1. 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: Fax:
Office hours: VG3-D101 – EEL TUE 11-13h, THU: 16-18h (Yellow zone)

2. 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

3. 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 = x10-19C
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)

4. 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

5. 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

6. 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

7. ...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}

8. ...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

9. ...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

10. 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

11. 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)

12. 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