act as resistors (to save space among other benefits). F. Najmabadi, ECE 65, Winter2013, Intro (6/15)

Currents and voltages are circuit variables Equations governing the circuits are: o Internal of each element: iv characteristic equation of each element: v = f(i) o How the elements are connected:

7 Currents and voltages are circuit variables Equations governing the circuits are: o Internal of each element: iv characteristic equation of each element: v = f(i) o How the elements are connected: KCL: (conservation of charge), and KVL: (topology) circuit with N two-terminal element has 2N variables and need 2N equations: o N o N iv characteristic equation KCL/KVL Node-voltage (or mesh current) method reduces the number of equations to be solved by automatically satisfying all KVLs (or KCLs). o Use node-voltage methods unless circuit is very simple! F. Najmabadi, ECE 65, Winter2013, Intro (7/15)

, v o in terms of v i ) If the network only contains linear elements, its function can be

8 We will analyze many functional circuits Two-terminal Networks Two-port Networks Function is defined by the iv equation Function is defined by the transfer function (e.g., v o in terms of v i ) If the network only contains linear elements, its function can be characterized by several parameters (or numbers) instead of an algebraic function F. Najmabadi, ECE 65, Winter2013, Intro (8/15)

linear two-terminal network can be represented by its Thevenin Equivalent Thevenin Theorem: If all elements inside a two-terminal network are linear, the iv equation of the two-terminal network would

9 linear two-terminal network can be represented by its Thevenin Equivalent Thevenin Theorem: If all elements inside a two-terminal network are linear, the iv equation of the two-terminal network would be linear: v + B i + C = 0 o linear two-terminal network can be modeled with two ideal circuit theory elements (v T = C/, R T = B/) v = v T R T i o If the two-terminal network does NOT contain an independent source, v T = 0 and it reduces to a resistor. o See Lecture note for examples of computing/measuring Thevenin equivalent circuit F. Najmabadi, ECE 65, Winter2013, Intro (9/15)

10 Functional circuit contains several twoterminal and two-port networks We divide the circuit into building blocks to simplify analysis and design Two-terminal network containing an independent source Two-terminal network containing NO independent source F. Najmabadi, ECE 65, Winter2013, Intro (10/15)

11 Source only sees a load resistor two-terminal network containing NO independent source We only need to analyze the response of a source ONCE with R L as a parameter. For a linear source, we only find the Thevenin parameters of the source. F. Najmabadi, ECE 65, Winter2013, Intro (11/15)

12 Two-port network two-terminal network containing N independent source two-terminal network containing NO independent source F. Najmabadi, ECE 65, Winter2013, Intro (12/15) Transfer function of a two-port network can be found by solving the above circuit once.

13 ccuracy Mathematical precision is neither possible nor required in practical systems!

ccuracy (or tolerance) in practical systems Measurement ccuracy: o Measuring instruments have a finite accuracy. o When a scope with an 2% read a voltage of 1.

14 ccuracy (or tolerance) in practical systems Measurement ccuracy: o Measuring instruments have a finite accuracy. o When a scope with an 2% read a voltage of V, it means that the real voltage is in the range of ± (or between and V). Component ccuracy: o Components are manufactured with a finite accuracy (tolerance). o 1k resistor with 5% accuracy has a resistance between and 1.050k. Modeling ccuracy o We approximate practical circuit elements with ideal circuit theory elements. (we will see this throughout the course for non-linear elements) nalysis ccuracy: o We make approximation in the analysis by ignoring terms. (next Slide) F. Najmabadi, ECE 65, Winter2013, Intro (14/15)

How accuracy affect analysis: F. Najmabadi, ECE 65, Winter2013, Intro (15/15) When a number has,, has a relative accuracy of ε, it means that its value is between (1 ε) and (1 + ε).

15 How accuracy affect analysis: F. Najmabadi, ECE 65, Winter2013, Intro (15/15) When a number has,, has a relative accuracy of ε, it means that its value is between (1 ε) and (1 + ε). lternatively, we are saying that all numbers in that range are approximately equal to each other. When we assume a <<, we mean: ) (1 ) 1 ( ε ε + B B ) (1 ) (1 a a a a ε ε ε ε ε ε a a ε <<

Chapter 10 AC Analysis Using Phasors

Chapter 10 AC Analysis Using Phasors 10.1 Introduction We would like to use our linear circuit theorems (Nodal analysis, Mesh analysis, Thevenin and Norton equivalent circuits, Superposition, etc.) to