  We examine dissipation effects in a multichannel quantum RC circuit,
comprising a cavity or single-electron box capacitively coupled to a gate and
connected to a reservoir lead via several conducting channels. Depending on the
engineering details of the quantum RC circuit, the number of channels
contributing to transport vary, as do the form of the interchannel couplings.
For low-frequency AC transport, the charge-relaxation resistance ($R_{q}$) is a
nontrivial function of the parameters of the system. However, in the vicinity
of the charge degeneracy points and for weak tunneling, we find as a result of
cross-mode mixing or channel asymmetry that $R_q$ becomes universal for a
metallic cavity at low temperatures, and equals the unit of quantum resistance.
To prove this universality we map the system to an effective one-channel Kondo
model, and construct an analogy with the Coulomb gas. Next, we probe the
opposite regime of near-perfect transmission using a bosonization approach.
Focussing on the two-channel case, we study the effect of backscattering at the
lead-dot interface, more specifically, the role of an asymmetry in the
backscattering amplitudes, and make a connection with the weak tunneling regime
near the charge degeneracy points.
