The elements of circuit theory Learning in Electronic Circuits The elements of circuit theory © 2013 Universitat Politècnica de Catalunya (EPSEVG) José Antonio Soria Pérez - Associate Professor at the EPSEVG (UPC). C/ V. Balaguer, 1. Vilanova i la G. (Barcelona - Spain). Telf: +34 93 896 77 27 Fax: +34 93 896 77 00 Office hours: VG3-D101 – EEL TUE 11-13h, THU: 16-18h (Yellow zone) E-mail: jose.antonio.soria@eel.upc.edu, jasoria@eel.upc.edu

In this lesson you will meet... ... the basic elements of circuit analysis Introductory concepts of electricity The ‘bipole’ concept: Definitions and properties Bipole specifications, i-v transfer characteristics and associations Circuit models for basic bipoles: Mechanical bipoles: switches and/or push buttons Passive bipoles: resistors, capacitors, inductors Active bipoles: voltage and current sources

Fundamentals of circuit theory Circuit theory lies on the principle of the electric charge Electric charge generates electric and magnetic fields Circuit analysis lies on the use of scalar voltage and current as primitive variables B1 B2 B3 B4 i2 i3 i4 Electric field Magnetic field iinN ioutM iin1 iout1 ++++++++++++++++++ - - - - - - - - - - - - - - - - - 1) Electric charge is quantified: qe = 1.601256x10-19C Q= i=1 N 𝑞 in i − j=1 M 𝑞 𝑜𝑢𝑡 𝑗 =0 2) Principle of charge conservation: i=1 N 𝑖 in i − j=1 M 𝑖 𝑜𝑢𝑡 𝑗 =0 v1 v2 + _ v4 + _ I B Bipole E Electrical System + _ VOLTAGE CURRENT Caused by the presence of electric charge Caused by the movement of electric charge i(t) First terminal Second Concentrated-parameter Networks 𝜕Ω 𝐸ds = Q 𝑉 ℰ 0 , Maxwell equations: 1) Gauss’s law 𝜕Ω 𝐵ds , 2) Gauss’s law for magnetissm 3) Faraday equation 𝜕Σ 𝐸dl=− Σ 𝜕𝐵 𝜕𝑡 ds, 𝜕Σ 𝐵dl= μ 0 𝐼+ ℰ 0 Σ 𝜕𝐸 𝜕𝑡 ds 4) Amper equation + _ v(t) Bipole Electric current: i(t) Ampers [A] Electric voltage: v(t) Volts [V] 𝑖 𝑡 = lim Δ𝑇→0 Δ𝑞 𝑡 Δ𝑇 = d𝑞 𝑡 d𝑡 𝑞 𝑡 = −∞ 𝑡 𝑖 𝑡 d 𝑡 Charge (q) + Stimulus (v) = current flow (i)

Fundamentals of circuit theory Circuit theory lies on the principle of the electric charge Electric charge generates electric and magnetic fields Circuit analysis lies on the use of scalar voltage and current as primitive variables Secondary variables: power vs. energy Conventions: Bipoles are electrically neutral Positive vs. negative voltage/current equivalences Consumption (absortion) vs. generated (transfer) power 𝑃 𝑡 = d𝑤 𝑡 d𝑡 = 𝑣 𝑡 d𝑞 𝑡 d𝑡 =𝑣 𝑡 𝑖 𝑡 Power: Energy: 𝑤 𝑡 = −∞ 𝑡 𝑃 𝑡 d 𝑡= −∞ 𝑡 𝑣 𝑡 𝑖 𝑡 d 𝑡 [Watts] [Joules] 𝑑q 𝑑𝑡 =0 𝑖 𝑎 − 𝑖 𝑏 = 𝑑q 𝑑𝑡 =0 ia ib Voltage Current t P(t) 1A + _ -1V Power absortion: P(t) > 0 + _ -1V 1V + _ 1V -1A i + _ v 𝑷 𝒕𝒓𝒂𝒏𝒔𝒇𝒆𝒓 =− 𝑷 𝒂𝒃𝒔𝒐𝒓𝒃𝒆𝒅 𝑖 𝑎 = 𝑖 𝑏 -1A 1A Absortion convention (positive current enter- ing the positive terminal) Power transfer: P(t) < 0

Bipole concepts Validity: Propagation delay much lower than the interval change Classification: Passive vs. Active. Lossy vs. Lossless. Memory vs. Memory-less Bipoles: several types depending on i-v relationship the short-circuit and the open-circuit bipole t w(t) lossless lossy Absorbed energy is returned to the electric medium absortion of energy Memory Memory-less Passive Active i(t) depends on v(t) at present time 𝑑 𝑐 = λ 2π dc.- maximum critical distance λ.- signal wavelength λ= 𝑐 (𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑙𝑖𝑔ℎ𝑡) 𝑓 (𝑠𝑖𝑔𝑛𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦) 𝑖 𝑡 =a· 𝑣 𝑡 +b· 𝑣 3 𝑡 Ex: t w(t) Active bipole 𝑤 ∞ = lim 𝑡→∞ 𝑤 𝑡 =0 Bipole Short-circuit Symbol Open-circuit Symbol 𝑣 ∞ =0 𝑜𝑟 𝑖 ∞ =0 generation of energy v i i-v characteristics (lossy) i(t) depends on previous v(t) and-or i(t) v i i-v characteristics i ≠ 0 𝑖 𝑡 =𝐾 d𝑣 𝑡 d𝑡 = lim Δ𝑡→0 𝑣 𝑡 −𝑣 𝑡−Δ𝑡 Δ𝑡 Ex: i ≠ 0 v = 0 + _ i = 0 v ≠ 0 + _ v ≠ 0 Push buttons and electronic switches implement open-circuits and shortcircuits

Bipole concepts Validity: Propagation delay much lower than the interval change Classification: Passive vs. Active. Lossy vs. lossless. Memory vs. memory-less Bipoles the short-circuit, the open-circit. the Resistor Capacitors and Inductors Signal sources (Active) v i i-v characteristics Voltage supply Current supply V V + I v i i-v characteristics t i(t) v(t) Capacitor Inductor Symbol L C R I Passive Lossless Memory Passive Lossy Memory-less 𝑖 𝑡 =𝐶 d𝑣 𝑡 d𝑡 𝑣 𝑡 =𝐿 d𝑖 𝑡 d𝑡 𝑖 𝑡 = 1 𝑅 𝑣 𝑡 Ohm’s Law i-v relationship

...more on bipoles Virtual short-circuit: the ideal switch (v = 0, i = 0) Resistors Available on several types of devices: bulbs, heaters, electronic resisters, temperature sensors, etc… Electric current flow in resistors and conductors depend on material resistivity Resistance vs. conductance Power consumption vs. Energy Series vs. parallel association R1 = R2 =… = Rn = R Two resistors in parallel i v i = 0 Operating point v = 0 + _ Bulbs Heaters Resistive Temperature Sensors Strain gauges length [m] i v 𝑅=𝜌 𝑙 𝐴 Resistive material [Ohms, Ω] Area [m2] Parallel: a b R1 R2 Rn 𝑃 𝑡 =𝑣 𝑡 𝑖 𝑡 =𝑅 𝑖 2 𝑡 = 𝑣 2 𝑡 𝑅 ≥0 𝑤 𝑡 = −∞ 𝑡 𝑃 𝑡 d 𝑡= 𝑣 2 𝑡 𝑅 𝑡≥0 i v + _ Resistivity Series: a b R1 R2 Rn R 1 1 𝑅 =𝐺 Rab Rab Series: 𝑅 𝑎𝑏 =𝑛𝑅 Parallel: 𝑅 𝑎𝑏 = 𝑅 𝑛 Junction (Connecting point) 𝑅 𝑎𝑏 = 𝑅 1 𝑅 2 𝑅 1 + 𝑅 2 Electronic Resistors Wires 𝑅 𝑎𝑏 = 𝑖=1 𝑛 𝑅 𝑖 1 𝐺 𝑎𝑏 = 𝑖=1 𝑛 1 𝐺 𝑖 1 𝑅 𝑎𝑏 = 𝑖=1 𝑛 1 𝑅 𝑖 𝐺 𝑎𝑏 = 𝑖=1 𝑛 𝐺 𝑖 𝑖 𝑡 = 1 𝑅 𝑣 𝑡 =𝐺𝑣 𝑡 G ≡ Conductance {Ω-1}

...more on bipoles Virtual short-circuit: the ideal switch (v = 0, i = 0) Resistors Signal sources: Symbols and conventions DC sources do not vary with the time AC sources do not possess comprehensive i-v characteristics Independent vs. controlled sources Source deactivation i-v characteristics v i t v(t) + Time response + DC Voltage source DC Current source Voltage Source H·vA Z·iA H.- Voltage gain (V/V) Z.- Tramsresistance (V/I) Current Source G·vA Y·iA G- Transconductance (I/V) Y.- Current gain (I/I) Current Source Voltage Source Independent sources i V i-v characteristics v i Controlled by voltage vA + _ + v(t1) v(t2) One variable (V) fixed and the other (i) depends on the circuit + V I V + I=0 V=0 Disconnecting a current source (I=0) means opening the circuit Disconnecting a voltage source (V=0) means shorting its terminals v(t1) v(t2) Controlled by current iA AC Voltage source AC Current source + These symbols also apply for generic sources

...more on bipoles Virtual short-circuit: the ideal switch (v = 0, i = 0) Resistors Signal sources: Symbols and conventions Capacitors and Inductors Static operation Symbol Conventions i-v relationship Electric Principle Energy (Electric Field) (Magnetic Field) 𝑞 𝑡 =𝐶·𝑣 𝑡 ϕ 𝑡 =𝐿·𝑖 𝑡 C + _ v(t) i(t) L 𝑖 𝑡 =𝐶 d𝑣 𝑡 d𝑡 𝑤 𝑡 = 1 2 𝑣 2 𝑡 𝑣 𝑡 =𝐿 d𝑖 𝑡 d𝑡 𝑤 𝑡 = 1 2 𝑖 2 𝑡 + _ v(∞) = V i(∞) = 0 V + C + _ v(∞) = 0 i(∞) = I I L

Absorbed power is sent back to the source ...more on bipoles Virtual short-circuit: the ideal switch (v = 0, i = 0) Resistors Signal sources: Symbols and conventions Capacitors and Inductors Static operation Dynamic operation: sinusoidal response and reactance P(t) >0 t i(t) v(t) v(t) Magnitude IP =VPωC Magnitude VP =IPωL ϕ Angle: ϕ =π/2 i(t) P(t) >0 i(t) 𝑋 𝐶 = 𝑣 𝑡 𝑖 𝑡 = 1 𝑗ω𝐶 = 1 𝑠𝐶 𝑋 𝐿 = 𝑣 𝑡 𝑖 𝑡 𝑗ω𝐿=𝑠𝐿 σ j 𝑤 ∞ = 0 ∞ 𝑃 𝑡 d𝑡 v(t) = Vpsin(ωt) + C + v(t) _ VPωC π/2 Capacitance Inductance VP =𝟎 + v(t) _ i(t) IPωL σ IP -π/2 j i(t) = Ipsin(ωt) Absorbed power is sent back to the source L 𝑠=𝑗𝜔 Frequency parameter

Voltage discontinuity Capacitor degradation Current discontinuity ...more on bipoles Virtual short-circuit: the ideal switch (v = 0, i = 0) Resistors Signal sources: Symbols and conventions Capacitors and Inductors Static operation Dynamic operation: sinusoidal response and reactance Other considerations: Series/parallel associations, continuity and models t v(t), i(t) t1- t1+ Time Domain: Frequency Domain: C1 C2 Cn a b L1 L2 Ln Series: 1 𝐶 𝑎𝑏 = 𝑖=1 𝑛 1 𝐶 𝑖 𝐿 𝑎𝑏 = 𝑖=1 𝑛 𝐿 𝑖 Parallel: 1 𝐿 𝑎𝑏 = 𝑖=1 𝑛 1 𝐿 𝑖 𝐶 𝑎𝑏 = 𝑖=1 𝑛 𝐶 𝑖 C V0 + _ vab (t) 𝑖 𝑡 =𝐶 d𝑣 𝑡 d𝑡 v(t) + _ vab (s) 𝑋 𝐶 𝑠 = 1 Cs 𝑉 𝑂 𝑠 = 𝑉 𝑂 s i(s) C + vab (t) _ i(t) a b 𝑞 𝑡 1 − ≠𝑞 𝑡 1 + 𝑖 𝑡 1 = d𝑞 𝑡 d𝑡 =∞ 𝑣 𝑡 1 − ≠𝑣 𝑡 1 + Voltage discontinuity (Impossible) Capacitor degradation Current discontinuity is possible) L + vab (t) _ a b i(t) L I0 + _ 𝑣 𝑎𝑏 𝑡 =𝐿 d𝑖 𝑡 d𝑡 i(t) ϕ 𝑡 1 − =ϕ 𝑡 1 + 𝑣 𝑡 1 = dϕ 𝑡 d𝑡 =∞ i(s) + _ 𝑣 𝑎𝑏 𝑠 𝐼 𝑂 𝑠 = 𝐼 𝑂 𝑠 𝑋 𝐿 𝑠 =𝐿𝑠 𝑖 𝑡 1 − =𝑖 𝑡 1 + Current discontinuity (Impossible) Inductor degradation (Voltage discontinuity is possible)

Summary 𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟:𝑍=Re 𝑍 +𝐼𝑚 𝑍 𝑃𝑜𝑙𝑎𝑟:𝑍=𝐴 𝑒 𝑗𝜃 𝑅𝑒 𝑍 =𝐴· cos 𝜃 Symbol Element type Units i-v relationship Frequency Domain Laplace Model * Observations Si Re{Z}<0 𝑠=𝑗𝜔 𝑠·𝑣 𝑠 → d𝑣 𝑡 d𝑡 𝑣 𝑠 𝑠 → 𝑣 𝑡 d𝑡 sL 𝐼 0 𝑠 + 𝑉 0 𝑠 1 𝑠𝐶 𝑗 𝜎 𝜃 A Re{Z} Im{Z} 𝑃𝑜𝑙𝑎𝑟:𝑍=𝐴 𝑒 𝑗𝜃 𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟:𝑍=Re 𝑍 +𝐼𝑚 𝑍 𝜔=2π𝑓 𝐴= 𝑅𝑒 2 𝑍 + 𝐼𝑚 2 𝑍 θ = tan −1 𝐼𝑚 𝑍 𝑅𝑒 𝑍 Si Re{Z}>0 π+tan −1 𝐼𝑚 𝑍 𝑅𝑒 𝑍 𝑅𝑒 𝑍 =𝐴· cos 𝜃 𝐼𝑚 𝑍 =𝐴· sin 𝜃 𝑣 𝑟𝑚𝑠 = 𝑉 𝑃 2 Resistor [Ω]. Ohm R 𝑣 𝑡 =𝑅·𝑖 𝑡 𝑉 𝑗𝜔 =𝑅·𝐼𝑗 𝜔 𝑉 𝑠 𝐼 𝑠 =𝑅 Capacitor [F]. Farad C 𝑖 𝑡 =𝐶 d𝑣 𝑡 d𝑡 𝑋 𝐶 𝑗𝜔 = 1 𝑗𝜔𝐶 ∗ 𝑋 𝐶 𝑠 = 1 𝑠𝐶 , 𝑖 𝑠 =Cs·𝑣 𝑠 Inductor [H]. Henry L 𝑣 𝑡 =𝐿 d𝑖 𝑡 d𝑡 𝑋 𝐿 𝑗𝜔 =𝑗𝜔𝐿 𝑋 𝐿 𝑠 =𝑠𝐿, 𝑣 𝑠 =𝐿𝑠·𝑖 𝑠 Capacitance Inductance Z Generic Impedance [Ω*]. Ohm (Reactance) 𝑉 𝑗𝜔 𝐼 𝑗𝜔 =𝑍 𝑗𝜔 𝑍 𝑠 = 𝑉 𝑠 𝐼 𝑠 V DC voltage Source [V]. Volts 𝑣 𝑡 = 𝑉 ∀𝑖,𝑡∈ −∞,∞ 𝑉 0 =𝑉 𝜔=0 𝑉 𝑠 = 𝑉 𝑠 I DC current Source [A]. Amper 𝑖 𝑡 = 𝐼 ∀𝑣,𝑡∈ −∞,∞ 𝐼 0 =𝐼 𝜔=0 𝐼 𝑠 = 𝐼 𝑠 v AC voltage Source [V]. Volts (rms1) 𝑣 𝑡 = 𝑉 𝑃 sin 𝜔𝑡+𝜃 ∗ 𝑉 𝑗𝜔 = 𝑉 𝑃 𝑉 𝑠 = 𝑉 𝑃 𝑠 sin 𝜃+𝜔 cos 𝜃 𝑠 2 + 𝜔 2 i AC current Source [A]. Amper (rms) 𝑖 𝑡 = 𝐼 𝑃 sin 𝜔𝑡+𝜃 𝐼 𝑗𝜔 = 𝐼 𝑃 𝐼 𝑠 = 𝐼 𝑃 𝑠 sin 𝜃+𝜔 cos 𝜃 𝑠 2 + 𝜔 2 vC Voltage controlled Source [V]. Volts* 𝒗 𝑪 = 𝑉 𝐶 + 𝑣 𝑐 𝑡 𝑉 𝐶 𝑗𝜔 𝑉 𝐶 𝑠 iC Current controlled Source [A]. Amper 𝒊 𝑪 = 𝐼 𝐶 + 𝑖 𝑐 𝑡 𝐼 𝐶 𝑗𝜔 𝐼 𝐶 𝑠