Basic Electric Circuits Session 15C

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Session 15C: Focus
 Ideal Capacitor Model
◦ Conduction Vs Displacement Currents
◦ Problem 1
◦ Voltage-Current Relationship
◦ Phase difference between V-I of a Capacitor
◦ Energy Stored in a Capacitor
◦ Home Work Problem

Ideal Capacitor Model

Ideal Model of a Capacitor
 Capacitor Symbol:
 C satisfies the conventions for a passive element
 Voltage-Current Relationship:

 This equation tells us that when the voltage doesn’t
change across the capacitor, current doesn’t flow;
 To have a current flow, the voltage must change.
 For a constant DC voltage source, capacitors act as
open circuits because there’s no current flow.
Note the signs of v across C

Relevance of KCL
 Consider one plate of a capacitor as a node
 Does KCL apply here?
 Apparently it looks like, it does not hold!!!
 Current is approaching the plate from the external circuit, but it
cannot flow out of the plate because of dielectric material is
inside
 This dilemma bothered a famous Scottish scientist, James
Clerk Maxwell a century ago

Relevance of KCL … continued
 The unified electromagnetic theory that he
subsequently developed hypothesizes a “displacement
current”
◦ Which is present wherever an electric field or a voltage is
varying with time.
 The displacement current (explained at the
beginning of this session) flowing internally between
the capacitor plates is exactly equal to the conduction
current flowing in the capacitor leads;
 Kirchhoff’s current law is therefore satisfied with
both conduction and displacement currents flowing.

V-I Relationship
of Capacitor

Problem 1:
 Find current i flowing though C for the two voltage waveforms:
Cosine waveform

V and I Relationship
 Voltage is a sine function and the current is cosine
 What is the value of i(t) when the v(t) is zero?
 What is the phase difference between them?
 Does the current lead the voltage or lag behind?
 Or the Voltage is lagging by 900 with respect to current.
90 Degrees
Maximum
Lead by 900

Phase Differences (V-I of Capacitor)
 In a sine wave, where is it changing most quickly?
◦ When it's crossing zero.
 Where is it changing least quickly?
◦ At the peaks, where the voltage is not changing at all.
 So a capacitor passes the most current when the sine
wave is changing most quickly, around zero,
 and vice versa it has no current flow at all at the peaks.

V-I Relationship of Capacitor
 We are aware, current (i) is:
 Since, for a capacitor:
 Now integrating this expression from t = - to some time t, and
assuming v(- ) = 0
◦ Which means that the capacitor is fully discharged initially
 v(t) indicates time dependence of voltage
 across the capacitor
 v(t0) is the charge accumulated on the plates of the capacitor
from t = - to time t = t0 (at t0, the capacitor is getting
charged)
∞
∞
∞

Energy Stored in a Capacitor
 The energy stored in the capacitor can be derived from the
power that is delivered to the it.
 This power is given by the expression
 Hence, the energy stored in the Electric field is:
 Electrical energy stored is wc(t) =
 Since, v(t = - ) = 0, and q(t) = C v(t) , wc(t):∞

Problem 2
 If the charge accumulated on two parallel
conductors charged to 12 V is 600 pC,
 What is the capacitance of the parallel conductors?
C = Q/V

Problem 3
 The voltage across a 5 μF capacitor has the waveform shown in
figure below.
 Determine the current waveform.

Home Work Problem
Note: This is Example 7.1 on page 219, Figure 7.3 in the Ref 1 book (by Hayt)

Session 15C: Summary
 Ideal Capacitor Model
◦ Conduction Vs Displacement Currents
◦ Problem 1
◦ Home Work Problem
◦ Voltage-Current Relationship

References
Ref 1 Ref 2

Basic Electric Circuits Session 15C

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